[
	{
		"chunk_id" : "00000001",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1  Introduction 7\n1 PREREQUISITES\nCredit: Andreas Kambanls\nChapter Outline\n1.1Real Numbers: Algebra Essentials\n1.2Exponents and Scientific Notation\n1.3Radicals and Rational Exponents\n1.4Polynomials\n1.5Factoring Polynomials\n1.6Rational Expressions\nIntroduction to Prerequisites\nIts a cold day in Antarctica. In fact, its always a cold day in Antarctica. Earths southernmost continent, Antarctica\nexperiences the coldest, driest, and windiest conditions known. The coldest temperature ever recorded, over one"
	},
	{
		"chunk_id" : "00000002",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hundred degrees below zero on the Celsius scale, was recorded by remote satellite. It is no surprise then, that no native\nhuman population can survive the harsh conditions. Only explorers and scientists brave the environment for any length\nof time.\nMeasuring and recording the characteristics of weather conditions in Antarctica requires a use of different kinds of\nnumbers. For tens of thousands of years, humans have undertaken methods to tally, track, and record numerical"
	},
	{
		"chunk_id" : "00000003",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "information. While we don't know much about their usage, the Lebombo Bone (dated to about 35,000 BCE) and the\nIshango Bone (dated to about 20,000 BCE) are among the earliest mathematical artifacts. Found in Africa, their clearly\ndeliberate groupings of notches may have been used to track time, moon cycles, or other information. Performing\ncalculations with them and using the results to make predictions requires an understanding of relationships among"
	},
	{
		"chunk_id" : "00000004",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers. In this chapter, we will review sets of numbers and properties of operations used to manipulate numbers. This\nunderstanding will serve as prerequisite knowledge throughout our study of algebra and trigonometry.\n1.1 Real Numbers: Algebra Essentials\nLearning Objectives\nIn this section, you will:\nClassify a real number as a natural, whole, integer, rational, or irrational number.\nPerform calculations using order of operations."
	},
	{
		"chunk_id" : "00000005",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Perform calculations using order of operations.\nUse the following properties of real numbers: commutative, associative, distributive, inverse, and identity.\nEvaluate algebraic expressions.\nSimplify algebraic expressions.\n8 1  Prerequisites\nIt is often said that mathematics is the language of science. If this is true, then an essential part of the language of\nmathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or"
	},
	{
		"chunk_id" : "00000006",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantitya\nsheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to\nimproved communications and the spread of civilization.\nThree to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they\nused them to represent the amount when a quantity was divided into equal parts."
	},
	{
		"chunk_id" : "00000007",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the\nexistence of nothing? From earliest times, people had thought of a base state while counting and used various\nsymbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added\nto the number system and used as a numeral in calculations."
	},
	{
		"chunk_id" : "00000008",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative\nnumbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting\nnumbers expanded the number system even further.\nBecause of the evolution of the number system, we can now perform complex calculations using these and other\ncategories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers,"
	},
	{
		"chunk_id" : "00000009",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and the use of numbers in expressions.\nClassifying a Real Number\nThe numbers we use for counting, or enumerating items, are thenatural numbers: 1, 2, 3, 4, 5, and so on. We describe\nthem in set notation as where the ellipsis () indicates that the numbers continue to infinity. The natural\nnumbers are, of course, also called thecounting numbers. Any time we enumerate the members of a team, count the"
	},
	{
		"chunk_id" : "00000010",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set ofwhole numbersis\nthe set of natural numbers plus zero:\nThe set ofintegersadds the opposites of the natural numbers to the set of whole numbers:\nIt is useful to note that the set of integers is made up of three distinct subsets: negative\nintegers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to"
	},
	{
		"chunk_id" : "00000011",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "think about it is that the natural numbers are a subset of the integers.\nThe set ofrational numbersis written as Notice from the definition that rational\nnumbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the\ndenominator is never 0. We can also see that every natural number, whole number, and integer is a rational number\nwith a denominator of 1."
	},
	{
		"chunk_id" : "00000012",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with a denominator of 1.\nBecause they are fractions, any rational number can also be expressed in decimal form. Any rational number can be\nrepresented as either:\n a terminating decimal: or  a repeating decimal:\nWe use a line drawn over the repeating block of numbers instead of writing the group multiple times.\nEXAMPLE1\nWriting Integers as Rational Numbers\nWrite each of the following as a rational number.\n 7  0  8\nSolution\nWrite a fraction with the integer in the numerator and 1 in the denominator."
	},
	{
		"chunk_id" : "00000013",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nTRY IT #1 Write each of the following as a rational number.\n 11  3  4\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 9\nEXAMPLE2\nIdentifying Rational Numbers\nWrite each of the following rational numbers as either a terminating or repeating decimal.\n  \nSolution\nWrite each fraction as a decimal by dividing the numerator by the denominator.\n a repeating decimal  (or 3.0), a terminating decimal\n a terminating decimal"
	},
	{
		"chunk_id" : "00000014",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " a terminating decimal\nTRY IT #2 Write each of the following rational numbers as either a terminating or repeating decimal.\n  \nIrrational Numbers\nAt some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for\ninstance, may have found that the diagonal of a square with unit sides was not 2 or even but was something else. Or\na garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit"
	},
	{
		"chunk_id" : "00000015",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "more than 3, but still not a rational number. Such numbers are said to beirrationalbecause they cannot be written as\nfractions. These numbers make up the set ofirrational numbers. Irrational numbers cannot be expressed as a fraction\nof two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if\nit is not rational. So we write this as shown.\nEXAMPLE3\nDifferentiating Rational and Irrational Numbers"
	},
	{
		"chunk_id" : "00000016",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Differentiating Rational and Irrational Numbers\nDetermine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a\nterminating or repeating decimal.\n    \nSolution\n This can be simplified as Therefore, is rational.\n Because it is a fraction of integers, is a rational number. Next, simplify and divide.\nSo, is rational and a repeating decimal.\n This cannot be simplified any further. Therefore, is an irrational number."
	},
	{
		"chunk_id" : "00000017",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Because it is a fraction of integers, is a rational number. Simplify and divide.\nSo, is rational and a terminating decimal.\n is not a terminating decimal. Also note that there is no repeating pattern because the group\nof 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational\nnumber. It is an irrational number.\n10 1  Prerequisites\nTRY IT #3 Determine whether each of the following numbers is rational or irrational. If it is rational,"
	},
	{
		"chunk_id" : "00000018",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determine whether it is a terminating or repeating decimal.\n    \nReal Numbers\nGiven any numbern, we know thatnis either rational or irrational. It cannot be both. The sets of rational and irrational\nnumbers together make up the set ofreal numbers. As we saw with integers, the real numbers can be divided into\nthree subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and"
	},
	{
		"chunk_id" : "00000019",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "irrational numbers according to their algebraic sign (+ or ). Zero is considered neither positive nor negative.\nThe real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative\nnumbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer\n(or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The"
	},
	{
		"chunk_id" : "00000020",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-\nto-one correspondence. We refer to this as thereal number lineas shown inFigure 1.\nFigure1 The real number line\nEXAMPLE4\nClassifying Real Numbers\nClassify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or\nthe right of 0 on the number line?\n    \nSolution"
	},
	{
		"chunk_id" : "00000021",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "    \nSolution\n is negative and rational. It lies to the left of 0 on the number line.\n is positive and irrational. It lies to the right of 0.\n is negative and rational. It lies to the left of 0.\n is negative and irrational. It lies to the left of 0.\n is a repeating decimal so it is rational and positive. It lies to the right of 0.\nTRY IT #4 Classify each number as either positive or negative and as either rational or irrational. Does the"
	},
	{
		"chunk_id" : "00000022",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number lie to the left or the right of 0 on the number line?\n    \nSets of Numbers as Subsets\nBeginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset\nrelationship between the sets of numbers we have encountered so far. These relationships become more obvious when\nseen as a diagram, such asFigure 2.\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 11\nFigure2 Sets of numbers\nN: the set of natural numbers"
	},
	{
		"chunk_id" : "00000023",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "N: the set of natural numbers\nW: the set of whole numbers\nI: the set of integers\nQ: the set of rational numbers\nQ: the set of irrational numbers\nSets of Numbers\nThe set ofnatural numbersincludes the numbers used for counting:\nThe set ofwhole numbersis the set of natural numbers plus zero:\nThe set ofintegersadds the negative natural numbers to the set of whole numbers:\nThe set ofrational numbersincludes fractions written as"
	},
	{
		"chunk_id" : "00000024",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The set ofirrational numbersis the set of numbers that are not rational, are nonrepeating, and are nonterminating:\nEXAMPLE5\nDifferentiating the Sets of Numbers\nClassify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or\nirrational number (Q).\n    \nSolution\nN W I Q Q\na. X X X X\nb. X\nc. X\nd. 6 X X\ne. 3.2121121112... X\n12 1  Prerequisites\nTRY IT #5 Classify each number as being a natural number (N), whole number (W), integer (I), rational"
	},
	{
		"chunk_id" : "00000025",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number (Q), and/or irrational number (Q).\n    \nPerforming Calculations Using the Order of Operations\nWhen we multiply a number by itself, we square it or raise it to a power of 2. For example, We can raise\nany number to any power. In general, theexponential notation means that the number or variable is used as a\nfactor times.\nIn this notation, is read as thenth power of or to the where is called thebaseand is called theexponent.A"
	},
	{
		"chunk_id" : "00000026",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "term in exponential notation may be part of a mathematical expression, which is a combination of numbers and\noperations. For example, is a mathematical expression.\nTo evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any\nrandom order. We use theorder of operations. This is a sequence of rules for evaluating such expressions.\nRecall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that"
	},
	{
		"chunk_id" : "00000027",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars\nare treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within\ngrouping symbols.\nThe next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right\nand finally addition and subtraction from left to right.\nLets take a look at the expression provided."
	},
	{
		"chunk_id" : "00000028",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lets take a look at the expression provided.\nThere are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so\nsimplify as 16.\nNext, perform multiplication or division, left to right.\nLastly, perform addition or subtraction, left to right.\nTherefore,\nFor some complicated expressions, several passes through the order of operations will be needed. For instance, there"
	},
	{
		"chunk_id" : "00000029",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following\nthe order of operations ensures that anyone simplifying the same mathematical expression will get the same result.\nOrder of Operations\nOperations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 13\nacronymPEMDAS:\nP(arentheses)\nE(xponents)"
	},
	{
		"chunk_id" : "00000030",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "acronymPEMDAS:\nP(arentheses)\nE(xponents)\nM(ultiplication) andD(ivision)\nA(ddition) andS(ubtraction)\n...\nHOW TO\nGiven a mathematical expression, simplify it using the order of operations.\nStep 1. Simplify any expressions within grouping symbols.\nStep 2. Simplify any expressions containing exponents or radicals.\nStep 3. Perform any multiplication and division in order, from left to right.\nStep 4. Perform any addition and subtraction in order, from left to right.\nEXAMPLE6\nUsing the Order of Operations"
	},
	{
		"chunk_id" : "00000031",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nUsing the Order of Operations\nUse the order of operations to evaluate each of the following expressions.\n   \n\nSolution\n\n\nNote that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the\nfraction bar is considered a grouping symbol so the numerator is considered to be grouped.\n\n14 1  Prerequisites\n\nIn this example, the fraction bar separates the numerator and denominator, which we simplify separately until the\nlast step.\n"
	},
	{
		"chunk_id" : "00000032",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "last step.\n\nTRY IT #6 Use the order of operations to evaluate each of the following expressions.\n  \n \nUsing Properties of Real Numbers\nFor some activities we perform, the order of certain operations does not matter, but the order of other operations does.\nFor example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does\nmatter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\nCommutative Properties"
	},
	{
		"chunk_id" : "00000033",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Commutative Properties\nThecommutative property of additionstates that numbers may be added in any order without affecting the sum.\nWe can better see this relationship when using real numbers.\nSimilarly, thecommutative property of multiplicationstates that numbers may be multiplied in any order without\naffecting the product.\nAgain, consider an example with real numbers.\nIt is important to note that neither subtraction nor division is commutative. For example, is not the same as\nSimilarly,"
	},
	{
		"chunk_id" : "00000034",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly,\nAssociative Properties\nTheassociative property of multiplicationtells us that it does not matter how we group numbers when multiplying.\nWe can move the grouping symbols to make the calculation easier, and the product remains the same.\nConsider this example.\nTheassociative property of additiontells us that numbers may be grouped differently without affecting the sum.\nThis property can be especially helpful when dealing with negative integers. Consider this example.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000035",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 15\nAre subtraction and division associative? Review these examples.\nAs we can see, neither subtraction nor division is associative.\nDistributive Property\nThedistributive propertystates that the product of a factor times a sum is the sum of the factor times each term in the\nsum.\nThis property combines both addition and multiplication (and is the only property to do so). Let us consider an example."
	},
	{
		"chunk_id" : "00000036",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by 7, and\nadding the products.\nTo be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not\ntrue, as we can see in this example.\nA special case of the distributive property occurs when a sum of terms is subtracted.\nFor example, consider the difference We can rewrite the difference of the two terms 12 and by"
	},
	{
		"chunk_id" : "00000037",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "turning the subtraction expression into addition of the opposite. So instead of subtracting we add the opposite.\nNow, distribute and simplify the result.\nThis seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce\nalgebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can\nrewrite the last example.\nIdentity Properties"
	},
	{
		"chunk_id" : "00000038",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rewrite the last example.\nIdentity Properties\nTheidentity property of additionstates that there is a unique number, called the additive identity (0) that, when added\nto a number, results in the original number.\nTheidentity property of multiplicationstates that there is a unique number, called the multiplicative identity (1) that,\nwhen multiplied by a number, results in the original number.\nFor example, we have and There are no exceptions for these properties; they work for every"
	},
	{
		"chunk_id" : "00000039",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "real number, including 0 and 1.\n16 1  Prerequisites\nInverse Properties\nTheinverse property of additionstates that, for every real numbera, there is a unique number, called the additive\ninverse (or opposite), denoted by (a), that, when added to the original number, results in the additive identity, 0.\nFor example, if the additive inverse is 8, since\nTheinverse property of multiplicationholds for all real numbers except 0 because the reciprocal of 0 is not defined."
	},
	{
		"chunk_id" : "00000040",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The property states that, for every real numbera, there is a unique number, called the multiplicative inverse (or\nreciprocal), denoted that, when multiplied by the original number, results in the multiplicative identity, 1.\nFor example, if the reciprocal, denoted is because\nProperties of Real Numbers\nThe following properties hold for real numbersa,b, andc.\nAddition Multiplication\nCommutative\nProperty\nAssociative\nProperty\nDistributive\nProperty"
	},
	{
		"chunk_id" : "00000041",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Associative\nProperty\nDistributive\nProperty\nIdentity There exists a unique real number called the There exists a unique real number called the\nProperty additive identity, 0, such that, for any real multiplicative identity, 1, such that, for any real\nnumbera numbera\nInverse Every real number a has an additive inverse, Every nonzero real numberahas a\nProperty or opposite, denoteda, such that multiplicative inverse, or reciprocal, denoted\nsuch that\nEXAMPLE7\nUsing Properties of Real Numbers"
	},
	{
		"chunk_id" : "00000042",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nUsing Properties of Real Numbers\nUse the properties of real numbers to rewrite and simplify each expression. State which properties apply.\n    \nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 17\nSolution\n\n\n\n\n\nTRY IT #7 Use the properties of real numbers to rewrite and simplify each expression. State which\nproperties apply.\n  \n \nEvaluating Algebraic Expressions"
	},
	{
		"chunk_id" : "00000043",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \n \nEvaluating Algebraic Expressions\nSo far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see\nexpressions such as or In the expression 5 is called aconstantbecause it does not vary\nandxis called avariablebecause it does. (In naming the variable, ignore any exponents or radicals containing the\nvariable.) Analgebraic expressionis a collection of constants and variables joined together by the algebraic operations"
	},
	{
		"chunk_id" : "00000044",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of addition, subtraction, multiplication, and division.\nWe have already seen some real number examples of exponential notation, a shorthand method of writing products of\nthe same factor. When variables are used, the constants and variables are treated the same way.\nIn each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or\nvariables."
	},
	{
		"chunk_id" : "00000045",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variables.\nAny variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the\nalgebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a\ngiven value of each variable in the expression. Replace each variable in the expression with the given value, then simplify\nthe resulting expression using the order of operations. If the algebraic expression contains more than one variable,"
	},
	{
		"chunk_id" : "00000046",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "replace each variable with its assigned value and simplify the expression as before.\n18 1  Prerequisites\nEXAMPLE8\nDescribing Algebraic Expressions\nList the constants and variables for each algebraic expression.\n x+ 5  \nSolution\nConstants Variables\na.x+ 5 5 x\nb.\nc. 2\nTRY IT #8 List the constants and variables for each algebraic expression.\n  2(L+W) \nEXAMPLE9\nEvaluating an Algebraic Expression at Different Values\nEvaluate the expression for each value forx.\n   \nSolution"
	},
	{
		"chunk_id" : "00000047",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "   \nSolution\n Substitute 0 for  Substitute 1 for  Substitute for  Substitute for\nTRY IT #9 Evaluate the expression for each value fory.\n   \nEXAMPLE10\nEvaluating Algebraic Expressions\nEvaluate each expression for the given values.\n for  for  for  for\n for\nSolution\n Substitute for  Substitute 10 for  Substitute 5 for\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 19\n Substitute 11 for and 8 for  Substitute 2 for and 3 for"
	},
	{
		"chunk_id" : "00000048",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #10 Evaluate each expression for the given values.\n for  for  for\n for  for\nFormulas\nAnequationis a mathematical statement indicating that two expressions are equal. The expressions can be numerical\nor algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true,\nthe solutions, are found using the properties of real numbers and other results. For example, the equation"
	},
	{
		"chunk_id" : "00000049",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has the solution of 3 because when we substitute 3 for in the equation, we obtain the true statement\nAformulais an equation expressing a relationship between constant and variable quantities. Very often, the equation is\na means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the\nmost common examples is the formula for finding the area of a circle in terms of the radius of the circle:"
	},
	{
		"chunk_id" : "00000050",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For any value of the area can be found by evaluating the expression\nEXAMPLE11\nUsing a Formula\nA right circular cylinder with radius and height has the surface area (in square units) given by the formula\nSeeFigure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in\nterms of\nFigure3 Right circular cylinder\nSolution\nEvaluate the expression for and\nThe surface area is square inches."
	},
	{
		"chunk_id" : "00000051",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The surface area is square inches.\nTRY IT #11 A photograph with lengthLand widthWis placed in a mat of width 8 centimeters (cm). The area\nof the mat (in square centimeters, or cm2) is found to be See\nFigure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.\n20 1  Prerequisites\nFigure4\nSimplifying Algebraic Expressions\nSometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so,"
	},
	{
		"chunk_id" : "00000052",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic\nexpressions.\nEXAMPLE12\nSimplifying Algebraic Expressions\nSimplify each algebraic expression.\n   \nSolution\n\n\n\n\nTRY IT #12 Simplify each algebraic expression.\n  \n\nEXAMPLE13\nSimplifying a Formula\nA rectangle with length and width has a perimeter given by Simplify this expression.\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 21\nSolution"
	},
	{
		"chunk_id" : "00000053",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nTRY IT #13 If the amount is deposited into an account paying simple interest for time the total value of\nthe deposit is given by Simplify the expression. (This formula will be explored in\nmore detail later in the course.)\nMEDIA\nAccess these online resources for additional instruction and practice with real numbers.\nSimplify an Expression.(http://openstax.org/l/simexpress)\nEvaluate an Expression 1.(http://openstax.org/l/ordofoper1)\nEvaluate an Expression 2.(http://openstax.org/l/ordofoper2)"
	},
	{
		"chunk_id" : "00000054",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.1 SECTION EXERCISES\nVerbal\n1. Is an example of a 2. What is the order of 3. What do the Associative\nrational terminating, operations? What acronym Properties allow us to do\nrational repeating, or is used to describe the order when following the order of\nirrational number? Tell why of operations, and what operations? Explain your\nit fits that category. does it stand for? answer.\nNumeric\nFor the following exercises, simplify the given expression.\n4. 5. 6.\n7. 8. 9.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18."
	},
	{
		"chunk_id" : "00000055",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7. 8. 9.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n22 1  Prerequisites\nAlgebraic\nFor the following exercises, evaluate the expressions using the given variable.\n28. for 29. for 30. for\n31. for 32. for 33. for\n34. For the 35. for 36. for\nfor\n37. for\nFor the following exercises, simplify the expression.\n38. 39. 40.\n41. 42. 43.\n44. 45. 46.\n47. 48. 49.\n50. 51. 52.\nReal-World Applications"
	},
	{
		"chunk_id" : "00000056",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. 48. 49.\n50. 51. 52.\nReal-World Applications\nFor the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a\nstreaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbors dog.\n53. Write the expression that represents the number 54. How much money does Fred keep?\nof dollars Fred keeps (and does not put in his\nsavings account). Remember the order of\noperations."
	},
	{
		"chunk_id" : "00000057",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "operations.\nFor the following exercises, solve the given problem.\n55. According to the U.S. Mint, the diameter of a 56. Jessica and her roommate, Adriana, have decided\nquarter is 0.955 inches. The circumference of the to share a change jar for joint expenses. Jessica\nquarter would be the diameter multiplied by Is put her loose change in the jar first, and then\nthe circumference of a quarter a whole number, a Adriana put her change in the jar. We know that it"
	},
	{
		"chunk_id" : "00000058",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rational number, or an irrational number? does not matter in which order the change was\nadded to the jar. What property of addition\ndescribes this fact?\nAccess for free at openstax.org\n1.1  Real Numbers: Algebra Essentials 23\nFor the following exercises, consider this scenario: There is a mound of pounds of gravel in a quarry. Throughout the\nday, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the"
	},
	{
		"chunk_id" : "00000059",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "mound. At the end of the day, the mound has 1,200 pounds of gravel.\n57. Write the equation that describes the situation. 58. Solve forg.\nFor the following exercise, solve the given problem.\n59. Ramon runs the marketing department at their\ncompany. Their department gets a budget every\nyear, and every year, they must spend the entire\nbudget without going over. If they spend less than\nthe budget, then the department gets a smaller\nbudget the following year. At the beginning of this"
	},
	{
		"chunk_id" : "00000060",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "year, Ramon got $2.5 million for the annual\nmarketing budget. They must spend the budget\nsuch that What property of\naddition tells us what the value ofxmust be?\nTechnology\nFor the following exercises, use a graphing calculator to solve forx. Round the answers to the nearest hundredth.\n60. 61.\nExtensions\n62. If a whole number is not a 63. Determine whether the 64. Determine whether the\nnatural number, what must statement is true or false: statement is true or false:"
	},
	{
		"chunk_id" : "00000061",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the number be? The multiplicative inverse The product of a rational\nof a rational number is also and irrational number is\nrational. always irrational.\n65. Determine whether the 66. Determine whether the 67. The division of two natural\nsimplified expression is simplified expression is numbers will always result\nrational or irrational: rational or irrational: in what type of number?\n68. What property of real\nnumbers would simplify\nthe following expression:\n24 1  Prerequisites"
	},
	{
		"chunk_id" : "00000062",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the following expression:\n24 1  Prerequisites\n1.2 Exponents and Scientific Notation\nLearning Objectives\nIn this section, you will:\nUse the product rule of exponents.\nUse the quotient rule of exponents.\nUse the power rule of exponents.\nUse the zero exponent rule of exponents.\nUse the negative rule of exponents.\nFind the power of a product and a quotient.\nSimplify exponential expressions.\nUse scientific notation."
	},
	{
		"chunk_id" : "00000063",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use scientific notation.\nMathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be\nobvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be\nperceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536\npixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per"
	},
	{
		"chunk_id" : "00000064",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information\nused to film a one-hour (3,600-second) digital film is then an extremely large number.\nUsing a calculator, we enter and press ENTER. The calculator displays 1.304596316E13.\nWhat does this mean? The E13 portion of the result represents the exponent 13 of ten, so there are a maximum of\napproximately bits of data in that one-hour film. In this section, we review rules of exponents first and then"
	},
	{
		"chunk_id" : "00000065",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "apply them to calculations involving very large or small numbers.\nUsing the Product Rule of Exponents\nConsider the product Both terms have the same base,x, but they are raised to different exponents. Expand each\nexpression, and then rewrite the resulting expression.\nThe result is that\nNotice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying"
	},
	{
		"chunk_id" : "00000066",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential expressions with the same base, we write the result with the common base and add the exponents. This is\ntheproduct rule of exponents.\nNow consider an example with real numbers.\nWe can always check that this is true by simplifying each exponential expression. We find that is 8, is 16, and is\n128. The product equals 128, so the relationship is true. We can use the product rule of exponents to simplify"
	},
	{
		"chunk_id" : "00000067",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expressions that are a product of two numbers or expressions with the same base but different exponents.\nThe Product Rule of Exponents\nFor any real number and natural numbers and the product rule of exponents states that\nEXAMPLE1\nUsing the Product Rule\nWrite each of the following products with a single base. Do not simplify further.\nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 25\n  \nSolution\nUse the product rule to simplify each expression.\n  "
	},
	{
		"chunk_id" : "00000068",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nAt first, it may appear that we cannot simplify a product of three factors. However, using the associative property of\nmultiplication, begin by simplifying the first two.\nNotice we get the same result by adding the three exponents in one step.\nTRY IT #1 Write each of the following products with a single base. Do not simplify further.\n  \nUsing the Quotient Rule of Exponents\nThequotient rule of exponentsallows us to simplify an expression that divides two numbers with the same base but"
	},
	{
		"chunk_id" : "00000069",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "different exponents. In a similar way to the product rule, we can simplify an expression such as where\nConsider the example Perform the division by canceling common factors.\nNotice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.\nIn other words, when dividing exponential expressions with the same base, we write the result with the common base\nand subtract the exponents."
	},
	{
		"chunk_id" : "00000070",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and subtract the exponents.\nFor the time being, we must be aware of the condition Otherwise, the difference could be zero or negative.\nThose possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify\nmatters and assume from here on that all variables represent nonzero real numbers.\nThe Quotient Rule of Exponents\nFor any real number and natural numbers and such that the quotient rule of exponents states that\nEXAMPLE2\nUsing the Quotient Rule"
	},
	{
		"chunk_id" : "00000071",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nUsing the Quotient Rule\nWrite each of the following products with a single base. Do not simplify further.\n26 1  Prerequisites\n  \nSolution\nUse the quotient rule to simplify each expression.\n  \nTRY IT #2 Write each of the following products with a single base. Do not simplify further.\n  \nUsing the Power Rule of Exponents\nSuppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the"
	},
	{
		"chunk_id" : "00000072",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "power rule of exponents. Consider the expression The expression inside the parentheses is multiplied twice\nbecause it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent\nof 3.\nThe exponent of the answer is the product of the exponents: In other words, when raising an\nexponential expression to a power, we write the result with the common base and the product of the exponents."
	},
	{
		"chunk_id" : "00000073",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different\nterms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a\nterm in exponential notation is raised to a power. In this case, you multiply the exponents.\nThe Power Rule of Exponents\nFor any real number and positive integers and the power rule of exponents states that\nEXAMPLE3\nUsing the Power Rule"
	},
	{
		"chunk_id" : "00000074",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nUsing the Power Rule\nWrite each of the following products with a single base. Do not simplify further.\n  \nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 27\nSolution\nUse the power rule to simplify each expression.\n  \nTRY IT #3 Write each of the following products with a single base. Do not simplify further.\n  \nUsing the Zero Exponent Rule of Exponents\nReturn to the quotient rule. We made the condition that so that the difference would never be zero or"
	},
	{
		"chunk_id" : "00000075",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "negative. What would happen if In this case, we would use thezero exponent rule of exponentsto simplify the\nexpression to 1. To see how this is done, let us begin with an example.\nIf we were to simplify the original expression using the quotient rule, we would have\nIf we equate the two answers, the result is This is true for any nonzero real number, or any variable representing\na real number.\nThe sole exception is the expression This appears later in more advanced courses, but for now, we will consider the"
	},
	{
		"chunk_id" : "00000076",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value to be undefined.\nThe Zero Exponent Rule of Exponents\nFor any nonzero real number the zero exponent rule of exponents states that\nEXAMPLE4\nUsing the Zero Exponent Rule\nSimplify each expression using the zero exponent rule of exponents.\n   \nSolution\nUse the zero exponent and other rules to simplify each expression.\n\n\n28 1  Prerequisites\n\n\nTRY IT #4 Simplify each expression using the zero exponent rule of exponents.\n   \nUsing the Negative Rule of Exponents"
	},
	{
		"chunk_id" : "00000077",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "   \nUsing the Negative Rule of Exponents\nAnother useful result occurs if we relax the condition that in the quotient rule even further. For example, can we\nsimplify When that is, where the difference is negativewe can use thenegative rule of exponentsto\nsimplify the expression to its reciprocal.\nDivide one exponential expression by another with a larger exponent. Use our example,\nIf we were to simplify the original expression using the quotient rule, we would have"
	},
	{
		"chunk_id" : "00000078",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Putting the answers together, we have This is true for any nonzero real number, or any variable representing\na nonzero real number.\nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 29\nA factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction\nbarfrom numerator to denominator or vice versa.\nWe have shown that the exponential expression is defined when is a natural number, 0, or the negative of a natural"
	},
	{
		"chunk_id" : "00000079",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number. That means that is defined for any integer Also, the product and quotient rules and all of the rules we will\nlook at soon hold for any integer\nThe Negative Rule of Exponents\nFor any nonzero real number and natural number the negative rule of exponents states that\nEXAMPLE5\nUsing the Negative Exponent Rule\nWrite each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.\n  \nSolution\n \n"
	},
	{
		"chunk_id" : "00000080",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nSolution\n \n\nTRY IT #5 Write each of the following quotients with a single base. Do not simplify further. Write answers\nwith positive exponents.\n  \nEXAMPLE6\nUsing the Product and Quotient Rules\nWrite each of the following products with a single base. Do not simplify further. Write answers with positive exponents.\n  \nSolution\n \n\nTRY IT #6 Write each of the following products with a single base. Do not simplify further. Write answers\nwith positive exponents.\n "
	},
	{
		"chunk_id" : "00000081",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with positive exponents.\n \nFinding the Power of a Product\nTo simplify the power of a product of two exponential expressions, we can use thepower of a product rule of exponents,\nwhich breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider\nWe begin by using the associative and commutative properties of multiplication to regroup the factors.\n30 1  Prerequisites\nIn other words,\nThe Power of a Product Rule of Exponents"
	},
	{
		"chunk_id" : "00000082",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Power of a Product Rule of Exponents\nFor any real numbers and and any integer the power of a product rule of exponents states that\nEXAMPLE7\nUsing the Power of a Product Rule\nSimplify each of the following products as much as possible using the power of a product rule. Write answers with\npositive exponents.\n    \nSolution\nUse the product and quotient rules and the new definitions to simplify each expression.\n \n \n"
	},
	{
		"chunk_id" : "00000083",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \n \n\nTRY IT #7 Simplify each of the following products as much as possible using the power of a product rule.\nWrite answers with positive exponents.\n    \nFinding the Power of a Quotient\nTo simplify the power of a quotient of two expressions, we can use thepower of a quotient rule,which states that the\npower of a quotient of factors is the quotient of the powers of the factors. For example, lets look at the following\nexample.\nLets rewrite the original problem differently and look at the result."
	},
	{
		"chunk_id" : "00000084",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "It appears from the last two steps that we can use the power of a product rule as a power of a quotient rule.\nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 31\nThe Power of a Quotient Rule of Exponents\nFor any real numbers and and any integer the power of a quotient rule of exponents states that\nEXAMPLE8\nUsing the Power of a Quotient Rule\nSimplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with\npositive exponents.\n    "
	},
	{
		"chunk_id" : "00000085",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "positive exponents.\n    \nSolution\n \n \n\nTRY IT #8 Simplify each of the following quotients as much as possible using the power of a quotient rule.\nWrite answers with positive exponents.\n    \nSimplifying Exponential Expressions\nRecall that to simplify an expression means to rewrite it by combing terms or exponents; in other words, to write the\nexpression more simply with fewer terms. The rules for exponents may be combined to simplify expressions.\nEXAMPLE9\nSimplifying Exponential Expressions"
	},
	{
		"chunk_id" : "00000086",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nSimplifying Exponential Expressions\nSimplify each expression and write the answer with positive exponents only.\n   \n \n32 1  Prerequisites\nSolution\n\n\n\n\n\n\nTRY IT #9 Simplify each expression and write the answer with positive exponents only.\n   \n \nUsing Scientific Notation\nRecall at the beginning of the section that we found the number when describing bits of information in digital"
	},
	{
		"chunk_id" : "00000087",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "images. Other extreme numbers include the width of a human hair, which is about 0.00005 m, and the radius of an\nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 33\nelectron, which is about 0.00000000000047 m. How can we effectively work read, compare, and calculate with numbers\nsuch as these?\nA shorthand method of writing very small and very large numbers is calledscientific notation, in which we express"
	},
	{
		"chunk_id" : "00000088",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers in terms of exponents of 10. To write a number in scientific notation, move the decimal point to the right of the\nfirst digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of placesnthat you\nmoved the decimal point. Multiply the decimal number by 10 raised to a power ofn. If you moved the decimal left as in a\nvery large number, is positive. If you moved the decimal right as in a small large number, is negative."
	},
	{
		"chunk_id" : "00000089",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which\nis 2.\nWe obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive\nbecause we moved the decimal point to the left. This is what we should expect for a large number.\nWorking with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the"
	},
	{
		"chunk_id" : "00000090",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same series of steps as above, except move the decimal point to the right.\nBe careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent\nof 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for\na small number.\nScientific Notation\nA number is written inscientific notationif it is written in the form where and is an integer.\nEXAMPLE10\nConverting Standard Notation to Scientific Notation"
	},
	{
		"chunk_id" : "00000091",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write each number in scientific notation.\n Distance to Andromeda Galaxy from Earth: 24,000,000,000,000,000,000,000 m\n Diameter of Andromeda Galaxy: 1,300,000,000,000,000,000,000 m\n Number of stars in Andromeda Galaxy: 1,000,000,000,000\n Diameter of electron: 0.00000000000094 m\n Probability of being struck by lightning in any single year: 0.00000143\nSolution\n\n\n34 1  Prerequisites\n\n\n\nAnalysis"
	},
	{
		"chunk_id" : "00000092",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\n\n\n34 1  Prerequisites\n\n\n\nAnalysis\nObserve that, if the given number is greater than 1, as in examples ac, the exponent of 10 is positive; and if the number\nis less than 1, as in examples de, the exponent is negative.\nTRY IT #10 Write each number in scientific notation.\n U.S. national debt per taxpayer (April 2014): $152,000\n World population (April 2014): 7,158,000,000\n World gross national income (April 2014): $85,500,000,000,000\n Time for light to travel 1 m: 0.00000000334 s"
	},
	{
		"chunk_id" : "00000093",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Time for light to travel 1 m: 0.00000000334 s\n Probability of winning lottery (match 6 of 49 possible numbers): 0.0000000715\nConverting from Scientific to Standard Notation\nTo convert a number inscientific notationto standard notation, simply reverse the process. Move the decimal places\nto the right if is positive or places to the left if is negative and add zeros as needed. Remember, if is positive, the\nvalue of the number is greater than 1, and if is negative, the value of the number is less than one."
	},
	{
		"chunk_id" : "00000094",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE11\nConverting Scientific Notation to Standard Notation\nConvert each number in scientific notation to standard notation.\n   \nSolution\n   \nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 35\nTRY IT #11 Convert each number in scientific notation to standard notation.\n   \nUsing Scientific Notation in Applications\nScientific notation, used with the rules of exponents, makes calculating with large or small numbers much easier than"
	},
	{
		"chunk_id" : "00000095",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "doing so using standard notation. For example, suppose we are asked to calculate the number of atoms in 1 L of water.\nEach water molecule contains 3 atoms (2 hydrogen and 1 oxygen). The average drop of water contains around\nmolecules of water and 1 L of water holds about average drops. Therefore, there are\napproximately atoms in 1 L of water. We simply multiply the decimal\nterms and add the exponents. Imagine having to perform the calculation without using scientific notation!"
	},
	{
		"chunk_id" : "00000096",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "When performing calculations with scientific notation, be sure to write the answer in proper scientific notation. For\nexample, consider the product The answer is not in proper scientific notation\nbecause 35 is greater than 10. Consider 35 as That adds a ten to the exponent of the answer.\nEXAMPLE12\nUsing Scientific Notation\nPerform the operations and write the answer in scientific notation.\n  \n \nSolution\n\n\n\n\n\nTRY IT #12 Perform the operations and write the answer in scientific notation."
	},
	{
		"chunk_id" : "00000097",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "36 1  Prerequisites\n \n \n\nEXAMPLE13\nApplying Scientific Notation to Solve Problems\nIn April 2014, the population of the United States was about 308,000,000 people. The national debt was about\n$17,547,000,000,000. Write each number in scientific notation, rounding figures to two decimal places, and find the\namount of the debt per U.S. citizen. Write the answer in both scientific and standard notations.\nSolution\nThe population was\nThe national debt was"
	},
	{
		"chunk_id" : "00000098",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe population was\nThe national debt was\nTo find the amount of debt per citizen, divide the national debt by the number of citizens.\nThe debt per citizen at the time was about or $57,000.\nTRY IT #13 An average human body contains around 30,000,000,000,000 red blood cells. Each cell measures\napproximately 0.000008 m long. Write each number in scientific notation and find the total length\nif the cells were laid end-to-end. Write the answer in both scientific and standard notations.\nMEDIA"
	},
	{
		"chunk_id" : "00000099",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with exponents and scientific notation.\nExponential Notation(http://openstax.org/l/exponnot)\nProperties of Exponents(http://openstax.org/l/exponprops)\nZero Exponent(http://openstax.org/l/zeroexponent)\nSimplify Exponent Expressions(http://openstax.org/l/exponexpres)\nQuotient Rule for Exponents(http://openstax.org/l/quotofexpon)\nScientific Notation(http://openstax.org/l/scientificnota)"
	},
	{
		"chunk_id" : "00000100",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Converting to Decimal Notation(http://openstax.org/l/decimalnota)\n1.2 SECTION EXERCISES\nVerbal\n1. Is the same as 2. When can you add two 3. What is the purpose of\nExplain. exponents? scientific notation?\n4. Explain what a negative\nexponent does.\nAccess for free at openstax.org\n1.2  Exponents and Scientific Notation 37\nNumeric\nFor the following exercises, simplify the given expression. Write answers with positive exponents.\n5. 6. 7.\n8. 9. 10.\n11. 12. 13.\n14."
	},
	{
		"chunk_id" : "00000101",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. 6. 7.\n8. 9. 10.\n11. 12. 13.\n14.\nFor the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive\nexponents.\n15. 16. 17.\n18. 19. 20.\nFor the following exercises, express the decimal in scientific notation.\n21. 0.0000314 22. 148,000,000\nFor the following exercises, convert each number in scientific notation to standard notation.\n23. 24.\nAlgebraic\nFor the following exercises, simplify the given expression. Write answers with positive exponents."
	},
	{
		"chunk_id" : "00000102",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n38 1  Prerequisites\n43.\nReal-World Applications\n44. To reach escape velocity, a 45. A dime is the thinnest coin 46. The average distance\nrocket must travel at the in U.S. currency. A dimes between Earth and the Sun\nrate of ft/min. thickness measures is 92,960,000 mi. Rewrite\nRewrite the rate in m. Rewrite the the distance using scientific\nstandard notation. number in standard notation.\nnotation."
	},
	{
		"chunk_id" : "00000103",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "notation.\n47. A terabyte is made of 48. The Gross Domestic 49. One picometer is\napproximately Product (GDP) for the approximately\n1,099,500,000,000 bytes. United States in the first in. Rewrite\nRewrite in scientific quarter of 2014 was this length using standard\nnotation. Rewrite notation.\nthe GDP in standard\nnotation.\n50. The value of the services\nsector of the U.S. economy\nin the first quarter of 2012\nwas $10,633.6 billion.\nRewrite this amount in\nscientific notation.\nTechnology"
	},
	{
		"chunk_id" : "00000104",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "scientific notation.\nTechnology\nFor the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.\n51. 52.\nExtensions\nFor the following exercises, simplify the given expression. Write answers with positive exponents.\n53. 54. 55.\n56. 57. 58. Avogadros constant is\nused to calculate the\nnumber of particles in a\nmole. A mole is a basic unit\nin chemistry to measure\nthe amount of a substance.\nThe constant is\nWrite\nAvogadros constant in\nstandard notation."
	},
	{
		"chunk_id" : "00000105",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write\nAvogadros constant in\nstandard notation.\nAccess for free at openstax.org\n1.3  Radicals and Rational Exponents 39\n59. Plancks constant is an\nimportant unit of measure\nin quantum physics. It\ndescribes the relationship\nbetween energy and\nfrequency. The constant is\nwritten as\nWrite\nPlancks constant in\nstandard notation.\n1.3 Radicals and Rational Exponents\nLearning Objectives\nIn this section, you will:\nEvaluate square roots.\nUse the product rule to simplify square roots."
	},
	{
		"chunk_id" : "00000106",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use the product rule to simplify square roots.\nUse the quotient rule to simplify square roots.\nAdd and subtract square roots.\nRationalize denominators.\nUse rational roots.\nA hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to\nbe purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of\nladder needed, we can draw a right triangle as shown inFigure 1, and use the Pythagorean Theorem."
	},
	{
		"chunk_id" : "00000107",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nNow, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we\nneed to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.\nEvaluating Square Roots\nWhen the square root of a number is squared, the result is the original number. Since the square root of is\nThe square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo"
	},
	{
		"chunk_id" : "00000108",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "squaring, we take the square root.\nIn general terms, if is a positive real number, then the square root of is a number that, when multiplied by itself,\ngives The square root could be positive or negative because multiplying two negative numbers gives a positive\nnumber. Theprincipal square rootis the nonnegative number that when multiplied by itself equals The square root\nobtained using a calculator is the principal square root."
	},
	{
		"chunk_id" : "00000109",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The principal square root of is written as The symbol is called aradical, the term under the symbol is called the\nradicand, and the entire expression is called aradical expression.\n40 1  Prerequisites\nPrincipal Square Root\nTheprincipal square rootof is the nonnegative number that, when multiplied by itself, equals It is written as a\nradical expression,with a symbol called aradicalover the term called theradicand:\nQ&A Does"
	},
	{
		"chunk_id" : "00000110",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Does\nNo. Although both and are the radical symbol implies only a nonnegative root, the principal\nsquare root. The principal square root of 25 is\nEXAMPLE1\nEvaluating Square Roots\nEvaluate each expression.\n   \nSolution\n because  because and\n because  because and\nQ&A For can we find the square roots before adding?\nNo. This is not equivalent to The order of operations\nrequires us to add the terms in the radicand before finding the square root.\nTRY IT #1 Evaluate each expression.\n   "
	},
	{
		"chunk_id" : "00000111",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #1 Evaluate each expression.\n   \nUsing the Product Rule to Simplify Square Roots\nTo simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several\nproperties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the\nproduct rule for simplifying square roots,which allows us to separate the square root of a product of two numbers into"
	},
	{
		"chunk_id" : "00000112",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the product of two separate rational expressions. For instance, we can rewrite as We can also use the\nproduct rule to express the product of multiple radical expressions as a single radical expression.\nThe Product Rule for Simplifying Square Roots\nIf and are nonnegative, the square root of the product is equal to the product of the square roots of and\n...\nHOW TO\nGiven a square root radical expression, use the product rule to simplify it.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000113",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n1.3  Radicals and Rational Exponents 41\n1. Factor any perfect squares from the radicand.\n2. Write the radical expression as a product of radical expressions.\n3. Simplify.\nEXAMPLE2\nUsing the Product Rule to Simplify Square Roots\nSimplify the radical expression.\n \nSolution\n\n\nTRY IT #2 Simplify\n...\nHOW TO\nGiven the product of multiple radical expressions, use the product rule to combine them into one radical\nexpression."
	},
	{
		"chunk_id" : "00000114",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expression.\n1. Express the product of multiple radical expressions as a single radical expression.\n2. Simplify.\nEXAMPLE3\nUsing the Product Rule to Simplify the Product of Multiple Square Roots\nSimplify the radical expression.\nSolution\nTRY IT #3 Simplify assuming\nUsing the Quotient Rule to Simplify Square Roots\nJust as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of"
	},
	{
		"chunk_id" : "00000115",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a quotient as a quotient of square roots, using thequotient rule for simplifying square roots.It can be helpful to\nseparate the numerator and denominator of a fraction under a radical so that we can take their square roots separately.\n42 1  Prerequisites\nWe can rewrite as\nThe Quotient Rule for Simplifying Square Roots\nThe square root of the quotient is equal to the quotient of the square roots of and where\n...\nHOW TO\nGiven a radical expression, use the quotient rule to simplify it."
	},
	{
		"chunk_id" : "00000116",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Write the radical expression as the quotient of two radical expressions.\n2. Simplify the numerator and denominator.\nEXAMPLE4\nUsing the Quotient Rule to Simplify Square Roots\nSimplify the radical expression.\nSolution\nTRY IT #4 Simplify\nEXAMPLE5\nUsing the Quotient Rule to Simplify an Expression with Two Square Roots\nSimplify the radical expression.\nSolution\nTRY IT #5 Simplify\nAccess for free at openstax.org\n1.3  Radicals and Rational Exponents 43\nAdding and Subtracting Square Roots"
	},
	{
		"chunk_id" : "00000117",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Adding and Subtracting Square Roots\nWe can add or subtract radical expressions only when they have the same radicand and when they have the same radical\ntype such as square roots. For example, the sum of and is However, it is often possible to simplify radical\nexpressions, and that may change the radicand. The radical expression can be written with a in the radicand, as\nso\n...\nHOW TO\nGiven a radical expression requiring addition or subtraction of square roots, simplify.\n1. Simplify each radical expression."
	},
	{
		"chunk_id" : "00000118",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Simplify each radical expression.\n2. Add or subtract expressions with equal radicands.\nEXAMPLE6\nAdding Square Roots\nAdd\nSolution\nWe can rewrite as According the product rule, this becomes The square root of is 2, so the\nexpression becomes which is Now the terms have the same radicand so we can add.\nTRY IT #6 Add\nEXAMPLE7\nSubtracting Square Roots\nSubtract\nSolution\nRewrite each term so they have equal radicands.\nNow the terms have the same radicand so we can subtract.\nTRY IT #7 Subtract"
	},
	{
		"chunk_id" : "00000119",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Subtract\n44 1  Prerequisites\nRationalizing Denominators\nWhen an expression involving square root radicals is written in simplest form, it will not contain a radical in the\ndenominator. We can remove radicals from the denominators of fractions using a process calledrationalizing the\ndenominator.\nWe know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to"
	},
	{
		"chunk_id" : "00000120",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions,\nmultiply by the form of 1 that will eliminate the radical.\nFor a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the\ndenominator is multiply by\nFor a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and"
	},
	{
		"chunk_id" : "00000121",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the\ndenominator. If the denominator is then the conjugate is\n...\nHOW TO\nGiven an expression with a single square root radical term in the denominator, rationalize the denominator.\na. Multiply the numerator and denominator by the radical in the denominator.\nb. Simplify.\nEXAMPLE8\nRationalizing a Denominator Containing a Single Term\nWrite in simplest form.\nSolution"
	},
	{
		"chunk_id" : "00000122",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write in simplest form.\nSolution\nThe radical in the denominator is So multiply the fraction by Then simplify.\nTRY IT #8 Write in simplest form.\n...\nHOW TO\nGiven an expression with a radical term and a constant in the denominator, rationalize the denominator.\n1. Find the conjugate of the denominator.\n2. Multiply the numerator and denominator by the conjugate.\n3. Use the distributive property.\n4. Simplify.\nAccess for free at openstax.org\n1.3  Radicals and Rational Exponents 45\nEXAMPLE9"
	},
	{
		"chunk_id" : "00000123",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.3  Radicals and Rational Exponents 45\nEXAMPLE9\nRationalizing a Denominator Containing Two Terms\nWrite in simplest form.\nSolution\nBegin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate\nof is Then multiply the fraction by\nTRY IT #9 Write in simplest form.\nUsing Rational Roots\nAlthough square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more."
	},
	{
		"chunk_id" : "00000124",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective\npower functions. These functions can be useful when we need to determine the number that, when raised to a certain\npower, gives a certain number.\nUnderstandingnth Roots\nSuppose we know that We want to find what number raised to the 3rd power is equal to 8. Since we say\nthat 2 is the cube root of 8."
	},
	{
		"chunk_id" : "00000125",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that 2 is the cube root of 8.\nThenth root of is a number that, when raised to thenth power, gives For example, is the 5th root of\nbecause If is a real number with at least onenth root, then theprincipalnth rootof is the number\nwith the same sign as that, when raised to thenth power, equals\nThe principalnth root of is written as where is a positive integer greater than or equal to 2. In the radical\nexpression, is called theindexof the radical.\nPrincipal th Root"
	},
	{
		"chunk_id" : "00000126",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Principal th Root\nIf is a real number with at least onenth root, then theprincipalnth rootof written as is the number with\nthe same sign as that, when raised to thenth power, equals Theindexof the radical is\nEXAMPLE10\nSimplifyingnth Roots\nSimplify each of the following:\n   \nSolution\n because\n First, express the product as a single radical expression. because\n\n46 1  Prerequisites\n\nTRY IT #10 Simplify.\n  \nUsing Rational Exponents"
	},
	{
		"chunk_id" : "00000127",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nUsing Rational Exponents\nRadical expressionscan also be written without using the radical symbol. We can use rational (fractional) exponents.\nThe index must be a positive integer. If the index is even, then cannot be negative.\nWe can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in\nlowest terms. We raise the base to a power and take annth root. The numerator tells us the power and the denominator\ntells us the root."
	},
	{
		"chunk_id" : "00000128",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tells us the root.\nAll of the properties of exponents that we learned for integer exponents also hold for rational exponents.\nRational Exponents\nRational exponents are another way to express principalnth roots. The general form for converting between a radical\nexpression with a radical symbol and one with a rational exponent is\n...\nHOW TO\nGiven an expression with a rational exponent, write the expression as a radical.\n1. Determine the power by looking at the numerator of the exponent."
	},
	{
		"chunk_id" : "00000129",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Determine the root by looking at the denominator of the exponent.\n3. Using the base as the radicand, raise the radicand to the power and use the root as the index.\nEXAMPLE11\nWriting Rational Exponents as Radicals\nWrite as a radical. Simplify.\nSolution\nThe 2 tells us the power and the 3 tells us the root.\nWe know that because Because the cube root is easy to find, it is easiest to find the cube root"
	},
	{
		"chunk_id" : "00000130",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "before squaring for this problem. In general, it is easier to find the root first and then raise it to a power.\nAccess for free at openstax.org\n1.3  Radicals and Rational Exponents 47\nTRY IT #11 Write as a radical. Simplify.\nEXAMPLE12\nWriting Radicals as Rational Exponents\nWrite using a rational exponent.\nSolution\nThe power is 2 and the root is 7, so the rational exponent will be We get Using properties of exponents, we get\nTRY IT #12 Write using a rational exponent.\nEXAMPLE13"
	},
	{
		"chunk_id" : "00000131",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE13\nSimplifying Rational Exponents\nSimplify:\n \nSolution\n\n\nTRY IT #13 Simplify\nMEDIA\nAccess these online resources for additional instruction and practice with radicals and rational exponents.\nRadicals(http://openstax.org/l/introradical)\nRational Exponents(http://openstax.org/l/rationexpon)\nSimplify Radicals(http://openstax.org/l/simpradical)\nRationalize Denominator(http://openstax.org/l/rationdenom)\n48 1  Prerequisites\n1.3 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000132",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "48 1  Prerequisites\n1.3 SECTION EXERCISES\nVerbal\n1. What does it mean when a 2. Where would radicals come 3. Every number will have two\nradical does not have an in the order of operations? square roots. What is the\nindex? Is the expression Explain why. principal square root?\nequal to the radicand?\nExplain.\n4. Can a radical with a negative\nradicand have a real square\nroot? Why or why not?\nNumeric\nFor the following exercises, simplify each expression.\n5. 6. 7.\n8. 9. 10.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19."
	},
	{
		"chunk_id" : "00000133",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8. 9. 10.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23. 24. 25.\n26. 27. 28.\n29. 30. 31.\n32. 33. 34.\nAccess for free at openstax.org\n1.3  Radicals and Rational Exponents 49\nAlgebraic\nFor the following exercises, simplify each expression.\n35. 36. 37.\n38. 39. 40.\n41. 42. 43.\n44. 45. 46.\n47. 48. 49.\n50. 51. 52.\n53. 54. 55.\n56. 57. 58.\n59. 60. 61.\n62. 63. 64.\nReal-World Applications\n65. A guy wire for a suspension bridge runs from the 66. A car accelerates at a rate of where"
	},
	{
		"chunk_id" : "00000134",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ground diagonally to the top of the closest pylon\ntis the time in seconds after the car moves from\nto make a triangle. We can use the Pythagorean\nrest. Simplify the expression.\nTheorem to find the length of guy wire needed.\nThe square of the distance between the wire on\nthe ground and the pylon on the ground is 90,000\nfeet. The square of the height of the pylon is\n160,000 feet. So the length of the guy wire can be\nfound by evaluating What is\nthe length of the guy wire?\n50 1  Prerequisites\nExtensions"
	},
	{
		"chunk_id" : "00000135",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "50 1  Prerequisites\nExtensions\nFor the following exercises, simplify each expression.\n67. 68. 69.\n70. 71. 72.\n73.\n1.4 Polynomials\nLearning Objectives\nIn this section, you will:\nIdentify the degree and leading coefficient of polynomials.\nAdd and subtract polynomials.\nMultiply polynomials.\nUse FOIL to multiply binomials.\nPerform operations with polynomials of several variables.\nMaahi is building a little free library (a small house-shaped book repository), whose front is in the shape of a square"
	},
	{
		"chunk_id" : "00000136",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "topped with a triangle. There will be a rectangular door through which people can take and donate books. Maahi wants\nto find the area of the front of the library so that they can purchase the correct amount of paint. Using the\nmeasurements of the front of the house, shown inFigure 1, we can create an expression that combines several variable\nterms, allowing us to solve this problem and others like it.\nFigure1\nFirst find the area of the square in square feet."
	},
	{
		"chunk_id" : "00000137",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "First find the area of the square in square feet.\nThen find the area of the triangle in square feet.\nNext find the area of the rectangular door in square feet.\nAccess for free at openstax.org\n1.4  Polynomials 51\nThe area of the front of the library can be found by adding the areas of the square and the triangle, and then subtracting\nthe area of the rectangle. When we do this, we get or ft2.\nIn this section, we will examine expressions such as this one, which combine several variable terms."
	},
	{
		"chunk_id" : "00000138",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying the Degree and Leading Coefficient of Polynomials\nThe formula just found is an example of apolynomial, which is a sum of or difference of terms, each consisting of a\nvariable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as\nis known as acoefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or"
	},
	{
		"chunk_id" : "00000139",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fractions. Each product such as is aterm of a polynomial. If a term does not contain a variable, it is called\naconstant.\nA polynomial containing only one term, such as is called amonomial. A polynomial containing two terms, such as\nis called abinomial. A polynomial containing three terms, such as is called atrinomial.\nWe can find thedegreeof a polynomial by identifying the highest power of the variable that occurs in the polynomial."
	},
	{
		"chunk_id" : "00000140",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The term with the highest degree is called theleading termbecause it is usually written first. The coefficient of the\nleading term is called theleading coefficient. When a polynomial is written so that the powers are descending, we say\nthat it is in standard form.\nPolynomials\nApolynomialis an expression that can be written in the form\nEach real numberais called acoefficient. The number that is not multiplied by a variable is called aconstant.\ni"
	},
	{
		"chunk_id" : "00000141",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "i\nEach product is aterm of a polynomial. The highest power of the variable that occurs in the polynomial is called\nthedegreeof a polynomial. Theleading termis the term with the highest power, and its coefficient is called the\nleading coefficient.\n...\nHOW TO\nGiven a polynomial expression, identify the degree and leading coefficient.\n1. Find the highest power ofxto determine the degree.\n2. Identify the term containing the highest power ofxto find the leading term."
	},
	{
		"chunk_id" : "00000142",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Identify the coefficient of the leading term.\nEXAMPLE1\nIdentifying the Degree and Leading Coefficient of a Polynomial\nFor the following polynomials, identify the degree, the leading term, and the leading coefficient.\n  \nSolution\n The highest power ofxis 3, so the degree is 3. The leading term is the term containing that degree, The\nleading coefficient is the coefficient of that term,\n The highest power oftis so the degree is The leading term is the term containing that degree, The"
	},
	{
		"chunk_id" : "00000143",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "leading coefficient is the coefficient of that term,\n52 1  Prerequisites\n The highest power ofpis so the degree is The leading term is the term containing that degree, The\nleading coefficient is the coefficient of that term,\nTRY IT #1 Identify the degree, leading term, and leading coefficient of the polynomial\nAdding and Subtracting Polynomials\nWe can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to"
	},
	{
		"chunk_id" : "00000144",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the same exponents. For example, and are like terms, and can be added to get but and are not\nlike terms, and therefore cannot be added.\n...\nHOW TO\nGiven multiple polynomials, add or subtract them to simplify the expressions.\n1. Combine like terms.\n2. Simplify and write in standard form.\nEXAMPLE2\nAdding Polynomials\nFind the sum.\nSolution\nAnalysis\nWe can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given"
	},
	{
		"chunk_id" : "00000145",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the\ngraphs. Using different windows can make the expressions seem equivalent when they are not.\nTRY IT #2 Find the sum.\nEXAMPLE3\nSubtracting Polynomials\nFind the difference.\nSolution\nAnalysis\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial\nAccess for free at openstax.org\n1.4  Polynomials 53\nto the first."
	},
	{
		"chunk_id" : "00000146",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.4  Polynomials 53\nto the first.\nTRY IT #3 Find the difference.\nMultiplying Polynomials\nMultiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive\nproperty to multiply each term in the first polynomial by each term in the second polynomial. We then combine like\nterms. We can also use a shortcut called theFOILmethod when multiplying binomials. Certain special products follow"
	},
	{
		"chunk_id" : "00000147",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a\nvariety of ways to multiply polynomials.\nMultiplying Polynomials Using the Distributive Property\nTo multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of\nthe polynomial. We can distribute the in to obtain the equivalent expression When multiplying"
	},
	{
		"chunk_id" : "00000148",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second.\nWe then add the products together and combine like terms to simplify.\n...\nHOW TO\nGiven the multiplication of two polynomials, use the distributive property to simplify the expression.\n1. Multiply each term of the first polynomial by each term of the second.\n2. Combine like terms.\n3. Simplify.\nEXAMPLE4\nMultiplying Polynomials Using the Distributive Property\nFind the product.\nSolution"
	},
	{
		"chunk_id" : "00000149",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the product.\nSolution\nAnalysis\nWe can use a table to keep track of our work, as shown inTable 1. Write one polynomial across the top and the other\ndown the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the\nterms together, combine like terms, and simplify.\nTable1\n54 1  Prerequisites\nTable1\nTRY IT #4 Find the product.\nUsing FOIL to Multiply Binomials"
	},
	{
		"chunk_id" : "00000150",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using FOIL to Multiply Binomials\nA shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the\nfirst terms, theouter terms, theinner terms, and then thelast terms of each binomial.\nThe FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by\neach term of the second binomial, and then combining like terms.\n...\nHOW TO\nGiven two binomials, use FOIL to simplify the expression."
	},
	{
		"chunk_id" : "00000151",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Multiply the first terms of each binomial.\n2. Multiply the outer terms of the binomials.\n3. Multiply the inner terms of the binomials.\n4. Multiply the last terms of each binomial.\n5. Add the products.\n6. Combine like terms and simplify.\nEXAMPLE5\nUsing FOIL to Multiply Binomials\nUse FOIL to find the product.\nSolution\nFind the product of the first terms.\nFind the product of the outer terms.\nFind the product of the inner terms.\nFind the product of the last terms.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000152",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n1.4  Polynomials 55\nTRY IT #5 Use FOIL to find the product.\nPerfect Square Trinomials\nCertain binomial products have special forms. When a binomial is squared, the result is called aperfect square\ntrinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of\nthese perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Lets"
	},
	{
		"chunk_id" : "00000153",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "look at a few perfect square trinomials to familiarize ourselves with the form.\nNotice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of\neach trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms.\nLastly, we see that the first sign of the trinomial is the same as the sign of the binomial.\nPerfect Square Trinomials"
	},
	{
		"chunk_id" : "00000154",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Perfect Square Trinomials\nWhen a binomial is squared, the result is the first term squared added to double the product of both terms and the\nlast term squared.\n...\nHOW TO\nGiven a binomial, square it using the formula for perfect square trinomials.\n1. Square the first term of the binomial.\n2. Square the last term of the binomial.\n3. For the middle term of the trinomial, double the product of the two terms.\n4. Add and simplify.\nEXAMPLE6\nExpanding Perfect Squares\nExpand\nSolution"
	},
	{
		"chunk_id" : "00000155",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Expanding Perfect Squares\nExpand\nSolution\nBegin by squaring the first term and the last term. For the middle term of the trinomial, double the product of the two\nterms.\nSimplify.\n56 1  Prerequisites\nTRY IT #6 Expand\nDifference of Squares\nAnother special product is called thedifference of squares, which occurs when we multiply a binomial by another\nbinomial with the same terms but the opposite sign. Lets see what happens when we multiply using the\nFOIL method."
	},
	{
		"chunk_id" : "00000156",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "FOIL method.\nThe middle term drops out, resulting in a difference of squares. Just as we did with the perfect squares, lets look at a few\nexamples.\nBecause the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only\nwith the square of the first term minus the square of the last term.\nQ&A Is there a special form for the sum of squares?\nNo. The difference of squares occurs because the opposite signs of the binomials cause the middle terms"
	},
	{
		"chunk_id" : "00000157",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to disappear. There are no two binomials that multiply to equal a sum of squares.\nDifference of Squares\nWhen a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the\nsquare of the first term minus the square of the last term.\n...\nHOW TO\nGiven a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of\nsquares.\n1. Square the first term of the binomials.\n2. Square the last term of the binomials."
	},
	{
		"chunk_id" : "00000158",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Square the last term of the binomials.\n3. Subtract the square of the last term from the square of the first term.\nEXAMPLE7\nMultiplying Binomials Resulting in a Difference of Squares\nMultiply\nSolution\nSquare the first term to get Square the last term to get Subtract the square of the last term from\nthe square of the first term to find the product of\nAccess for free at openstax.org\n1.4  Polynomials 57\nTRY IT #7 Multiply\nPerforming Operations with Polynomials of Several Variables"
	},
	{
		"chunk_id" : "00000159",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of\nthe same rules apply when working with polynomials containing several variables. Consider an example:\nEXAMPLE8\nMultiplying Polynomials Containing Several Variables\nMultiply\nSolution\nFollow the same steps that we used to multiply polynomials containing only one variable.\nTRY IT #8 Multiply\nMEDIA\nAccess these online resources for additional instruction and practice with polynomials."
	},
	{
		"chunk_id" : "00000160",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Adding and Subtracting Polynomials(http://openstax.org/l/addsubpoly)\nMultiplying Polynomials(http://openstax.org/l/multiplpoly)\nSpecial Products of Polynomials(http://openstax.org/l/specialpolyprod)\n1.4 SECTION EXERCISES\nVerbal\n1. Evaluate the following 2. Many times, multiplying two 3. You can multiply\nstatement: The degree of a binomials with two variables polynomials with any\npolynomial in standard form results in a trinomial. This is number of terms and any"
	},
	{
		"chunk_id" : "00000161",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the exponent of the not the case when there is a number of variables using\nleading term. Explain why difference of two squares. four basic steps over and\nthe statement is true or Explain why the product in over until you reach the\nfalse. this case is also a binomial. expanded polynomial. What\nare the four steps?\n4. State whether the following\nstatement is true and\nexplain why or why not: A\ntrinomial is always a higher\ndegree than a monomial.\n58 1  Prerequisites\nAlgebraic"
	},
	{
		"chunk_id" : "00000162",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "58 1  Prerequisites\nAlgebraic\nFor the following exercises, identify the degree of the polynomial.\n5. 6. 7.\n8. 9. 10.\nFor the following exercises, find the sum or difference.\n11. 12.\n13. 14.\n15. 16.\nFor the following exercises, find the product.\n17. 18. 19.\n20. 21. 22.\n23.\nFor the following exercises, expand the binomial.\n24. 25. 26.\n27. 28. 29.\n30.\nFor the following exercises, multiply the binomials.\n31. 32. 33.\n34. 35. 36.\n37.\nFor the following exercises, multiply the polynomials.\n38. 39. 40.\n41. 42. 43."
	},
	{
		"chunk_id" : "00000163",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "38. 39. 40.\n41. 42. 43.\n44. 45. 46.\nAccess for free at openstax.org\n1.5  Factoring Polynomials 59\n47. 48. 49.\n50. 51. 52.\nReal-World Applications\n53. A developer wants to purchase a plot of land to 54. A prospective buyer wants to know how much\nbuild a house. The area of the plot can be grain a specific silo can hold. The area of the floor\ndescribed by the following expression: of the silo is The height of the silo is\nwherexis measured in meters. wherexis measured in feet. Expand the"
	},
	{
		"chunk_id" : "00000164",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Multiply the binomials to find the area of the plot square and multiply by the height to find the\nin standard form. expression that shows how much grain the silo\ncan hold.\nExtensions\nFor the following exercises, perform the given operations.\n55. 56. 57.\n1.5 Factoring Polynomials\nLearning Objectives\nIn this section, you will:\nFactor the greatest common factor of a polynomial.\nFactor a trinomial.\nFactor by grouping.\nFactor a perfect square trinomial.\nFactor a difference of squares."
	},
	{
		"chunk_id" : "00000165",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Factor a difference of squares.\nFactor the sum and difference of cubes.\nFactor expressions using fractional or negative exponents.\nImagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The\nlawn is the green portion inFigure 1.\nFigure1\nThe area of the entire region can be found using the formula for the area of a rectangle.\nThe areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The"
	},
	{
		"chunk_id" : "00000166",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "60 1  Prerequisites\ntwo square regions each have an area of units2. The other rectangular region has one side of length\nand one side of length giving an area of units2. So the region that must be\nsubtracted has an area of units2.\nThe area of the region that requires grass seed is found by subtracting units2. This area can also be\nexpressed in factored form as units2. We can confirm that this is an equivalent expression by multiplying."
	},
	{
		"chunk_id" : "00000167",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of\nmethods that can be used to factor polynomial expressions.\nFactoring the Greatest Common Factor of a Polynomial\nWhen we study fractions, we learn that thegreatest common factor(GCF) of two numbers is the largest number that\ndivides evenly into both numbers. For instance, is the GCF of and because it is the largest number that divides"
	},
	{
		"chunk_id" : "00000168",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "evenly into both and The GCF of polynomials works the same way: is the GCF of and because it is the\nlargest polynomial that divides evenly into both and\nWhen factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients,\nand then look for the GCF of the variables.\nGreatest Common Factor\nThegreatest common factor(GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000169",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a polynomial expression, factor out the greatest common factor.\n1. Identify the GCF of the coefficients.\n2. Identify the GCF of the variables.\n3. Combine to find the GCF of the expression.\n4. Determine what the GCF needs to be multiplied by to obtain each term in the expression.\n5. Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.\nEXAMPLE1\nFactoring the Greatest Common Factor\nFactor\nSolution"
	},
	{
		"chunk_id" : "00000170",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Factor\nSolution\nFirst, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of , and is . (Note that the GCF of a\nset of expressions in the form will always be the exponent of lowest degree.) And the GCF of , and is .\nCombine these to find the GCF of the polynomial, .\nNext, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that\n, and"
	},
	{
		"chunk_id" : "00000171",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : ", and\nFinally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.\nAnalysis\nAfter factoring, we can check our work by multiplying. Use the distributive property to confirm that\nTRY IT #1 Factor by pulling out the GCF.\nAccess for free at openstax.org\n1.5  Factoring Polynomials 61\nFactoring a Trinomial with Leading Coefficient 1\nAlthough we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial"
	},
	{
		"chunk_id" : "00000172",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expressions can be factored. The polynomial has a GCF of 1, but it can be written as the product of the\nfactors and\nTrinomials of the form can be factored by finding two numbers with a product of and a sum of The\ntrinomial for example, can be factored using the numbers and because the product of those numbers\nis and their sum is The trinomial can be rewritten as the product of and\nFactoring a Trinomial with Leading Coefficient 1\nA trinomial of the form can be written in factored form as where and"
	},
	{
		"chunk_id" : "00000173",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Can every trinomial be factored as a product of binomials?\nNo. Some polynomials cannot be factored. These polynomials are said to be prime.\n...\nHOW TO\nGiven a trinomial in the form factor it.\n1. List factors of\n2. Find and a pair of factors of with a sum of\n3. Write the factored expression\nEXAMPLE2\nFactoring a Trinomial with Leading Coefficient 1\nFactor\nSolution\nWe have a trinomial with leading coefficient and We need to find two numbers with a product of"
	},
	{
		"chunk_id" : "00000174",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and a sum of In the table below, we list factors until we find a pair with the desired sum.\nFactors of Sum of Factors\n14\n2\nNow that we have identified and as and write the factored form as\nAnalysis\nWe can check our work by multiplying. Use FOIL to confirm that\nQ&A Does the order of the factors matter?\nNo. Multiplication is commutative, so the order of the factors does not matter.\n62 1  Prerequisites\nTRY IT #2 Factor\nFactoring by Grouping"
	},
	{
		"chunk_id" : "00000175",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Factor\nFactoring by Grouping\nTrinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can\nfactor by groupingby dividing thexterm into the sum of two terms, factoring each portion of the expression\nseparately, and then factoring out the GCF of the entire expression. The trinomial can be rewritten as\nusing this process. We begin by rewriting the original expression as and then factor"
	},
	{
		"chunk_id" : "00000176",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each portion of the expression to obtain We then pull out the GCF of to find the factored\nexpression.\nFactor by Grouping\nTo factor a trinomial in the form by grouping, we find two numbers with a product of and a sum of\nWe use these numbers to divide the term into the sum of two terms and factor each portion of the expression\nseparately, then factor out the GCF of the entire expression.\n...\nHOW TO\nGiven a trinomial in the form factor by grouping.\n1. List factors of"
	},
	{
		"chunk_id" : "00000177",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. List factors of\n2. Find and a pair of factors of with a sum of\n3. Rewrite the original expression as\n4. Pull out the GCF of\n5. Pull out the GCF of\n6. Factor out the GCF of the expression.\nEXAMPLE3\nFactoring a Trinomial by Grouping\nFactor by grouping.\nSolution\nWe have a trinomial with and First, determine We need to find two numbers with a\nproduct of and a sum of In the table below, we list factors until we find a pair with the desired sum.\nFactors of Sum of Factors\n29\n13\n7\nSo and"
	},
	{
		"chunk_id" : "00000178",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Factors of Sum of Factors\n29\n13\n7\nSo and\nAccess for free at openstax.org\n1.5  Factoring Polynomials 63\nAnalysis\nWe can check our work by multiplying. Use FOIL to confirm that\nTRY IT #3 Factor\n \nFactoring a Perfect Square Trinomial\nA perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is\nsquared, the result is the square of the first term added to twice the product of the two terms and the square of the last\nterm."
	},
	{
		"chunk_id" : "00000179",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "term.\nWe can use this equation to factor any perfect square trinomial.\nPerfect Square Trinomials\nA perfect square trinomial can be written as the square of a binomial:\n...\nHOW TO\nGiven a perfect square trinomial, factor it into the square of a binomial.\n1. Confirm that the first and last term are perfect squares.\n2. Confirm that the middle term is twice the product of\n3. Write the factored form as\nEXAMPLE4\nFactoring a Perfect Square Trinomial\nFactor\nSolution"
	},
	{
		"chunk_id" : "00000180",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Factor\nSolution\nNotice that and are perfect squares because and Then check to see if the middle term is\ntwice the product of and The middle term is, indeed, twice the product: Therefore, the trinomial is\na perfect square trinomial and can be written as\nTRY IT #4 Factor\nFactoring a Difference of Squares\nA difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be"
	},
	{
		"chunk_id" : "00000181",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when\n64 1  Prerequisites\nthe two factors are multiplied.\nWe can use this equation to factor any differences of squares.\nDifferences of Squares\nA difference of squares can be rewritten as two factors containing the same terms but opposite signs.\n...\nHOW TO\nGiven a difference of squares, factor it into binomials.\n1. Confirm that the first and last term are perfect squares."
	},
	{
		"chunk_id" : "00000182",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Write the factored form as\nEXAMPLE5\nFactoring a Difference of Squares\nFactor\nSolution\nNotice that and are perfect squares because and The polynomial represents a difference of\nsquares and can be rewritten as\nTRY IT #5 Factor\nQ&A Is there a formula to factor the sum of squares?\nNo. A sum of squares cannot be factored.\nFactoring the Sum and Difference of Cubes\nNow, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be"
	},
	{
		"chunk_id" : "00000183",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "factored, the sum of cubes can be factored into a binomial and a trinomial.\nSimilarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.\nWe can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of\neach word relates to the signs:SameOppositeAlwaysPositive. For example, consider the following example.\nThe sign of the first 2 is thesameas the sign between The sign of the term isoppositethe sign between"
	},
	{
		"chunk_id" : "00000184",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "And the sign of the last term, 4, isalways positive.\nSum and Difference of Cubes\nWe can factor the sum of two cubes as\nAccess for free at openstax.org\n1.5  Factoring Polynomials 65\nWe can factor the difference of two cubes as\n...\nHOW TO\nGiven a sum of cubes or difference of cubes, factor it.\n1. Confirm that the first and last term are cubes, or\n2. For a sum of cubes, write the factored form as For a difference of cubes, write the\nfactored form as\nEXAMPLE6\nFactoring a Sum of Cubes\nFactor\nSolution"
	},
	{
		"chunk_id" : "00000185",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nFactoring a Sum of Cubes\nFactor\nSolution\nNotice that and are cubes because Rewrite the sum of cubes as\nAnalysis\nAfter writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored\nfurther. However, the trinomial portion cannot be factored, so we do not need to check.\nTRY IT #6 Factor the sum of cubes:\nEXAMPLE7\nFactoring a Difference of Cubes\nFactor\nSolution\nNotice that and are cubes because and Write the difference of cubes as\nAnalysis"
	},
	{
		"chunk_id" : "00000186",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nJust as with the sum of cubes, we will not be able to further factor the trinomial portion.\nTRY IT #7 Factor the difference of cubes:\nFactoring Expressions with Fractional or Negative Exponents\nExpressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or\nexponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest"
	},
	{
		"chunk_id" : "00000187",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "power. These expressions follow the same factoring rules as those with integer exponents. For instance, can\nbe factored by pulling out and being rewritten as\n66 1  Prerequisites\nEXAMPLE8\nFactoring an Expression with Fractional or Negative Exponents\nFactor\nSolution\nFactor out the term with the lowest value of the exponent. In this case, that would be\nTRY IT #8 Factor\nMEDIA\nAccess these online resources for additional instruction and practice with factoring polynomials."
	},
	{
		"chunk_id" : "00000188",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identify GCF(http://openstax.org/l/findgcftofact)\nFactor Trinomials when a Equals 1(http://openstax.org/l/facttrinom1)\nFactor Trinomials when a is not equal to 1(http://openstax.org/l/facttrinom2)\nFactor Sum or Difference of Cubes(http://openstax.org/l/sumdifcube)\n1.5 SECTION EXERCISES\nVerbal\n1. If the terms of a polynomial 2. A polynomial is factorable, 3. How do you factor by\ndo not have a GCF, does that but it is not a perfect square grouping?\nmean it is not factorable? trinomial or a difference of"
	},
	{
		"chunk_id" : "00000189",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Explain. two squares. Can you factor\nthe polynomial without\nfinding the GCF?\nAlgebraic\nFor the following exercises, find the greatest common factor.\n4. 5. 6.\n7. 8. 9.\nFor the following exercises, factor by grouping.\n10. 11. 12.\n13. 14. 15.\nAccess for free at openstax.org\n1.5  Factoring Polynomials 67\nFor the following exercises, factor the polynomial.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\nFor the following exercises, factor the polynomials.\n37. 38. 39."
	},
	{
		"chunk_id" : "00000190",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "37. 38. 39.\n40. 41. 42.\n43. 44. 45.\n46. 47. 48.\n49. 50.\nReal-World Applications\nFor the following exercises, consider this scenario:\nCharlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains\nin one of the citys parks. The park is a rectangle with an area of m2, as shown in the figure below. The\nlength and width of the park are perfect factors of the area.\n68 1  Prerequisites"
	},
	{
		"chunk_id" : "00000191",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "68 1  Prerequisites\n51. Factor by grouping to find 52. A statue is to be placed in 53. At the northwest corner of\nthe length and width of the the center of the park. The the park, the city is going\npark. area of the base of the to install a fountain. The\nstatue is area of the base of the\nFactor the area to find the fountain is\nlengths of the sides of the Factor the area to find the\nstatue. lengths of the sides of the\nfountain.\nFor the following exercise, consider the following scenario:"
	},
	{
		"chunk_id" : "00000192",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure\nbelow. The flagpole will take up a square plot with area yd2.\n54. Find the length of the base of the flagpole by\nfactoring.\nExtensions\nFor the following exercises, factor the polynomials completely.\n55. 56. 57.\n58. 59.\n1.6 Rational Expressions\nLearning Objectives\nIn this section, you will:\nSimplify rational expressions.\nMultiply rational expressions.\nDivide rational expressions."
	},
	{
		"chunk_id" : "00000193",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Divide rational expressions.\nAdd and subtract rational expressions.\nSimplify complex rational expressions.\nA pastry shop has fixed costs of per week and variable costs of per box of pastries. The shops costs per week in\nterms of the number of boxes made, is We can divide the costs per week by the number of boxes made to\ndetermine the cost per box of pastries.\nNotice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will"
	},
	{
		"chunk_id" : "00000194",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "explore quotients of polynomial expressions.\nSimplifying Rational Expressions\nThe quotient of two polynomial expressions is called arational expression. We can apply the properties of fractions to\nAccess for free at openstax.org\n1.6  Rational Expressions 69\nrational expressions, such as simplifying the expressions by canceling common factors from the numerator and the\ndenominator. To do this, we first need to factor both the numerator and denominator. Lets start with the rational\nexpression shown."
	},
	{
		"chunk_id" : "00000195",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expression shown.\nWe can factor the numerator and denominator to rewrite the expression.\nThen we can simplify that expression by canceling the common factor\n...\nHOW TO\nGiven a rational expression, simplify it.\n1. Factor the numerator and denominator.\n2. Cancel any common factors.\nEXAMPLE1\nSimplifying Rational Expressions\nSimplify\nSolution\nAnalysis\nWe can cancel the common factor because any expression divided by itself is equal to 1.\nQ&A Can the term be cancelled inExample 1?"
	},
	{
		"chunk_id" : "00000196",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Can the term be cancelled inExample 1?\nNo. A factor is an expression that is multiplied by another expression. The term is not a factor of the\nnumerator or the denominator.\nTRY IT #1 Simplify\nMultiplying Rational Expressions\nMultiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the\nnumerators to find the numerator of the product, and then multiply the denominators to find the denominator of the"
	},
	{
		"chunk_id" : "00000197",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying\nrational expressions. We are often able to simplify the product of rational expressions.\n...\nHOW TO\nGiven two rational expressions, multiply them.\n1. Factor the numerator and denominator.\n70 1  Prerequisites\n2. Multiply the numerators.\n3. Multiply the denominators.\n4. Simplify.\nEXAMPLE2\nMultiplying Rational Expressions\nMultiply the rational expressions and show the product in simplest form:"
	},
	{
		"chunk_id" : "00000198",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nTRY IT #2 Multiply the rational expressions and show the product in simplest form:\nDividing Rational Expressions\nDivision of rational expressions works the same way as division of other fractions. To divide a rational expression by\nanother rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would\nrewrite as the product Once the division expression has been rewritten as a multiplication expression,\nwe can multiply as we did before.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000199",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we can multiply as we did before.\n...\nHOW TO\nGiven two rational expressions, divide them.\n1. Rewrite as the first rational expression multiplied by the reciprocal of the second.\n2. Factor the numerators and denominators.\n3. Multiply the numerators.\n4. Multiply the denominators.\n5. Simplify.\nEXAMPLE3\nDividing Rational Expressions\nDivide the rational expressions and express the quotient in simplest form:\nAccess for free at openstax.org\n1.6  Rational Expressions 71\nSolution"
	},
	{
		"chunk_id" : "00000200",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.6  Rational Expressions 71\nSolution\nTRY IT #3 Divide the rational expressions and express the quotient in simplest form:\nAdding and Subtracting Rational Expressions\nAdding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions,\nwe need to find a common denominator. Lets look at an example of fraction addition.\nWe have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same"
	},
	{
		"chunk_id" : "00000201",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thing when adding or subtracting rational expressions.\nThe easiest common denominator to use will be theleast common denominator, or LCD. The LCD is the smallest\nmultiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions\nand multiply all of the distinct factors. For instance, if the factored denominators were and\nthen the LCD would be\nOnce we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the"
	},
	{
		"chunk_id" : "00000202",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "LCD. We would need to multiply the expression with a denominator of by and the expression with a\ndenominator of by\n...\nHOW TO\nGiven two rational expressions, add or subtract them.\n1. Factor the numerator and denominator.\n2. Find the LCD of the expressions.\n3. Multiply the expressions by a form of 1 that changes the denominators to the LCD.\n4. Add or subtract the numerators.\n5. Simplify.\nEXAMPLE4\nAdding Rational Expressions\nAdd the rational expressions:\nSolution\n72 1  Prerequisites"
	},
	{
		"chunk_id" : "00000203",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\n72 1  Prerequisites\nFirst, we have to find the LCD. In this case, the LCD will be We then multiply each expression by the appropriate form\nof 1 to obtain as the denominator for each fraction.\nNow that the expressions have the same denominator, we simply add the numerators to find the sum.\nAnalysis\nMultiplying by or does not change the value of the original expression because any number divided by itself is 1,\nand multiplying an expression by 1 gives the original expression.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00000204",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE5\nSubtracting Rational Expressions\nSubtract the rational expressions:\nSolution\nQ&A Do we have to use the LCD to add or subtract rational expressions?\nNo. Any common denominator will work, but it is easiest to use the LCD.\nTRY IT #4 Subtract the rational expressions:\nSimplifying Complex Rational Expressions\nA complex rational expression is a rational expression that contains additional rational expressions in the numerator, the"
	},
	{
		"chunk_id" : "00000205",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as\nsingle rational expressions and dividing. The complex rational expression can be simplified by rewriting the\nnumerator as the fraction and combining the expressions in the denominator as We can then rewrite the\nexpression as a multiplication problem using the reciprocal of the denominator. We get which is equal to\n...\nHOW TO\nGiven a complex rational expression, simplify it."
	},
	{
		"chunk_id" : "00000206",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given a complex rational expression, simplify it.\nAccess for free at openstax.org\n1.6  Rational Expressions 73\n1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.\n2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.\n3. Rewrite as the numerator divided by the denominator.\n4. Rewrite as multiplication.\n5. Multiply.\n6. Simplify.\nEXAMPLE6\nSimplifying Complex Rational Expressions\nSimplify: .\nSolution"
	},
	{
		"chunk_id" : "00000207",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Simplify: .\nSolution\nBegin by combining the expressions in the numerator into one expression.\nNow the numerator is a single rational expression and the denominator is a single rational expression.\nWe can rewrite this as division, and then multiplication.\nTRY IT #5 Simplify:\nQ&A Can a complex rational expression always be simplified?\nYes. We can always rewrite a complex rational expression as a simplified rational expression.\nMEDIA"
	},
	{
		"chunk_id" : "00000208",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with rational expressions.\nSimplify Rational Expressions(http://openstax.org/l/simpratexpress)\nMultiply and Divide Rational Expressions(http://openstax.org/l/multdivratex)\nAdd and Subtract Rational Expressions(http://openstax.org/l/addsubratex)\nSimplify a Complex Fraction(http://openstax.org/l/complexfract)\n74 1  Prerequisites\n1.6 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000209",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "74 1  Prerequisites\n1.6 SECTION EXERCISES\nVerbal\n1. How can you use factoring 2. How do you use the LCD to 3. Tell whether the following\nto simplify rational combine two rational statement is true or false\nexpressions? expressions? and explain why: You only\nneed to find the LCD when\nadding or subtracting\nrational expressions.\nAlgebraic\nFor the following exercises, simplify the rational expressions.\n4. 5. 6.\n7. 8. 9.\n10. 11. 12.\n13."
	},
	{
		"chunk_id" : "00000210",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. 5. 6.\n7. 8. 9.\n10. 11. 12.\n13.\nFor the following exercises, multiply the rational expressions and express the product in simplest form.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23.\nFor the following exercises, divide the rational expressions.\n24. 25. 26.\n27. 28. 29.\n30. 31. 32.\nAccess for free at openstax.org\n1.6  Rational Expressions 75\nFor the following exercises, add and subtract the rational expressions, and then simplify.\n33. 34. 35.\n36. 37. 38.\n39. 40. 41."
	},
	{
		"chunk_id" : "00000211",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "33. 34. 35.\n36. 37. 38.\n39. 40. 41.\nFor the following exercises, simplify the rational expression.\n42. 43. 44.\n45. 46. 47.\n48. 49. 50.\nReal-World Applications\n51. Brenda is placing tile on her 52. The area of Lijuan's yard is 53. Elroi wants to mulch his\nbathroom floor. The area of the ft2. A patch of garden. His garden is\nfloor is ft2. The area sod has an area of ft2. One bag\nof one tile is To find ft2. Divide of mulch covers"
	},
	{
		"chunk_id" : "00000212",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the number of tiles needed, the two areas and simplify ft2. Divide the expressions\nsimplify the rational expression: to find how many pieces of and simplify to find how\nsod Lijuan needs to cover many bags of mulch Elroi\nher yard. needs to mulch his garden.\nExtensions\nFor the following exercises, perform the given operations and simplify.\n54. 55. 56.\n57.\n76 1  Chapter Review\nChapter Review\nKey Terms"
	},
	{
		"chunk_id" : "00000213",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "76 1  Chapter Review\nChapter Review\nKey Terms\nalgebraic expression constants and variables combined using addition, subtraction, multiplication, and division\nassociative property of addition the sum of three numbers may be grouped differently without affecting the result;\nin symbols,\nassociative property of multiplication the product of three numbers may be grouped differently without affecting\nthe result; in symbols,\nbase in exponential notation, the expression that is being multiplied"
	},
	{
		"chunk_id" : "00000214",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "binomial a polynomial containing two terms\ncoefficient any real number in a polynomial in the form\ncommutative property of addition two numbers may be added in either order without affecting the result; in\nsymbols,\ncommutative property of multiplication two numbers may be multiplied in any order without affecting the result; in\nsymbols,\nconstant a quantity that does not change value\ndegree the highest power of the variable that occurs in a polynomial"
	},
	{
		"chunk_id" : "00000215",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference of squares the binomial that results when a binomial is multiplied by a binomial with the same terms, but\nthe opposite sign\ndistributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in\nsymbols,\nequation a mathematical statement indicating that two expressions are equal\nexponent in exponential notation, the raised number or variable that indicates how many times the base is being\nmultiplied"
	},
	{
		"chunk_id" : "00000216",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplied\nexponential notation a shorthand method of writing products of the same factor\nfactor by grouping a method for factoring a trinomial in the form by dividing thexterm into the sum of\ntwo terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire\nexpression\nformula an equation expressing a relationship between constant and variable quantities\ngreatest common factor the largest polynomial that divides evenly into each polynomial"
	},
	{
		"chunk_id" : "00000217",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "identity property of addition there is a unique number, called the additive identity, 0, which, when added to a\nnumber, results in the original number; in symbols,\nidentity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when\nmultiplied by a number, results in the original number; in symbols,\nindex the number above the radical sign indicating thenth root\nintegers the set consisting of the natural numbers, their opposites, and 0:"
	},
	{
		"chunk_id" : "00000218",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inverse property of addition for every real number there is a unique number, called the additive inverse (or\nopposite), denoted which, when added to the original number, results in the additive identity, 0; in symbols,\ninverse property of multiplication for every non-zero real number there is a unique number, called the\nmultiplicative inverse (or reciprocal), denoted which, when multiplied by the original number, results in the\nmultiplicative identity, 1; in symbols,"
	},
	{
		"chunk_id" : "00000219",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplicative identity, 1; in symbols,\nirrational numbers the set of all numbers that are not rational; they cannot be written as either a terminating or\nrepeating decimal; they cannot be expressed as a fraction of two integers\nleading coefficient the coefficient of the leading term\nleading term the term containing the highest degree\nleast common denominator the smallest multiple that two denominators have in common\nmonomial a polynomial containing one term\nnatural numbers the set of counting numbers:"
	},
	{
		"chunk_id" : "00000220",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "natural numbers the set of counting numbers:\norder of operations a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to\noperations\nperfect square trinomial the trinomial that results when a binomial is squared\npolynomial a sum of terms each consisting of a variable raised to a nonnegative integer power\nprincipalnth root the number with the same sign as that when raised to thenth power equals"
	},
	{
		"chunk_id" : "00000221",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "principal square root the nonnegative square root of a number that, when multiplied by itself, equals\nradical the symbol used to indicate a root\nradical expression an expression containing a radical symbol\nradicand the number under the radical symbol\nAccess for free at openstax.org\n1  Chapter Review 77\nrational expression the quotient of two polynomial expressions\nrational numbers the set of all numbers of the form where and are integers and Any rational number"
	},
	{
		"chunk_id" : "00000222",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "may be written as a fraction or a terminating or repeating decimal.\nreal number line a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent\n0; positive numbers lie to the right of 0 and negative numbers to the left.\nreal numbers the sets of rational numbers and irrational numbers taken together\nscientific notation a shorthand notation for writing very large or very small numbers in the form where\nand is an integer"
	},
	{
		"chunk_id" : "00000223",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and is an integer\nterm of a polynomial any of a polynomial in the form\ntrinomial a polynomial containing three terms\nvariable a quantity that may change value\nwhole numbers the set consisting of 0 plus the natural numbers:\nKey Equations\nRules of Exponents\nFor nonzero real numbers and and integers and\nProduct rule\nQuotient rule\nPower rule\nZero exponent rule\nNegative rule\nPower of a product rule\nPower of a quotient rule\nperfect square trinomial\ndifference of squares\ndifference of squares"
	},
	{
		"chunk_id" : "00000224",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference of squares\ndifference of squares\nperfect square trinomial\nsum of cubes\ndifference of cubes\nKey Concepts\n1.1Real Numbers: Algebra Essentials\n Rational numbers may be written as fractions or terminating or repeating decimals. SeeExample 1andExample 2.\n Determine whether a number is rational or irrational by writing it as a decimal. SeeExample 3.\n The rational numbers and irrational numbers make up the set of real numbers. SeeExample 4. A number can be"
	},
	{
		"chunk_id" : "00000225",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "classified as natural, whole, integer, rational, or irrational. SeeExample 5.\n78 1  Chapter Review\n The order of operations is used to evaluate expressions. SeeExample 6.\n The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of\nreal numbers. These are the commutative properties, the associative properties, the distributive property, the\nidentity properties, and the inverse properties. SeeExample 7."
	},
	{
		"chunk_id" : "00000226",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Algebraic expressions are composed of constants and variables that are combined using addition, subtraction,\nmultiplication, and division. SeeExample 8. They take on a numerical value when evaluated by replacing variables\nwith constants. SeeExample 9,Example 10, andExample 12\n Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or\nevaluated as any mathematical expression. SeeExample 11andExample 13.\n1.2Exponents and Scientific Notation"
	},
	{
		"chunk_id" : "00000227",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.2Exponents and Scientific Notation\n Products of exponential expressions with the same base can be simplified by adding exponents. SeeExample 1.\n Quotients of exponential expressions with the same base can be simplified by subtracting exponents. SeeExample\n2.\n Powers of exponential expressions with the same base can be simplified by multiplying exponents. SeeExample 3.\n An expression with exponent zero is defined as 1. SeeExample 4."
	},
	{
		"chunk_id" : "00000228",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " An expression with a negative exponent is defined as a reciprocal. SeeExample 5andExample 6.\n The power of a product of factors is the same as the product of the powers of the same factors. SeeExample 7.\n The power of a quotient of factors is the same as the quotient of the powers of the same factors. SeeExample 8.\n The rules for exponential expressions can be combined to simplify more complicated expressions. SeeExample 9."
	},
	{
		"chunk_id" : "00000229",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Scientific notation uses powers of 10 to simplify very large or very small numbers. SeeExample 10andExample 11.\n Scientific notation may be used to simplify calculations with very large or very small numbers. SeeExample 12and\nExample 13.\n1.3Radicals and Rational Exponents\n The principal square root of a number is the nonnegative number that when multiplied by itself equals See\nExample 1.\n If and are nonnegative, the square root of the product is equal to the product of the square roots of and"
	},
	{
		"chunk_id" : "00000230",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 2andExample 3.\n If and are nonnegative, the square root of the quotient is equal to the quotient of the square roots of and\nSeeExample 4andExample 5.\n We can add and subtract radical expressions if they have the same radicand and the same index. SeeExample 6and\nExample 7.\n Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square\nroot radical from the denominator, multiply both the numerator and the denominator by the conjugate of the"
	},
	{
		"chunk_id" : "00000231",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator. SeeExample 8andExample 9.\n The principalnth root of is the number with the same sign as that when raised to thenth power equals These\nroots have the same properties as square roots.SeeExample 10.\n Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. SeeExample 11\nandExample 12.\n The properties of exponents apply to rational exponents. SeeExample 13.\n1.4Polynomials"
	},
	{
		"chunk_id" : "00000232",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.4Polynomials\n A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is\nthe highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest\ndegree, and the leading coefficient is the coefficient of that term. SeeExample 1.\n We can add and subtract polynomials by combining like terms. SeeExample 2andExample 3."
	},
	{
		"chunk_id" : "00000233",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in\nthe second. Then add the products. SeeExample 4.\n FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. SeeExample 5.\n Perfect square trinomials and difference of squares are special products. SeeExample 6andExample 7.\n Follow the same rules to work with polynomials containing several variables. SeeExample 8.\n1.5Factoring Polynomials"
	},
	{
		"chunk_id" : "00000234",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1.5Factoring Polynomials\n The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first\nstep in any factoring problem. SeeExample 1.\n Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a\nsum of the second term. SeeExample 2.\nAccess for free at openstax.org\n1  Exercises 79\n Trinomials can be factored using a process called factoring by grouping. SeeExample 3."
	},
	{
		"chunk_id" : "00000235",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Perfect square trinomials and the difference of squares are special products and can be factored using equations.\nSeeExample 4andExample 5.\n The sum of cubes and the difference of cubes can be factored using equations. SeeExample 6andExample 7.\n Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. SeeExample 8.\n1.6Rational Expressions\n Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See\nExample 1."
	},
	{
		"chunk_id" : "00000236",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 1.\n We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See\nExample 2.\n To divide rational expressions, multiply by the reciprocal of the second expression. SeeExample 3.\n Adding or subtracting rational expressions requires finding a common denominator. SeeExample 4andExample 5.\n Complex rational expressions have fractions in the numerator or the denominator. These expressions can be\nsimplified. SeeExample 6.\nExercises\nReview Exercises"
	},
	{
		"chunk_id" : "00000237",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Exercises\nReview Exercises\nReal Numbers: Algebra Essentials\nFor the following exercises, perform the given operations.\n1. 2. 3.\nFor the following exercises, solve the equation.\n4. 5.\nFor the following exercises, simplify the expression.\n6. 7.\nFor the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive\nanswer.\n8. 11 9. 0 10.\n11.\nExponents and Scientific Notation\nFor the following exercises, simplify the expression.\n12. 13. 14.\n15. 16. 17."
	},
	{
		"chunk_id" : "00000238",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16. 17.\n18. 19. 20. Write the number in\nstandard notation:\n80 1  Exercises\n21. Write the number in\nscientific notation:\n16,340,000\nRadicals and Rational Expressions\nFor the following exercises, find the principal square root.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34.\nPolynomials\nFor the following exercises, perform the given operations and simplify.\n35. 36. 37.\n38. 39. 40.\n41. 42. 43.\n44.\nFactoring Polynomials\nFor the following exercises, find the greatest common factor."
	},
	{
		"chunk_id" : "00000239",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "45. 46. 47.\nFor the following exercises, factor the polynomial.\n48. 49. 50.\n51. 52. 53.\n54. 55. 56.\n57. 58. 59.\nAccess for free at openstax.org\n1  Exercises 81\n60.\nRational Expressions\nFor the following exercises, simplify the expression.\n61. 62. 63.\n64. 65. 66.\n67. 68. 69.\n70.\nPractice Test\nFor the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive\nanswer.\n1. 2.\nFor the following exercises, evaluate the expression.\n3. 4. 5. Write the number in"
	},
	{
		"chunk_id" : "00000240",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. 4. 5. Write the number in\nstandard notation:\n6. Write the number in\nscientific notation:\n0.0000000212.\nFor the following exercises, simplify the expression.\n7. 8. 9.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19. 20.\n82 1  Exercises\n21. 22.\nFor the following exercises, factor the polynomial.\n23. 24. 25.\n26.\nFor the following exercises, simplify the expression.\n27. 28. 29.\nAccess for free at openstax.org\n2  Introduction 83\n2 EQUATIONS AND INEQUALITIES"
	},
	{
		"chunk_id" : "00000241",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2  Introduction 83\n2 EQUATIONS AND INEQUALITIES\nFrom the air, a landscape of circular crop fields may seem random, but they are laid out and irrigated very precisely.\nFarmers and irrigation providers combining agricultural science, engineering, and mathematics to achieve the most\nproductive and efficient array. (Credit: Modification of \"Aerial Phot of Center Pivot Irrigations Systems (1)\"\" by Soil"
	},
	{
		"chunk_id" : "00000242",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.1The Rectangular Coordinate Systems and Graphs\n2.2Linear Equations in One Variable\n2.3Models and Applications\n2.4Complex Numbers\n2.5Quadratic Equations\n2.6Other Types of Equations\n2.7Linear Inequalities and Absolute Value Inequalities\nIntroduction to Equations and Inequalities\nIrrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally"
	},
	{
		"chunk_id" : "00000243",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and\nproductive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related\nequipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation\nsystems and conduits can be installed within the plotted area.\n2.1 The Rectangular Coordinate Systems and Graphs\nLearning Objectives"
	},
	{
		"chunk_id" : "00000244",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nPlot ordered pairs in a Cartesian coordinate system.\nGraph equations by plotting points.\nGraph equations with a graphing utility.\nFind x-intercepts and y-intercepts.\nUse the distance formula.\nUse the midpoint formula.\n84 2  Equations and Inequalities\nFigure1\nTracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is"
	},
	{
		"chunk_id" : "00000245",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "indicated by a red dot inFigure 1. Laying a rectangular coordinate grid over the map, we can see that each stop aligns\nwith an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in\nlocations.\nPlotting Ordered Pairs in the Cartesian Coordinate System\nAn old story describes how seventeenth-century philosopher/mathematician Ren Descartes, while sick in bed, invented"
	},
	{
		"chunk_id" : "00000246",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on\nthe ceiling when he realized that he could describe the flys location in relation to the perpendicular lines formed by the\nadjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each\naxis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just"
	},
	{
		"chunk_id" : "00000247",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "two numbersthe displacement from the horizontal axis and the displacement from the vertical axis.\nWhile there is evidence that ideas similar to Descartes grid system existed centuries earlier, it was Descartes who\nintroduced the components that comprise theCartesian coordinate system, a grid system having perpendicular axes.\nDescartes named the horizontal axis thex-axisand the vertical axis they-axis."
	},
	{
		"chunk_id" : "00000248",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane\nconsisting of thex-axis and they-axis. Perpendicular to each other, the axes divide the plane into four sections. Each\nsection is called aquadrant; the quadrants are numbered counterclockwise as shown inFigure 2\nFigure2\nThe center of the plane is the point at which the two axes cross. It is known as theorigin, or point From the origin,"
	},
	{
		"chunk_id" : "00000249",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each axis is further divided into equal units: increasing, positive numbers to the right on thex-axis and up they-axis;\ndecreasing, negative numbers to the left on thex-axis and down they-axis. The axes extend to positive and negative\ninfinity as shown by the arrowheads inFigure 3.\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 85\nFigure3\nEach point in the plane is identified by itsx-coordinate, or horizontal displacement from the origin, and its"
	},
	{
		"chunk_id" : "00000250",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-coordinate, or vertical displacement from the origin. Together, we write them as anordered pairindicating the\ncombined distance from the origin in the form An ordered pair is also known as a coordinate pair because it\nconsists ofx-andy-coordinates. For example, we can represent the point in the plane by moving three units to\nthe right of the origin in the horizontal direction, and one unit down in the vertical direction. SeeFigure 4.\nFigure4"
	},
	{
		"chunk_id" : "00000251",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\nWhen dividing the axes into equally spaced increments, note that thex-axis may be considered separately from the\ny-axis. In other words, while thex-axis may be divided and labeled according to consecutive integers, they-axis may be\ndivided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against\nthe balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system"
	},
	{
		"chunk_id" : "00000252",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "primarily as a method for showing the relationship between two quantities.\nCartesian Coordinate System\nA two-dimensional plane where the\n x-axis is the horizontal axis\n y-axis is the vertical axis\nA point in the plane is defined as an ordered pair, such thatxis determined by its horizontal distance from the\norigin andyis determined by its vertical distance from the origin.\nEXAMPLE1\nPlotting Points in a Rectangular Coordinate System\nPlot the points and in the plane.\nSolution"
	},
	{
		"chunk_id" : "00000253",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Plot the points and in the plane.\nSolution\nTo plot the point begin at the origin. Thex-coordinate is 2, so move two units to the left. They-coordinate is 4,\nso then move four units up in the positiveydirection.\n86 2  Equations and Inequalities\nTo plot the point begin again at the origin. Thex-coordinate is 3, so move three units to the right. The\ny-coordinate is also 3, so move three units up in the positiveydirection."
	},
	{
		"chunk_id" : "00000254",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To plot the point begin again at the origin. Thex-coordinate is 0. This tells us not to move in either direction\nalong thex-axis. They-coordinate is 3, so move three units down in the negativeydirection. See the graph inFigure 5.\nFigure5\nAnalysis\nNote that when either coordinate is zero, the point must be on an axis. If thex-coordinate is zero, the point is on the\ny-axis. If they-coordinate is zero, the point is on thex-axis.\nGraphing Equations by Plotting Points"
	},
	{
		"chunk_id" : "00000255",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Equations by Plotting Points\nWe can plot a set of points to represent an equation. When such an equation contains both anxvariable and ay\nvariable, it is called anequation in two variables. Its graph is called agraph in two variables. Any graph on a two-\ndimensional plane is a graph in two variables.\nSuppose we want to graph the equation We can begin by substituting a value forxinto the equation and"
	},
	{
		"chunk_id" : "00000256",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determining the resulting value ofy. Each pair ofx- andy-values is an ordered pair that can be plotted.Table 1lists\nvalues ofxfrom 3 to 3 and the resulting values fory.\nTable1\nWe can plot the points in the table. The points for this particular equation form a line, so we can connect them. See\nFigure 6.This is not true for all equations.\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 87\nFigure6"
	},
	{
		"chunk_id" : "00000257",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nNote that thex-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some\nsituations may require particular values ofxto be plotted in order to see a particular result. Otherwise, it is logical to\nchoose values that can be calculated easily, and it is always a good idea to choose values that are both negative and\npositive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind,"
	},
	{
		"chunk_id" : "00000258",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "however, that the more points we plot, the more accurately we can sketch the graph.\n...\nHOW TO\nGiven an equation, graph by plotting points.\n1. Make a table with one column labeledx, a second column labeled with the equation, and a third column listing\nthe resulting ordered pairs.\n2. Enterx-values down the first column using positive and negative values. Selecting thex-values in numerical\norder will make the graphing simpler."
	},
	{
		"chunk_id" : "00000259",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "order will make the graphing simpler.\n3. Selectx-values that will yieldy-values with little effort, preferably ones that can be calculated mentally.\n4. Plot the ordered pairs.\n5. Connect the points if they form a line.\nEXAMPLE2\nGraphing an Equation in Two Variables by Plotting Points\nGraph the equation by plotting points.\nSolution\nFirst, we construct a table similar toTable 2. Choosexvalues and calculatey.\nTable2\n88 2  Equations and Inequalities\nTable2"
	},
	{
		"chunk_id" : "00000260",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table2\n88 2  Equations and Inequalities\nTable2\nNow, plot the points. Connect them if they form a line. SeeFigure 7\nFigure7\nTRY IT #1 Construct a table and graph the equation by plotting points:\nGraphing Equations with a Graphing Utility\nMost graphing calculators require similar techniques to graph an equation. The equations sometimes have to be\nmanipulated so they are written in the style The TI-84 Plus, and many other calculator makes and models,"
	},
	{
		"chunk_id" : "00000261",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of\na graph can be seen.\nFor example, the equation has been entered in the TI-84 Plus shown inFigure 8a.InFigure 8b,the resulting\ngraph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen\non the TI-84 Plus shows and SeeFigure 8c."
	},
	{
		"chunk_id" : "00000262",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on the TI-84 Plus shows and SeeFigure 8c.\nFigure8 a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.\nBy changing the window to show more of the positivex-axis and more of the negativey-axis, we have a much better\nview of the graph and thex-andy-intercepts. SeeFigure 9aandFigure 9b.\nFigure9 a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000263",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 89\nEXAMPLE3\nUsing a Graphing Utility to Graph an Equation\nUse a graphing utility to graph the equation:\nSolution\nEnter the equation in they=function of the calculator. Set the window settings so that both thex-andy-intercepts are\nshowing in the window. SeeFigure 10.\nFigure10\nFindingx-intercepts andy-intercepts\nTheinterceptsof a graph are points at which the graph crosses the axes. Thex-interceptis the point at which the"
	},
	{
		"chunk_id" : "00000264",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph crosses thex-axis. At this point, they-coordinate is zero. They-interceptis the point at which the graph crosses\nthey-axis. At this point, thex-coordinate is zero.\nTo determine thex-intercept, we setyequal to zero and solve forx. Similarly, to determine they-intercept, we setx\nequal to zero and solve fory. For example, lets find the intercepts of the equation\nTo find thex-intercept, set\nTo find they-intercept, set"
	},
	{
		"chunk_id" : "00000265",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To find they-intercept, set\nWe can confirm that our results make sense by observing a graph of the equation as inFigure 11. Notice that the graph\ncrosses the axes where we predicted it would.\nFigure11\nGiven an equation, find the intercepts.\n Find thex-intercept by setting and solving for\n90 2  Equations and Inequalities\n Find they-intercept by setting and solving for\nEXAMPLE4\nFinding the Intercepts of the Given Equation\nFind the intercepts of the equation Then sketch the graph using only the intercepts."
	},
	{
		"chunk_id" : "00000266",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSet to find thex-intercept.\nSet to find they-intercept.\nPlot both points, and draw a line passing through them as inFigure 12.\nFigure12\nTRY IT #2 Find the intercepts of the equation and sketch the graph:\nUsing the Distance Formula\nDerived from thePythagorean Theorem, thedistance formulais used to find the distance between two points in the\nplane. The Pythagorean Theorem, is based on a right triangle whereaandbare the lengths of the legs"
	},
	{
		"chunk_id" : "00000267",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "adjacent to the right angle, andcis the length of the hypotenuse. SeeFigure 13.\nFigure13\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 91\nThe relationship of sides and to sidedis the same as that of sidesaandbto sidec.We use the\nabsolute value symbol to indicate that the length is a positive number because the absolute value of any number is\npositive. (For example, ) The symbols and indicate that the lengths of the sides of the"
	},
	{
		"chunk_id" : "00000268",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "triangle are positive. To find the lengthc, take the square root of both sides of the Pythagorean Theorem.\nIt follows that the distance formula is given as\nWe do not have to use the absolute value symbols in this definition because any number squared is positive.\nThe Distance Formula\nGiven endpoints and the distance between two points is given by\nEXAMPLE5\nFinding the Distance between Two Points\nFind the distance between the points and\nSolution"
	},
	{
		"chunk_id" : "00000269",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the distance between the points and\nSolution\nLet us first look at the graph of the two points. Connect the points to form a right triangle as inFigure 14.\nFigure14\nThen, calculate the length ofdusing the distance formula.\nTRY IT #3 Find the distance between two points: and\nEXAMPLE6\nFinding the Distance between Two Locations\nLets return to the situation introduced at the beginning of this section."
	},
	{
		"chunk_id" : "00000270",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is\n92 2  Equations and Inequalities\nindicated by a red dot inFigure 1. Find the total distance that Tracie traveled. Compare this with the distance between\nher starting and final positions.\nSolution\nThe first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the"
	},
	{
		"chunk_id" : "00000271",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the\nfirst stop is 1 block east and 1 block north, so it is at The next stop is 5 blocks to the east, so it is at After\nthat, she traveled 3 blocks east and 2 blocks north to Lastly, she traveled 4 blocks north to We can label\nthese points on the grid as inFigure 15.\nFigure15\nNext, we can calculate the distance. Note that each grid unit represents 1,000 feet."
	},
	{
		"chunk_id" : "00000272",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " From her starting location to her first stop at Tracie might have driven north 1,000 feet and then east 1,000\nfeet, or vice versa. Either way, she drove 2,000 feet to her first stop.\n Her second stop is at So from to Tracie drove east 4,000 feet.\n Her third stop is at There are a number of routes from to Whatever route Tracie decided to use,\nthe distance is the same, as there are no angular streets between the two points. Lets say she drove east 3,000 feet"
	},
	{
		"chunk_id" : "00000273",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and then north 2,000 feet for a total of 5,000 feet.\n Tracies final stop is at This is a straight drive north from for a total of 4,000 feet.\nNext, we will add the distances listed inTable 3.\nFrom/To Number of Feet Driven\nto 2,000\nto 4,000\nto 5,000\nto 4,000\nTotal 15,000\nTable3\nThe total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 93"
	},
	{
		"chunk_id" : "00000274",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and ending positions. To find this distance, we can use the distance formula between the points and\nAt 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The\ndistance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight\ndiagonal from the origin to the point Perhaps you have heard the saying as the crow flies, which means the"
	},
	{
		"chunk_id" : "00000275",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shortest distance between two points because a crow can fly in a straight line even though a person on the ground has\nto travel a longer distance on existing roadways.\nUsing the Midpoint Formula\nWhen the endpoints of a line segment are known, we can find the point midway between them. This point is known as\nthe midpoint and the formula is known as themidpoint formula. Given the endpoints of a line segment, and\nthe midpoint formula states how to find the coordinates of the midpoint"
	},
	{
		"chunk_id" : "00000276",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A graphical view of a midpoint is shown inFigure 16. Notice that the line segments on either side of the midpoint are\ncongruent.\nFigure16\nEXAMPLE7\nFinding the Midpoint of the Line Segment\nFind the midpoint of the line segment with the endpoints and\nSolution\nUse the formula to find the midpoint of the line segment.\nTRY IT #4 Find the midpoint of the line segment with endpoints and\nEXAMPLE8\nFinding the Center of a Circle\nThe diameter of a circle has endpoints and Find the center of the circle."
	},
	{
		"chunk_id" : "00000277",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "94 2  Equations and Inequalities\nSolution\nThe center of a circle is the center, or midpoint, of its diameter. Thus, the midpoint formula will yield the center point.\nMEDIA\nAccess these online resources for additional instruction and practice with the Cartesian coordinate system.\nPlotting points on the coordinate plane(http://openstax.org/l/coordplotpnts)\nFind x and y intercepts based on the graph of a line(http://openstax.org/l/xyintsgraph)\n2.1 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000278",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.1 SECTION EXERCISES\nVerbal\n1. Is it possible for a point 2. Describe the process for 3. Describe in your own words\nplotted in the Cartesian finding thex-intercept and what they-intercept of a\ncoordinate system to not lie they-intercept of a graph graph is.\nin one of the four algebraically.\nquadrants? Explain.\n4. When using the distance\nformula\nexplain the correct order of\noperations that are to be\nperformed to obtain the\ncorrect answer.\nAlgebraic"
	},
	{
		"chunk_id" : "00000279",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "performed to obtain the\ncorrect answer.\nAlgebraic\nFor each of the following exercises, find thex-intercept and they-intercept without graphing. Write the coordinates of\neach intercept.\n5. 6. 7.\n8. 9. 10.\nFor each of the following exercises, solve the equation foryin terms ofx.\n11. 12. 13.\n14. 15. 16.\nFor each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact\nanswer in simplest radical form for irrational answers.\n17. and 18. and 19. and"
	},
	{
		"chunk_id" : "00000280",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "17. and 18. and 19. and\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 95\n20. and 21. Find the distance between\nthe two points given using\nyour calculator, and round\nyour answer to the nearest\nhundredth.\nand\nFor each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given\npoints.\n22. and 23. and 24. and\n25. and 26. and\nGraphical\nFor each of the following exercises, identify the information requested."
	},
	{
		"chunk_id" : "00000281",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "27. What are the coordinates 28. If a point is located on the 29. If a point is located on the\nof the origin? y-axis, what is the x-axis, what is the\nx-coordinate? y-coordinate?\nFor each of the following exercises, plot the three points on the given coordinate plane. State whether the three points\nyou plotted appear to be collinear (on the same line).\n30. 31. 32.\n33. Name the coordinates of the 34. Name the quadrant in\npoints graphed. which the following points\nwould be located. If the"
	},
	{
		"chunk_id" : "00000282",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would be located. If the\npoint is on an axis, name\nthe axis.\n \n \n\n96 2  Equations and Inequalities\nFor each of the following exercises, construct a table and graph the equation by plotting at least three points.\n35. 36. 37.\nNumeric\nFor each of the following exercises, find and plot thex-andy-intercepts, and graph the straight line based on those two\npoints.\n38. 39. 40.\n41. 42.\nFor each of the following exercises, use the graph in the figure below."
	},
	{
		"chunk_id" : "00000283",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "43. Find the distance between 44. Find the coordinates of the 45. Find the distance that\nthe two endpoints using midpoint of the line is from the origin.\nthe distance formula. segment connecting the\nRound to three decimal two points.\nplaces.\n46. Find the distance that 47. Which point is closer to the\nis from the origin. Round to origin?\nthree decimal places.\nTechnology\nFor the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu."
	},
	{
		"chunk_id" : "00000284",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "After graphing it, use the 2ndCALC button and 1:value button, hit enter. At the lower part of the screen you will see x=\nand a blinking cursor. You may enter any number forxand it will display theyvalue for anyxvalue you input. Use this\nand plug inx= 0, thus finding they-intercept, for each of the following graphs.\n48. 49. 50.\nAccess for free at openstax.org\n2.1  The Rectangular Coordinate Systems and Graphs 97"
	},
	{
		"chunk_id" : "00000285",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.\nAfter graphing it, use the 2ndCALCbutton and 2:zero button, hitENTER. At the lower part of the screen you will see left\nbound? and a blinking cursor on the graph of the line. Move this cursor to the left of thex-intercept, hitENTER. Now it\nsays right bound? Move the cursor to the right of thex-intercept, hitENTER. Now it says guess? Move your cursor to"
	},
	{
		"chunk_id" : "00000286",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the left somewhere in between the left and right bound near thex-intercept. HitENTER. At the bottom of your screen it\nwill display the coordinates of thex-intercept or the zero to they-value. Use this to find thex-intercept.\nNote: With linear/straight line functions the zero is not really a guess, but it is necessary to enter a guess so it will\nsearch and find the exactx-intercept between your right and left boundaries. With other types of functions (more than"
	},
	{
		"chunk_id" : "00000287",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "onex-intercept), they may be irrational numbers so guess is more appropriate to give it the correct limits to find a very\nclose approximation between the left and right boundaries.\n51. 52. 53. Round your\nanswer to the nearest\nthousandth.\nExtensions\n54. Someone drove 10 mi 55. If the road was made in the 56. Given these four points:\ndirectly east from their previous exercise, how , , ,\nhome, made a left turn at much shorter would the and find the"
	},
	{
		"chunk_id" : "00000288",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "an intersection, and then persons one-way trip be coordinates of the\ntraveled 5 mi north to their every day? midpoint of line segments\nplace of work. If a road was and\nmade directly from the\nhome to the place of work,\nwhat would its distance be\nto the nearest tenth of a\nmile?\n57. After finding the two 58. Given the graph of the rectangle 59. In the previous exercise,\nmidpoints in the previous shown and the coordinates of its find the coordinates of the"
	},
	{
		"chunk_id" : "00000289",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exercise, find the distance vertices, prove that the diagonals of midpoint for each\nbetween the two midpoints the rectangle are of equal length. diagonal.\nto the nearest thousandth.\n98 2  Equations and Inequalities\nReal-World Applications\n60. The coordinates on a map 61. If San Joses coordinates 62. A small craft in Lake\nfor San Francisco are are , where the Ontario sends out a\nand those for coordinates represent distress signal. The\nSacramento are . miles, find the distance coordinates of the boat in"
	},
	{
		"chunk_id" : "00000290",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Note that coordinates between San Jose and San trouble were One\nrepresent miles. Find the Francisco to the nearest rescue boat is at the\ndistance between the cities mile. coordinates and a\nto the nearest mile. second Coast Guard craft is\nat coordinates .\nAssuming both rescue craft\ntravel at the same rate,\nwhich one would get to the\ndistressed boat the fastest?\n63. A person on the top of a 64. If we rent a truck and pay a\nbuilding wants to have a $75/day fee plus $.20 for"
	},
	{
		"chunk_id" : "00000291",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "guy wire extend to a point every mile we travel, write\non the ground 20 ft from a linear equation that\nthe building. To the nearest would express the total\nfoot, how long will the wire cost per day using to\nhave to be if the building is represent the number of\n50 ft tall? miles we travel. Graph this\nfunction on your graphing\ncalculator and find the total\ncost for one day if we travel\n70 mi.\n2.2 Linear Equations in One Variable\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00000292",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nSolve equations in one variable algebraically.\nSolve a rational equation.\nFind a linear equation.\nGiven the equations of two lines, determine whether their graphs are parallel or perpendicular.\nWrite the equation of a line parallel or perpendicular to a given line.\nCaroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has"
	},
	{
		"chunk_id" : "00000293",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of\n$400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip\nwill cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can\nonly work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section,"
	},
	{
		"chunk_id" : "00000294",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we will investigate problems like this and others, which generate graphs like the line inFigure 1.\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 99\nFigure1\nSolving Linear Equations in One Variable\nAlinear equationis an equation of a straight line, written in one variable. The only power of the variable is 1. Linear\nequations in one variable may take the form and are solved using basic algebraic operations."
	},
	{
		"chunk_id" : "00000295",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An\nidentity equationis true for all values of the variable. Here is an example of an identity equation.\nThesolution setconsists of all values that make the equation true. For this equation, the solution set is all real numbers\nbecause any real number substituted for will make the equation true."
	},
	{
		"chunk_id" : "00000296",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Aconditional equationis true for only some values of the variable. For example, if we are to solve the equation\nwe have the following:\nThe solution set consists of one number: It is the only solution and, therefore, we have solved a conditional\nequation.\nAninconsistent equationresults in a false statement. For example, if we are to solve we have the\nfollowing:\nIndeed, There is no solution because this is an inconsistent equation."
	},
	{
		"chunk_id" : "00000297",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations.\nA brief review of those operations follows.\nLinear Equation in One Variable\nA linear equation in one variable can be written in the form\nwhereaandbare real numbers,\n...\nHOW TO\nGiven a linear equation in one variable, use algebra to solve it.\nThe following steps are used to manipulate an equation and isolate the unknown variable, so that the last line reads"
	},
	{
		"chunk_id" : "00000298",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ifxis the unknown. There is no set order, as the steps used depend on what is given:\n100 2  Equations and Inequalities\n1. We may add, subtract, multiply, or divide an equation by a number or an expression as long as we do the same\nthing to both sides of the equal sign. Note that we cannot divide by zero.\n2. Apply the distributive property as needed:\n3. Isolate the variable on one side of the equation."
	},
	{
		"chunk_id" : "00000299",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. When the variable is multiplied by a coefficient in the final stage, multiply both sides of the equation by the\nreciprocal of the coefficient.\nEXAMPLE1\nSolving an Equation in One Variable\nSolve the following equation:\nSolution\nThis equation can be written in the form by subtracting from both sides. However, we may proceed to\nsolve the equation in its original form by performing algebraic operations.\nThe solution is 6.\nTRY IT #1 Solve the linear equation in one variable:\nEXAMPLE2"
	},
	{
		"chunk_id" : "00000300",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nSolving an Equation Algebraically When the Variable Appears on Both Sides\nSolve the following equation:\nSolution\nApply standard algebraic properties.\nAnalysis\nThis problem requires the distributive property to be applied twice, and then the properties of algebra are used to reach\nthe final line,\nTRY IT #2 Solve the equation in one variable:\nSolving a Rational Equation\nIn this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation"
	},
	{
		"chunk_id" : "00000301",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "contains at least one rational expression, it is a considered arational equation.\nRecall that arational numberis the ratio of two numbers, such as or Arational expressionis the ratio, or quotient,\nof two polynomials. Here are three examples.\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 101\nRational equations have a variable in the denominator in at least one of the terms. Our goal is to perform algebraic"
	},
	{
		"chunk_id" : "00000302",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both\nsides of the equation by theleast common denominator(LCD).\nFinding the LCD is identifying an expression that contains the highest power of all of the factors in all of the\ndenominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in\neach denominator will equal one and will cancel out.\nEXAMPLE3\nSolving a Rational Equation"
	},
	{
		"chunk_id" : "00000303",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nSolving a Rational Equation\nSolve the rational equation:\nSolution\nWe have three denominators; and 3. The LCD must contain and 3. An LCD of contains all three\ndenominators. In other words, each denominator can be divided evenly into the LCD. Next, multiply both sides of the\nequation by the LCD\nA common mistake made when solving rational equations involves finding the LCD when one of the denominators is a"
	},
	{
		"chunk_id" : "00000304",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "binomialtwo terms added or subtractedsuch as Always consider a binomial as an individual factorthe\nterms cannot be separated. For example, suppose a problem has three terms and the denominators are and\nFirst, factor all denominators. We then have and as the denominators. (Note the parentheses\nplaced around the second denominator.) Only the last two denominators have a common factor of The in the\nfirst denominator is separate from the in the denominators. An effective way to remember this is to write"
	},
	{
		"chunk_id" : "00000305",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate\nfactor. The LCD in this instance is found by multiplying together the one factor of and the 3. Thus, the LCD is\nthe following:\nSo, both sides of the equation would be multiplied by Leave the LCD in factored form, as this makes it easier\nto see how each denominator in the problem cancels out.\nAnother example is a problem with two denominators, such as and Once the second denominator is factored"
	},
	{
		"chunk_id" : "00000306",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as there is a common factor ofxin both denominators and the LCD is\nSometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction\nand there are no other terms in the equation.\nWe can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by\ncrossing over the equal sign.\n102 2  Equations and Inequalities\nMultiply and which results in"
	},
	{
		"chunk_id" : "00000307",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Multiply and which results in\nAny solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.\nRational Equations\nArational equationcontains at least one rational expression where the variable appears in at least one of the\ndenominators.\n...\nHOW TO\nGiven a rational equation, solve it.\n1. Factor all denominators in the equation.\n2. Find and exclude values that set each denominator equal to zero.\n3. Find the LCD."
	},
	{
		"chunk_id" : "00000308",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Find the LCD.\n4. Multiply the whole equation by the LCD. If the LCD is correct, there will be no denominators left.\n5. Solve the remaining equation.\n6. Make sure to check solutions back in the original equations to avoid a solution producing zero in a denominator.\nEXAMPLE4\nSolving a Rational Equation without Factoring\nSolve the following rational equation:\nSolution\nWe have three denominators: and No factoring is required. The product of the first two denominators is equal"
	},
	{
		"chunk_id" : "00000309",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to the third denominator, so, the LCD is Only one value is excluded from a solution set, 0. Next, multiply the whole\nequation (both sides of the equal sign) by\nThe proposed solution is 1, which is not an excluded value, so the solution set contains one number, or\nwritten in set notation.\nTRY IT #3 Solve the rational equation:\nEXAMPLE5\nSolving a Rational Equation by Factoring the Denominator\nSolve the following rational equation:\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 103"
	},
	{
		"chunk_id" : "00000310",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.2  Linear Equations in One Variable 103\nSolution\nFirst find the common denominator. The three denominators in factored form are and The\nsmallest expression that is divisible by each one of the denominators is Only is an excluded value. Multiply\nthe whole equation by\nThe solution is\nTRY IT #4 Solve the rational equation:\nEXAMPLE6\nSolving Rational Equations with a Binomial in the Denominator\nSolve the following rational equations and state the excluded values:\n  \nSolution\n"
	},
	{
		"chunk_id" : "00000311",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nSolution\n\nThe denominators and have nothing in common. Therefore, the LCD is the product However, for\nthis problem, we can cross-multiply.\nThe solution is 15. The excluded values are and\n\nThe LCD is Multiply both sides of the equation by\nThe solution is The excluded value is\n104 2  Equations and Inequalities\n\nThe least common denominator is Multiply both sides of the equation by\nThe solution is 4. The excluded value is\nTRY IT #5 Solve State the excluded values.\nEXAMPLE7"
	},
	{
		"chunk_id" : "00000312",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nSolving a Rational Equation with Factored Denominators and Stating Excluded Values\nSolve the rational equation after factoring the denominators: State the excluded values.\nSolution\nWe must factor the denominator We recognize this as the difference of squares, and factor it as\nThus, the LCD that contains each denominator is Multiply the whole equation by the LCD, cancel out the\ndenominators, and solve the remaining equation.\nThe solution is The excluded values are and"
	},
	{
		"chunk_id" : "00000313",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solution is The excluded values are and\nTRY IT #6 Solve the rational equation:\nFinding a Linear Equation\nPerhaps the most familiar form of a linear equation is the slope-intercept form, written as where\nand Let us begin with the slope.\nThe Slope of a Line\nTheslopeof a line refers to the ratio of the vertical change inyover the horizontal change inxbetween any two points\non a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise\nover run."
	},
	{
		"chunk_id" : "00000314",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "over run.\nIf the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope\nincreases, the line becomes steeper. Some examples are shown inFigure 2. The lines indicate the following slopes:\nand\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 105\nFigure2\nThe Slope of a Line\nThe slope of a line,m, represents the change inyover the change inx.Given two points, and the"
	},
	{
		"chunk_id" : "00000315",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following formula determines the slope of a line containing these points:\nEXAMPLE8\nFinding the Slope of a Line Given Two Points\nFind the slope of a line that passes through the points and\nSolution\nWe substitute they-values and thex-values into the formula.\nThe slope is\nAnalysis\nIt does not matter which point is called or As long as we are consistent with the order of theyterms\nand the order of thexterms in the numerator and denominator, the calculation will yield the same result."
	},
	{
		"chunk_id" : "00000316",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Find the slope of the line that passes through the points and\nEXAMPLE9\nIdentifying the Slope andy-intercept of a Line Given an Equation\nIdentify the slope andy-intercept, given the equation\n106 2  Equations and Inequalities\nSolution\nAs the line is in form, the given line has a slope of They-intercept is\nAnalysis\nThey-intercept is the point at which the line crosses they-axis. On they-axis, We can always identify the"
	},
	{
		"chunk_id" : "00000317",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-intercept when the line is in slope-intercept form, as it will always equalb.Or, just substitute and solve fory.\nThe Point-Slope Formula\nGiven the slope and one point on a line, we can find the equation of the line using the point-slope formula.\nThis is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation\nof a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and"
	},
	{
		"chunk_id" : "00000318",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.\nThe Point-Slope Formula\nGiven one point and the slope, the point-slope formula will lead to the equation of a line:\nEXAMPLE10\nFinding the Equation of a Line Given the Slope and One Point\nWrite the equation of the line with slope and passing through the point Write the final equation in slope-\nintercept form.\nSolution\nUsing the point-slope formula, substitute formand the point for\nAnalysis"
	},
	{
		"chunk_id" : "00000319",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNote that any point on the line can be used to find the equation. If done correctly, the same final equation will be\nobtained.\nTRY IT #8 Given find the equation of the line in slope-intercept form passing through the point\nEXAMPLE11\nFinding the Equation of a Line Passing Through Two Given Points\nFind the equation of the line passing through the points and Write the final equation in slope-intercept\nform.\nSolution\nFirst, we calculate the slope using the slope formula and two points."
	},
	{
		"chunk_id" : "00000320",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n2.2  Linear Equations in One Variable 107\nNext, we use the point-slope formula with the slope of and either point. Lets pick the point for\nIn slope-intercept form, the equation is written as\nAnalysis\nTo prove that either point can be used, let us use the second point and see if we get the same equation.\nWe see that the same line will be obtained using either point. This makes sense because we used both points to calculate\nthe slope.\nStandard Form of a Line"
	},
	{
		"chunk_id" : "00000321",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the slope.\nStandard Form of a Line\nAnother way that we can represent the equation of a line is instandard form. Standard form is given as\nwhere and are integers. Thex-andy-terms are on one side of the equal sign and the constant term is on the\nother side.\nEXAMPLE12\nFinding the Equation of a Line and Writing It in Standard Form\nFind the equation of the line with and passing through the point Write the equation in standard form.\nSolution\nWe begin using the point-slope formula."
	},
	{
		"chunk_id" : "00000322",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe begin using the point-slope formula.\nFrom here, we multiply through by 2, as no fractions are permitted in standard form, and then move both variables to\nthe left aside of the equal sign and move the constants to the right.\nThis equation is now written in standard form.\nTRY IT #9 Find the equation of the line in standard form with slope and passing through the point\nVertical and Horizontal Lines"
	},
	{
		"chunk_id" : "00000323",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Vertical and Horizontal Lines\nThe equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the\nformulas to prove that the equations are correct. The equation of avertical lineis given as\nwherecis a constant. The slope of a vertical line is undefined, and regardless of they-value of any point on the line, the\nx-coordinate of the point will bec.\n108 2  Equations and Inequalities"
	},
	{
		"chunk_id" : "00000324",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "108 2  Equations and Inequalities\nSuppose that we want to find the equation of a line containing the following points: and\nFirst, we will find the slope.\nZero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula.\nHowever, we can plot the points. Notice that all of thex-coordinates are the same and we find a vertical line through\nSeeFigure 3.\nThe equation of ahorizontal lineis given as"
	},
	{
		"chunk_id" : "00000325",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The equation of ahorizontal lineis given as\nwherecis a constant. The slope of a horizontal line is zero, and for anyx-value of a point on the line, they-coordinate\nwill bec.\nSuppose we want to find the equation of a line that contains the following set of points: and\nWe can use the point-slope formula. First, we find the slope using any two points on the line.\nUse any point for in the formula, or use they-intercept."
	},
	{
		"chunk_id" : "00000326",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The graph is a horizontal line through Notice that all of they-coordinates are the same. SeeFigure 3.\nFigure3 The linex= 3 is a vertical line. The liney= 2 is a horizontal line.\nEXAMPLE13\nFinding the Equation of a Line Passing Through the Given Points\nFind the equation of the line passing through the given points: and\nSolution\nThex-coordinate of both points is 1. Therefore, we have a vertical line,\nTRY IT #10 Find the equation of the line passing through and"
	},
	{
		"chunk_id" : "00000327",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determining Whether Graphs of Lines are Parallel or Perpendicular\nParallel lines have the same slope and differenty-intercepts. Lines that areparallelto each other will never intersect. For\nexample,Figure 4shows the graphs of various lines with the same slope,\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 109\nFigure4 Parallel lines\nAll of the lines shown in the graph are parallel because they have the same slope and differenty-intercepts."
	},
	{
		"chunk_id" : "00000328",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lines that areperpendicularintersect to form a -angle. The slope of one line is the negativereciprocalof the other.\nWe can show that two lines are perpendicular if the product of the two slopes is For example,Figure\n5shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of\nFigure5 Perpendicular lines\nEXAMPLE14\nGraphing Two Equations, and Determining Whether the Lines are Parallel, Perpendicular, or Neither"
	},
	{
		"chunk_id" : "00000329",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph the equations of the given lines, and state whether they are parallel, perpendicular, or neither: and\nSolution\nThe first thing we want to do is rewrite the equations so that both equations are in slope-intercept form.\nFirst equation:\n110 2  Equations and Inequalities\nSecond equation:\nSee the graph of both lines inFigure 6\nFigure6\nFrom the graph, we can see that the lines appear perpendicular, but we must compare the slopes."
	},
	{
		"chunk_id" : "00000330",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The slopes are negative reciprocals of each other, confirming that the lines are perpendicular.\nTRY IT #11 Graph the two lines and determine whether they are parallel, perpendicular, or neither:\nand\nWriting the Equations of Lines Parallel or Perpendicular to a Given Line\nAs we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To"
	},
	{
		"chunk_id" : "00000331",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding\nthe equation of any line. After finding the slope, use thepoint-slope formulato write the equation of the new line.\n...\nHOW TO\nGiven an equation for a line, write the equation of a line parallel or perpendicular to it.\n1. Find the slope of the given line. The easiest way to do this is to write the equation in slope-intercept form."
	},
	{
		"chunk_id" : "00000332",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Use the slope and the given point with the point-slope formula.\n3. Simplify the line to slope-intercept form and compare the equation to the given line.\nEXAMPLE15\nWriting the Equation of a Line Parallel to a Given Line Passing Through a Given Point\nWrite the equation of line parallel to a and passing through the point\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 111\nSolution\nFirst, we will write the equation in slope-intercept form to find the slope."
	},
	{
		"chunk_id" : "00000333",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The slope is They-intercept is but that really does not enter into our problem, as the only thing we need for\ntwo lines to be parallel is the same slope. The one exception is that if they-intercepts are the same, then the two lines\nare the same line. The next step is to use this slope and the given point with the point-slope formula.\nThe equation of the line is SeeFigure 7.\nFigure7\nTRY IT #12 Find the equation of the line parallel to and passing through the point\nEXAMPLE16"
	},
	{
		"chunk_id" : "00000334",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE16\nFinding the Equation of a Line Perpendicular to a Given Line Passing Through a Given Point\nFind the equation of the line perpendicular to and passing through the point\nSolution\nThe first step is to write the equation in slope-intercept form.\nWe see that the slope is This means that the slope of the line perpendicular to the given line is the negative\nreciprocal, or Next, we use the point-slope formula with this new slope and the given point.\n112 2  Equations and Inequalities\nMEDIA"
	},
	{
		"chunk_id" : "00000335",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "112 2  Equations and Inequalities\nMEDIA\nAccess these online resources for additional instruction and practice with linear equations.\nSolving rational equations(http://openstax.org/l/rationaleqs)\nEquation of a line given two points(http://openstax.org/l/twopointsline)\nFinding the equation of a line perpendicular to another line through a given point(http://openstax.org/l/\nfindperpline)\nFinding the equation of a line parallel to another line through a given point(http://openstax.org/l/findparaline)"
	},
	{
		"chunk_id" : "00000336",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.2 SECTION EXERCISES\nVerbal\n1. What does it mean when we 2. What is the relationship 3. How do we recognize when\nsay that two lines are between the slopes of an equation, for example\nparallel? perpendicular lines will be a straight\n(assuming neither is line (linear) when graphed?\nhorizontal nor vertical)?\n4. What does it mean when we 5. When solving the following\nsay that a linear equation is equation:\ninconsistent?\nexplain why we must\nexclude and\nas possible solutions from\nthe solution set.\nAlgebraic"
	},
	{
		"chunk_id" : "00000337",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the solution set.\nAlgebraic\nFor the following exercises, solve the equation for\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15.\nFor the following exercises, solve each rational equation for State allx-values that are excluded from the solution set.\n16. 17. 18.\n19. 20. 21.\nAccess for free at openstax.org\n2.2  Linear Equations in One Variable 113\nFor the following exercises, find the equation of the line using the point-slope formula. Write all the final equations using\nthe slope-intercept form."
	},
	{
		"chunk_id" : "00000338",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the slope-intercept form.\n22. with a slope of 23. with a slope of 24. x-intercept is 1, and\n25. y-intercept is 2, and 26. and 27.\n28. parallel to and 29. perpendicular to\npasses through the point and passes\nthrough the point .\nFor the following exercises, find the equation of the line using the given information.\n30. and 31. and 32. The slope is undefined and\nit passes through the point\n33. The slope equals zero and 34. Theslopeis 35. and\nit passes through the point anditpassesthroughthepoint\n.\nGraphical"
	},
	{
		"chunk_id" : "00000339",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : ".\nGraphical\nFor the following exercises, graph the pair of equations on the same axes, and state whether they are parallel,\nperpendicular, or neither.\n36. 37. 38.\n39.\nNumeric\nFor the following exercises, find the slope of the line that passes through the given points.\n40. and 41. and 42. and\n43. and 44. and\nFor the following exercises, find the slope of the lines that pass through each pair of points and determine whether the\nlines are parallel or perpendicular.\n45. 46.\n114 2  Equations and Inequalities"
	},
	{
		"chunk_id" : "00000340",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "45. 46.\n114 2  Equations and Inequalities\nTechnology\nFor the following exercises, express the equations in slope intercept form (rounding each number to the thousandths\nplace). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include\nwhere they-intercept occurs. State your ymin and ymax values.\n47. 48. 49.\nExtensions\n50. Starting with the point- 51. Starting with the standard 52. Use the above derived\nslope formula form of an equation formula to put the"
	},
	{
		"chunk_id" : "00000341",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solve solve this following standard\nthis expression for in expression for in terms of equation in slope intercept\nterms of and . and . Then put the form:\nexpression in slope-\nintercept form.\n53. Given that the following 54. Find the slopes of the\ncoordinates are the diagonals in the previous\nvertices of a rectangle, exercise. Are they\nprove that this truly is a perpendicular?\nrectangle by showing the\nslopes of the sides that\nmeet are perpendicular.\nand\nReal-World Applications"
	},
	{
		"chunk_id" : "00000342",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and\nReal-World Applications\n55. The slope for a wheelchair ramp for a home has to 56. If the profit equation for a small business selling\nbe If the vertical distance from the ground to number of item one and number of item two is\nthe door bottom is 2.5 ft, find the distance the find the value when\nramp has to extend from the home in order to\ncomply with the needed slope.\nFor the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week."
	},
	{
		"chunk_id" : "00000343",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "An equation to represent the cost would be where is the number of miles traveled.\n57. What is your cost if you 58. If your cost were 59. Suppose you have a\ntravel 50 mi? how many miles were you maximum of $100 to spend\ncharged for traveling? for the car rental. What\nwould be the maximum\nnumber of miles you could\ntravel?\nAccess for free at openstax.org\n2.3  Models and Applications 115\n2.3 Models and Applications\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00000344",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nSet up a linear equation to solve a real-world application.\nUse a formula to solve a real-world application.\nFigure1 Credit: Kevin Dooley\nNeka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only\nthe final exam remains, and the maximum of points that can be earned is 100. Is it possible for Neka to end the course\nwith an A? A simple linear equation will give Neka his answer."
	},
	{
		"chunk_id" : "00000345",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a\nmonthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to producex\nwidgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile\ndriven. These are examples of applications we come across every day that are modeled by linear equations. In this"
	},
	{
		"chunk_id" : "00000346",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "section, we will set up and use linear equations to solve such problems.\nSetting up a Linear Equation to Solve a Real-World Application\nTo set up or model a linear equation to fit a real-world application, we must first determine the known quantities and\ndefine the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using\nmathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is"
	},
	{
		"chunk_id" : "00000347",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplied by an unknown quantity, the number of miles driven. Therefore, we can write This expression\nrepresents a variable cost because it changes according to the number of miles driven.\nIf a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts\ndo not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We"
	},
	{
		"chunk_id" : "00000348",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can use these quantities to model an equation that can be used to find the daily car rental cost\nWhen dealing with real-world applications, there are certain expressions that we can translate directly into math.Table 1\nlists some common verbal expressions and their equivalent mathematical expressions.\nVerbal Translation to Math Operations\nOne number exceeds another bya\nTwice a number\nOne number isamore than another number\nOne number isaless than twice another number"
	},
	{
		"chunk_id" : "00000349",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "One number isaless than twice another number\nThe product of a number anda, decreased byb\nThe quotient of a number and the number plusais three times the number\nTable1\n116 2  Equations and Inequalities\nVerbal Translation to Math Operations\nThe product of three times a number and the number decreased bybisc\nTable1\n...\nHOW TO\nGiven a real-world problem, model a linear equation to fit it.\n1. Identify known quantities.\n2. Assign a variable to represent the unknown quantity."
	},
	{
		"chunk_id" : "00000350",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.\n4. Write an equation interpreting the words as mathematical operations.\n5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.\nEXAMPLE1\nModeling a Linear Equation to Solve an Unknown Number Problem\nFind a linear equation to solve for the following unknown quantities: One number exceeds another number by and\ntheir sum is Find the two numbers.\nSolution"
	},
	{
		"chunk_id" : "00000351",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "their sum is Find the two numbers.\nSolution\nLet equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as\nThe sum of the two numbers is 31. We usually interpret the wordisas an equal sign.\nThe two numbers are and\nTRY IT #1 Find a linear equation to solve for the following unknown quantities: One number is three more\nthan twice another number. If the sum of the two numbers is find the numbers.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00000352",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nSetting Up a Linear Equation to Solve a Real-World Application\nThere are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus\n$.05/min talk-time. Company B charges a monthly service fee of $40 plus $.04/min talk-time.\n Write a linear equation that models the packages offered by both companies.\n If the average number of minutes used each month is 1,160, which company offers the better plan?"
	},
	{
		"chunk_id" : "00000353",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If the average number of minutes used each month is 420, which company offers the better plan?\n How many minutes of talk-time would yield equal monthly statements from both companies?\nAccess for free at openstax.org\n2.3  Models and Applications 117\nSolution\n The model for CompanyAcan be written as This includes the variable cost of plus the\nmonthly service charge of $34. CompanyBs package charges a higher monthly fee of $40, but a lower variable cost\nof CompanyBs model can be written as\n"
	},
	{
		"chunk_id" : "00000354",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of CompanyBs model can be written as\n\nIf the average number of minutes used each month is 1,160, we have the following:\nSo, CompanyBoffers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by CompanyA\nwhen the average number of minutes used each month is 1,160.\n\nIf the average number of minutes used each month is 420, we have the following:\nIf the average number of minutes used each month is 420, then CompanyAoffers a lower monthly cost of $55"
	},
	{
		"chunk_id" : "00000355",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compared to CompanyBs monthly cost of $56.80.\n\nTo answer the question of how many talk-time minutes would yield the same bill from both companies, we should\nthink about the problem in terms of coordinates: At what point are both thex-value and they-value equal? We\ncan find this point by setting the equations equal to each other and solving forx.\nCheck thex-value in each equation.\nTherefore, a monthly average of 600 talk-time minutes renders the plans equal. SeeFigure 2\n118 2  Equations and Inequalities"
	},
	{
		"chunk_id" : "00000356",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "118 2  Equations and Inequalities\nFigure2\nTRY IT #2 Find a linear equation to model this real-world application: It costs ABC electronics company $2.50\nper unit to produce a part used in a popular brand of desktop computers. The company has\nmonthly operating expenses of $350 for utilities and $3,300 for salaries. What are the companys\nmonthly expenses?\nUsing a Formula to Solve a Real-World Application"
	},
	{
		"chunk_id" : "00000357",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using a Formula to Solve a Real-World Application\nMany applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities\nare substituted into the formula, the equation is solved for the unknown, and the problems question is answered.\nTypically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples\nof formulas include theareaof a rectangular region, theperimeterof a rectangle, and the"
	},
	{
		"chunk_id" : "00000358",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "volumeof a rectangular solid, When there are two unknowns, we find a way to write one in terms of the\nother because we can solve for only one variable at a time.\nEXAMPLE3\nSolving an Application Using a Formula\nIt takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer,\nand he averages 10 mi/h less than in the morning. How far does Andrew drive to work?\nSolution"
	},
	{
		"chunk_id" : "00000359",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThis is a distance problem, so we can use the formula where distance equals rate multiplied by time. Note that\nwhen rate is given in mi/h, time must be expressed in hours. Consistent units of measurement are key to obtaining a\ncorrect solution.\nFirst, we identify the known and unknown quantities. Andrews morning drive to work takes 30 min, or h at rate His\ndrive home takes 40 min, or h, and his speed averages 10 mi/h less than the morning drive. Both trips cover distance"
	},
	{
		"chunk_id" : "00000360",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A table, such asTable 2, is often helpful for keeping track of information in these types of problems.\nTo Work\nTo Home\nTable2\nWrite two equations, one for each trip.\nAccess for free at openstax.org\n2.3  Models and Applications 119\nAs both equations equal the same distance, we set them equal to each other and solve forr.\nWe have solved for the rate of speed to work, 40 mph. Substituting 40 into the rate on the return trip yields 30 mi/h. Now"
	},
	{
		"chunk_id" : "00000361",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we can answer the question. Substitute the rate back into either equation and solve ford.\nThe distance between home and work is 20 mi.\nAnalysis\nNote that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to\nsolve for\nTRY IT #3 On Saturday morning, it took Jennifer 3.6 h to drive to her mothers house for the weekend. On\nSunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi/h"
	},
	{
		"chunk_id" : "00000362",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "slower on Sunday than on Saturday. What was her speed on Sunday?\nEXAMPLE4\nSolving a Perimeter Problem\nThe perimeter of a rectangular outdoor patio is ft. The length is ft greater than the width. What are the dimensions\nof the patio?\nSolution\nThe perimeter formula is standard: We have two unknown quantities, length and width. However, we can\nwrite the length in terms of the width as Substitute the perimeter value and the expression for length into"
	},
	{
		"chunk_id" : "00000363",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the formula. It is often helpful to make a sketch and label the sides as inFigure 3.\nFigure3\nNow we can solve for the width and then calculate the length.\n120 2  Equations and Inequalities\nThe dimensions are ft and ft.\nTRY IT #4 Find the dimensions of a rectangle given that the perimeter is cm and the length is 1 cm more\nthan twice the width.\nEXAMPLE5\nSolving an Area Problem\nThe perimeter of a tablet of graph paper is 48 in. The length is in. more than the width. Find the area of the graph\npaper.\nSolution"
	},
	{
		"chunk_id" : "00000364",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "paper.\nSolution\nThe standard formula for area is however, we will solve the problem using the perimeter formula. The reason\nwe use the perimeter formula is because we know enough information about the perimeter that the formula will allow\nus to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often\nused together to solve a problem such as this one."
	},
	{
		"chunk_id" : "00000365",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We know that the length is 6 in. more than the width, so we can write length as Substitute the value of the\nperimeter and the expression for length into the perimeter formula and find the length.\nNow, we find the area given the dimensions of in. and in.\nThe area is in.2.\nTRY IT #5 A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft2\nof new carpeting should be ordered?\nEXAMPLE6\nSolving a Volume Problem"
	},
	{
		"chunk_id" : "00000366",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nSolving a Volume Problem\nFind the dimensions of a shipping box given that the length is twice the width, the height is inches, and the volume is\nAccess for free at openstax.org\n2.3  Models and Applications 121\n1,600 in.3.\nSolution\nThe formula for the volume of a box is given as the product of length, width, and height. We are given that\nand The volume is cubic inches.\nThe dimensions are in., in., and in.\nAnalysis"
	},
	{
		"chunk_id" : "00000367",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The dimensions are in., in., and in.\nAnalysis\nNote that the square root of would result in a positive and a negative value. However, because we are describing\nwidth, we can use only the positive result.\nMEDIA\nAccess these online resources for additional instruction and practice with models and applications of linear\nequations.\nProblem solving using linear equations(http://openstax.org/l/lineqprobsolve)\nProblem solving using equations(http://openstax.org/l/equationprsolve)"
	},
	{
		"chunk_id" : "00000368",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the dimensions of area given the perimeter(http://openstax.org/l/permareasolve)\nFind the distance between the cities using the distance = rate * time formula(http://openstax.org/l/ratetimesolve)\nLinear equation application (Write a cost equation)(http://openstax.org/l/lineqappl)\n2.3 SECTION EXERCISES\nVerbal\n1. To set up a model linear 2. Use your own words to 3. If the total amount of\nequation to fit real-world describe this equation money you had to invest"
	},
	{
		"chunk_id" : "00000369",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "applications, what should wherenis a number: was $2,000 and you deposit\nalways be the first step? amount in one\ninvestment, how can you\nrepresent the remaining\namount?\n4. If a carpenter sawed a 10-ft 5. If Bill was traveling mi/h,\nboard into two sections and how would you represent\none section was ft long, Daemons speed if he was\nhow long would the other traveling 10 mi/h faster?\nsection be in terms of ?\n122 2  Equations and Inequalities\nReal-World Applications"
	},
	{
		"chunk_id" : "00000370",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Real-World Applications\nFor the following exercises, use the information to find a linear algebraic equation model to use to answer the question\nbeing asked.\n6. Mark and Don are planning 7. Beth and Ann are joking that 8. Ruden originally filled out 8\nto sell each of their marble their combined ages equal more applications than\ncollections at a garage sale. Sams age. If Beth is twice Hanh. Then each boy filled\nIf Don has 1 more than 3 Anns age and Sam is 69 yr out 3 additional"
	},
	{
		"chunk_id" : "00000371",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "times the number of old, what are Beth and applications, bringing the\nmarbles Mark has, how Anns ages? total to 28. How many\nmany does each boy have to applications did each boy\nsell if the total number of originally fill out?\nmarbles is 113?\nFor the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is\nconsidering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5"
	},
	{
		"chunk_id" : "00000372",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and charges $.10/min for calls.\n9. Find the model of the total 10. Find the model of the total 11. Find out how many\ncost of Company As plan, cost of Company Bs plan, minutes of calling would\nusing for the minutes. using for the minutes. make the two plans equal.\n12. If the person makes a\nmonthly average of 200\nmin of calls, which plan\nshould for the person\nchoose?\nFor the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering."
	},
	{
		"chunk_id" : "00000373",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up\nto 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all\nthe lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use for the number of devices\nthat need data plans as part of their cost.\n13. Find the model of the total 14. Find the model of the total 15. Assuming they stay under"
	},
	{
		"chunk_id" : "00000374",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cost of the Family Plan. cost of the Mobile Share their data limit, find the\nPlan. number of devices that\nwould make the two plans\nequal in cost.\n16. If a family has 3 smart\nphones, which plan should\nthey choose?\nAccess for free at openstax.org\n2.3  Models and Applications 123\nFor exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from"
	},
	{
		"chunk_id" : "00000375",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to\nput in a CD that pays 7%.\n17. If we let be the amount 18. Set up and solve the 19. Two planes fly in opposite\nthe woman invests in the equation for how much the directions. One travels 450\n15% bond, how much will woman should invest in mi/h and the other 550 mi/\nshe be able to invest in the each option to sustain a h. How long will it take"
	},
	{
		"chunk_id" : "00000376",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "CD? $6,000 annual return. before they are 4,000 mi\napart?\n20. Ben starts walking along a 21. Fiora starts riding her bike 22. A chemistry teacher needs\npath at 4 mi/h. One and a at 20 mi/h. After a while, to mix a 30% salt solution\nhalf hours after Ben leaves, she slows down to 12 mi/h, with a 70% salt solution to\nhis sister Amanda begins and maintains that speed make 20 qt of a 40% salt\njogging along the same for the rest of the trip. The solution. How many quarts"
	},
	{
		"chunk_id" : "00000377",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "path at 6 mi/h. How long whole trip of 70 mi takes of each solution should the\nwill it be before Amanda her 4.5 h. For what distance teacher mix to get the\ncatches up to Ben? did she travel at 20 mi/h? desired result?\n23. Ral has $20,000 to invest.\nHis intent is to earn 11%\ninterest on his investment.\nHe can invest part of his\nmoney at 8% interest and\npart at 12% interest. How\nmuch does Ral need to\ninvest in each option to\nmake get a total 11%\nreturn on his $20,000?"
	},
	{
		"chunk_id" : "00000378",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "make get a total 11%\nreturn on his $20,000?\nFor the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus\n$.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.\n24. Write the model equation 25. Write the model equation 26. Find the number of miles\nfor the cost of renting a for the cost of renting a that would generate the\ntruck with plan A. truck with plan B. same cost for both plans.\n27. If Tim knows he has to\ntravel 300 mi, which plan"
	},
	{
		"chunk_id" : "00000379",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "travel 300 mi, which plan\nshould he choose?\nFor the following exercises, use the formula given to solve for the required value.\n28. is used to 29. The formula 30. indicates that\nfind the principal amountP relates force , velocity force (F) equals mass (m)\ndeposited, earningr% , mass , and resistance times acceleration (a). Find\ninterest, fortyears. Use . Find when the acceleration of a mass\nthis to find what principal and of 50 kg if a force of 12 N is\namountPDavid invested at exerted on it."
	},
	{
		"chunk_id" : "00000380",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "amountPDavid invested at exerted on it.\na 3% rate for 20 yr if\n124 2  Equations and Inequalities\n31. is the formula\nfor an infinite series sum. If\nthe sum is 5, find\nFor the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you\nwill use it to solve a question.\n32. Solve forW: 33. Use the formula from the 34. Solve for\nprevious question to find\nthe width, of a\nrectangle whose length is\n15 and whose perimeter is\n58."
	},
	{
		"chunk_id" : "00000381",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "15 and whose perimeter is\n58.\n35. Use the formula from the 36. Solve for in the slope- 37. Use the formula from the\nprevious question to find intercept formula: previous question to find\nwhen when the coordinates of\nthe point are and\n38. The area of a trapezoid is 39. Solve forh: 40. Use the formula from the\ngiven by previous question to find\nUse the formula to find the the height of a trapezoid\narea of a trapezoid with with ,\nand"
	},
	{
		"chunk_id" : "00000382",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "area of a trapezoid with with ,\nand\n41. Find the dimensions of an 42. Distance equals rate times 43. Using the formula in the\nAmerican football field. The time, Find the previous exercise, find the\nlength is 200 ft more than distance Tom travels if he is distance that Susan travels\nthe width, and the moving at a rate of 55 mi/h if she is moving at a rate of\nperimeter is 1,040 ft. Find for 3.5 h. 60 mi/h for 6.75 h.\nthe length and width. Use\nthe perimeter formula"
	},
	{
		"chunk_id" : "00000383",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the length and width. Use\nthe perimeter formula\n44. What is the total distance 45. If the area model for a 46. Solve forh:\nthat two people travel in 3 triangle is find\nh if one of them is riding a the area of a triangle with a\nbike at 15 mi/h and the height of 16 in. and a base\nother is walking at 3 mi/h? of 11 in.\n47. Use the formula from the 48. The volume formula for a 49. Solve forh:\nprevious question to find cylinder is Using\nthe height to the nearest the symbol in your"
	},
	{
		"chunk_id" : "00000384",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the height to the nearest the symbol in your\ntenth of a triangle with a answer, find the volume of\nbase of 15 and an area of a cylinder with a radius,\n215. of 4 cm and a height of 14\ncm.\nAccess for free at openstax.org\n2.4  Complex Numbers 125\n50. Use the formula from the 51. Solve forr: 52. Use the formula from the\nprevious question to find previous question to find\nthe height of a cylinder the radius of a cylinder\nwith a radius of 8 and a with a height of 36 and a\nvolume of volume of"
	},
	{
		"chunk_id" : "00000385",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "volume of volume of\n53. The formula for the 54. Solve the formula from the\ncircumference of a circle is previous question for\nFind the Notice why is sometimes\ncircumference of a circle defined as the ratio of the\nwith a diameter of 12 in. circumference to its\n(diameter = 2r). Use the diameter.\nsymbol in your final\nanswer.\n2.4 Complex Numbers\nLearning Objectives\nIn this section, you will:\nAdd and subtract complex numbers.\nMultiply and divide complex numbers.\nSimplify powers of\nFigure1"
	},
	{
		"chunk_id" : "00000386",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Simplify powers of\nFigure1\nDiscovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The\nimage is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many\nsurprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that\ngenerates this image turns out to be rather simple."
	},
	{
		"chunk_id" : "00000387",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of\nmathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers.\nThe set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the\nset of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set"
	},
	{
		"chunk_id" : "00000388",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of numbers that fills voids in the set of real numbers and find out how to work within it.\nExpressing Square Roots of Negative Numbers as Multiples of\nWe know how to find the square root of any positive real number. In a similar way, we can find the square root of any\nnegative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be\nan imaginary number.The imaginary number is defined as the square root of\nSo, using properties of radicals,"
	},
	{
		"chunk_id" : "00000389",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So, using properties of radicals,\nWe can write the square root of any negative number as a multiple of Consider the square root of\n126 2  Equations and Inequalities\nWe use and not because the principal root of is the positive root.\nA complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard\nform when written where is the real part and is the imaginary part. For example, is a complex number.\nSo, too, is"
	},
	{
		"chunk_id" : "00000390",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So, too, is\nImaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number.\nRecall that when a positive real number is squared, the result is a positive real number and when a negative real number\nis squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.\nImaginary and Complex Numbers\nAcomplex numberis a number of the form where\n is the real part of the complex number."
	},
	{
		"chunk_id" : "00000391",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " is the real part of the complex number.\n is the imaginary part of the complex number.\nIf then is a real number. If and is not equal to 0, the complex number is called a pure imaginary\nnumber. Animaginary numberis an even root of a negative number.\n...\nHOW TO\nGiven an imaginary number, express it in the standard form of a complex number.\n1. Write as\n2. Express as\n3. Write in simplest form.\nEXAMPLE1\nExpressing an Imaginary Number in Standard Form\nExpress in standard form.\nSolution"
	},
	{
		"chunk_id" : "00000392",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Express in standard form.\nSolution\nIn standard form, this is\nTRY IT #1 Express in standard form.\nPlotting a Complex Number on the Complex Plane\nWe cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them\ngraphically. To represent a complex number, we need to address the two components of the number. We use the\ncomplex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical"
	},
	{
		"chunk_id" : "00000393",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs\nAccess for free at openstax.org\n2.4  Complex Numbers 127\nwhere represents the coordinate for the horizontal axis and represents the coordinate for the vertical axis.\nLets consider the number The real part of the complex number is and the imaginary part is 3. We plot the\nordered pair to represent the complex number as shown inFigure 2.\nFigure2\nComplex Plane"
	},
	{
		"chunk_id" : "00000394",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nComplex Plane\nIn the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown inFigure\n3.\nFigure3\n...\nHOW TO\nGiven a complex number, represent its components on the complex plane.\n1. Determine the real part and the imaginary part of the complex number.\n2. Move along the horizontal axis to show the real part of the number.\n3. Move parallel to the vertical axis to show the imaginary part of the number.\n4. Plot the point.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00000395",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Plot the point.\nEXAMPLE2\nPlotting a Complex Number on the Complex Plane\nPlot the complex number on the complex plane.\nSolution\nThe real part of the complex number is and the imaginary part is 4. We plot the ordered pair as shown in\nFigure 4.\n128 2  Equations and Inequalities\nFigure4\nTRY IT #2 Plot the complex number on the complex plane.\nAdding and Subtracting Complex Numbers\nJust as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex"
	},
	{
		"chunk_id" : "00000396",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers, we combine the real parts and then combine the imaginary parts.\nComplex Numbers: Addition and Subtraction\nAdding complex numbers:\nSubtracting complex numbers:\n...\nHOW TO\nGiven two complex numbers, find the sum or difference.\n1. Identify the real and imaginary parts of each number.\n2. Add or subtract the real parts.\n3. Add or subtract the imaginary parts.\nEXAMPLE3\nAdding and Subtracting Complex Numbers\nAdd or subtract as indicated.\n \nSolution\nWe add the real parts and add the imaginary parts.\n "
	},
	{
		"chunk_id" : "00000397",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nAccess for free at openstax.org\n2.4  Complex Numbers 129\nTRY IT #3 Subtract from\nMultiplying Complex Numbers\nMultiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and\nimaginary parts separately.\nMultiplying a Complex Number by a Real Number\nLets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a\nbinomial. Consider, for example, :\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000398",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "binomial. Consider, for example, :\n...\nHOW TO\nGiven a complex number and a real number, multiply to find the product.\n1. Use the distributive property.\n2. Simplify.\nEXAMPLE4\nMultiplying a Complex Number by a Real Number\nFind the product\nSolution\nDistribute the 4.\nTRY IT #4 Find the product:\nMultiplying Complex Numbers Together\nNow, lets multiply two complex numbers. We can use either the distributive property or more specifically the FOIL"
	},
	{
		"chunk_id" : "00000399",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and\nLast terms together. The difference with complex numbers is that when we get a squared term, it equals\n...\nHOW TO\nGiven two complex numbers, multiply to find the product.\n1. Use the distributive property or the FOIL method.\n2. Remember that\n3. Group together the real terms and the imaginary terms\n130 2  Equations and Inequalities\nEXAMPLE5\nMultiplying a Complex Number by a Complex Number"
	},
	{
		"chunk_id" : "00000400",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Multiplying a Complex Number by a Complex Number\nMultiply:\nSolution\nTRY IT #5 Multiply:\nDividing Complex Numbers\nDividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide\nby an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in\nstandard form We need to find a term by which we can multiply the numerator and the denominator that will"
	},
	{
		"chunk_id" : "00000401",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term\nis called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the\ncomplex number. In other words, the complex conjugate of is For example, the product of and\nis\nThe result is a real number.\nNote that complex conjugates have an opposite relationship: The complex conjugate of is and the complex"
	},
	{
		"chunk_id" : "00000402",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "conjugate of is Further, when a quadratic equation with real coefficients has complex solutions, the\nsolutions are always complex conjugates of one another.\nSuppose we want to divide by where neither nor equals zero. We first write the division as a fraction,\nthen find the complex conjugate of the denominator, and multiply.\nMultiply the numerator and denominator by the complex conjugate of the denominator.\nApply the distributive property.\nSimplify, remembering that\nThe Complex Conjugate"
	},
	{
		"chunk_id" : "00000403",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Simplify, remembering that\nThe Complex Conjugate\nThecomplex conjugateof a complex number is It is found by changing the sign of the imaginary part\nof the complex number. The real part of the number is left unchanged.\n When a complex number is multiplied by its complex conjugate, the result is a real number.\nAccess for free at openstax.org\n2.4  Complex Numbers 131\n When a complex number is added to its complex conjugate, the result is a real number.\nEXAMPLE6\nFinding Complex Conjugates"
	},
	{
		"chunk_id" : "00000404",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nFinding Complex Conjugates\nFind the complex conjugate of each number.\n \nSolution\n The number is already in the form The complex conjugate is or\n We can rewrite this number in the form as The complex conjugate is or This can be\nwritten simply as\nAnalysis\nAlthough we have seen that we can find the complex conjugate of an imaginary number, in practice we generally find\nthe complex conjugates of only complex numbers with both a real and an imaginary component. To obtain a real"
	},
	{
		"chunk_id" : "00000405",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number from an imaginary number, we can simply multiply by\nTRY IT #6 Find the complex conjugate of\n...\nHOW TO\nGiven two complex numbers, divide one by the other.\n1. Write the division problem as a fraction.\n2. Determine the complex conjugate of the denominator.\n3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.\n4. Simplify.\nEXAMPLE7\nDividing Complex Numbers\nDivide: by\nSolution\nWe begin by writing the problem as a fraction."
	},
	{
		"chunk_id" : "00000406",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We begin by writing the problem as a fraction.\nThen we multiply the numerator and denominator by the complex conjugate of the denominator.\nTo multiply two complex numbers, we expand the product as we would with polynomials (using FOIL).\n132 2  Equations and Inequalities\nNote that this expresses the quotient in standard form.\nSimplifying Powers ofi\nThe powers of are cyclic. Lets look at what happens when we raise to increasing powers."
	},
	{
		"chunk_id" : "00000407",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can see that when we get to the fifth power of it is equal to the first power. As we continue to multiply by\nincreasing powers, we will see a cycle of four. Lets examine the next four powers of\nThe cycle is repeated continuously: every four powers.\nEXAMPLE8\nSimplifying Powers of\nEvaluate:\nSolution\nSince we can simplify the problem by factoring out as many factors of as possible. To do so, first determine\nhow many times 4 goes into 35:\nTRY IT #7 Evaluate:\nQ&A Can we write in other helpful ways?"
	},
	{
		"chunk_id" : "00000408",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Can we write in other helpful ways?\nAs we saw inExample 8, we reduced to by dividing the exponent by 4 and using the remainder to\nfind the simplified form. But perhaps another factorization of may be more useful.Table 1shows\nsome other possible factorizations.\nFactorization of\nReduced form\nSimplified form\nTable1\nEach of these will eventually result in the answer we obtained above but may require several more steps\nthan our earlier method.\nMEDIA"
	},
	{
		"chunk_id" : "00000409",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "than our earlier method.\nMEDIA\nAccess these online resources for additional instruction and practice with complex numbers.\nAccess for free at openstax.org\n2.4  Complex Numbers 133\nAdding and Subtracting Complex Numbers(http://openstax.org/l/addsubcomplex)\nMultiply Complex Numbers(http://openstax.org/l/multiplycomplex)\nMultiplying Complex Conjugates(http://openstax.org/l/multcompconj)\nRaisingito Powers(http://openstax.org/l/raisingi)\n2.4 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000410",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.4 SECTION EXERCISES\nVerbal\n1. Explain how to add complex 2. What is the basic principle in 3. Give an example to show\nnumbers. multiplication of complex that the product of two\nnumbers? imaginary numbers is not\nalways imaginary.\n4. What is a characteristic of\nthe plot of a real number in\nthe complex plane?\nAlgebraic\nFor the following exercises, evaluate the algebraic expressions.\n5. If evaluate 6. If evaluate 7. If evaluate\ngiven given given\n8. If evaluate 9. If evaluate given 10. If evaluate\ngiven given"
	},
	{
		"chunk_id" : "00000411",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given given\nGraphical\nFor the following exercises, plot the complex numbers on the complex plane.\n11. 12. 13.\n14.\nNumeric\nFor the following exercises, perform the indicated operation and express the result as a simplified complex number.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23.\n24. 25. 26.\n134 2  Equations and Inequalities\n27. 28. 29.\n30. 31. 32.\n33. 34. 35.\n36. 37. 38.\n39. 40. 41.\nTechnology\nFor the following exercises, use a calculator to help answer the questions."
	},
	{
		"chunk_id" : "00000412",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "42. Evaluate for 43. Evaluate for 44. Evaluate\nand . Predict and . Predict for and . Predict\nthe value if the value if the value for\n45. Show that a solution of 46. Show that a solution of\nis is\nExtensions\nFor the following exercises, evaluate the expressions, writing the result as a simplified complex number.\n47. 48. 49.\n50. 51. 52.\n53. 54. 55.\n56.\n2.5 Quadratic Equations\nLearning Objectives\nIn this section, you will:\nSolve quadratic equations by factoring."
	},
	{
		"chunk_id" : "00000413",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve quadratic equations by factoring.\nSolve quadratic equations by the square root property.\nSolve quadratic equations by completing the square.\nSolve quadratic equations by using the quadratic formula.\nAccess for free at openstax.org\n2.5  Quadratic Equations 135\nFigure1\nThe computer monitor on the left inFigure 1is a 23.6-inch model and the one on the right is a 27-inch model.\nProportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor"
	},
	{
		"chunk_id" : "00000414",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using\nfour different methods.\nSolving Quadratic Equations by Factoring\nAn equation containing a second-degree polynomial is called aquadratic equation. For example, equations such as\nand are quadratic equations. They are used in countless ways in the fields of engineering,\narchitecture, finance, biological science, and, of course, mathematics."
	},
	{
		"chunk_id" : "00000415",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Often the easiest method of solving a quadratic equation isfactoring. Factoring means finding expressions that can be\nmultiplied together to give the expression on one side of the equation.\nIf a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the\nzero-product property, which states that if then or whereaandbare real numbers or algebraic"
	},
	{
		"chunk_id" : "00000416",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or\none of the expressions must equal zero because zero multiplied by anything equals zero.\nMultiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the\noperation of multiplication undoes the operation of factoring. For example, expand the factored expression\nby multiplying the two factors together."
	},
	{
		"chunk_id" : "00000417",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by multiplying the two factors together.\nThe product is a quadratic expression. Set equal to zero, is a quadratic equation. If we were to factor the\nequation, we would get back the factors we multiplied.\nThe process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We\nwill look at both situations; but first, we want to confirm that the equation is written in standard form,\nwherea,b, andcare real numbers, and The equation is in standard form."
	},
	{
		"chunk_id" : "00000418",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can use the zero-product property to solve quadratic equations in which we first have to factor out thegreatest\ncommon factor(GCF), and for equations that have special factoring formulas as well, such as the difference of squares,\nboth of which we will see later in this section.\nThe Zero-Product Property and Quadratic Equations\nThezero-product propertystates\nwhereaandbare real numbers or algebraic expressions.\nAquadratic equationis an equation containing a second-degree polynomial; for example"
	},
	{
		"chunk_id" : "00000419",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "wherea,b, andcare real numbers, and if it is in standard form.\nSolving Quadratics with a Leading Coefficient of 1\nIn the quadratic equation the leading coefficient, or the coefficient of is 1. We have one method of\n136 2  Equations and Inequalities\nfactoring quadratic equations in this form.\n...\nHOW TO\nGiven a quadratic equation with the leading coefficient of 1, factor it.\n1. Find two numbers whose product equalscand whose sum equalsb."
	},
	{
		"chunk_id" : "00000420",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Use those numbers to write two factors of the form wherekis one of the numbers found in\nstep 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and the factors are\n3. Solve using the zero-product property by setting each factor equal to zero and solving for the variable.\nEXAMPLE1\nFactoring and Solving a Quadratic with Leading Coefficient of 1\nFactor and solve the equation:\nSolution"
	},
	{
		"chunk_id" : "00000421",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Factor and solve the equation:\nSolution\nTo factor we look for two numbers whose product equals and whose sum equals 1. Begin by looking\nat the possible factors of\nThe last pair, sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the\nfactors.\nTo solve this equation, we use the zero-product property. Set each factor equal to zero and solve.\nThe two solutions are and We can see how the solutions relate to the graph inFigure 2. The solutions are the\nx-intercepts of"
	},
	{
		"chunk_id" : "00000422",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-intercepts of\nAccess for free at openstax.org\n2.5  Quadratic Equations 137\nFigure2\nTRY IT #1 Factor and solve the quadratic equation:\nEXAMPLE2\nSolve the Quadratic Equation by Factoring\nSolve the quadratic equation by factoring:\nSolution\nFind two numbers whose product equals and whose sum equals List the factors of\nThe numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.\nThe solutions are and\nTRY IT #2 Solve the quadratic equation by factoring:"
	},
	{
		"chunk_id" : "00000423",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "138 2  Equations and Inequalities\nEXAMPLE3\nUsing the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares\nSolve the difference of squares equation using the zero-product property:\nSolution\nRecognizing that the equation represents the difference of squares, we can write the two factors by taking the square\nroot of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve\nusing the zero-factor property."
	},
	{
		"chunk_id" : "00000424",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "using the zero-factor property.\nThe solutions are and\nTRY IT #3 Solve by factoring:\nSolving a Quadratic Equation by Factoring when the Leading Coefficient is not 1\nWhen the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires\nfour terms. With the equation in standard form, lets review the grouping procedures:\n1. With the quadratic in standard form, multiply\n2. Find two numbers whose product equals and whose sum equals"
	},
	{
		"chunk_id" : "00000425",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Rewrite the equation replacing the term with two terms using the numbers found in step 1 as coefficients ofx.\n4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the\nsame to use grouping.\n5. Factor out the expression in parentheses.\n6. Set the expressions equal to zero and solve for the variable.\nEXAMPLE4\nSolving a Quadratic Equation Using Grouping\nUse grouping to factor and solve the quadratic equation:\nSolution"
	},
	{
		"chunk_id" : "00000426",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nFirst, multiply Then list the factors of\nThe only pair of factors that sums to is Rewrite the equation replacing thebterm, with two terms using 3\nand 12 as coefficients ofx. Factor the first two terms, and then factor the last two terms.\nSolve using the zero-product property.\nAccess for free at openstax.org\n2.5  Quadratic Equations 139\nThe solutions are and SeeFigure 3.\nFigure3\nTRY IT #4 Solve using factoring by grouping:\nEXAMPLE5\nSolving a Polynomial of Higher Degree by Factoring"
	},
	{
		"chunk_id" : "00000427",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the equation by factoring:\nSolution\nThis equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do\nwhen solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out from all of the\nterms and then proceed with grouping.\nUse grouping on the expression in parentheses.\nNow, we use the zero-product property. Notice that we have three factors.\n140 2  Equations and Inequalities\nThe solutions are and"
	},
	{
		"chunk_id" : "00000428",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solutions are and\nTRY IT #5 Solve by factoring:\nUsing the Square Root Property\nWhen there is no linear term in the equation, another method of solving a quadratic equation is by using thesquare\nroot property, in which we isolate the term and take the square root of the number on the other side of the equals\nsign. Keep in mind that sometimes we may have to manipulate the equation to isolate the term so that the square\nroot property can be used.\nThe Square Root Property"
	},
	{
		"chunk_id" : "00000429",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Square Root Property\nWith the term isolated, the square root property states that:\nwherekis a nonzero real number.\n...\nHOW TO\nGiven a quadratic equation with an term but no term, use the square root property to solve it.\n1. Isolate the term on one side of the equal sign.\n2. Take the square root of both sides of the equation, putting a sign before the expression on the side opposite\nthe squared term.\n3. Simplify the numbers on the side with the sign.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00000430",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nSolving a Simple Quadratic Equation Using the Square Root Property\nSolve the quadratic using the square root property:\nSolution\nTake the square root of both sides, and then simplify the radical. Remember to use a sign before the radical symbol.\nThe solutions are\nAccess for free at openstax.org\n2.5  Quadratic Equations 141\nEXAMPLE7\nSolving a Quadratic Equation Using the Square Root Property\nSolve the quadratic equation:\nSolution\nFirst, isolate the term. Then take the square root of both sides."
	},
	{
		"chunk_id" : "00000431",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solutions are and\nTRY IT #6 Solve the quadratic equation using the square root property:\nCompleting the Square\nNot all quadratic equations can be factored or can be solved in their original form using the square root property. In\nthese cases, we may use a method for solving aquadratic equationknown ascompleting the square. Using this\nmethod, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of"
	},
	{
		"chunk_id" : "00000432",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equal sign. We then apply the square root property. To complete the square, the leading coefficient,a, must equal 1.\nIf it does not, then divide the entire equation bya. Then, we can use the following procedures to solve a quadratic\nequation by completing the square.\nWe will use the example to illustrate each step.\n1. Given a quadratic equation that cannot be factored, and with first add or subtract the constant term to the\nright side of the equal sign.\n2. Multiply thebterm by and square it."
	},
	{
		"chunk_id" : "00000433",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Multiply thebterm by and square it.\n3. Add to both sides of the equal sign and simplify the right side. We have\n4. The left side of the equation can now be factored as a perfect square.\n5. Use the square root property and solve.\n6. The solutions are and\n142 2  Equations and Inequalities\nEXAMPLE8\nSolving a Quadratic by Completing the Square\nSolve the quadratic equation by completing the square:\nSolution\nFirst, move the constant term to the right side of the equal sign."
	},
	{
		"chunk_id" : "00000434",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then, take of thebterm and square it.\nAdd the result to both sides of the equal sign.\nFactor the left side as a perfect square and simplify the right side.\nUse the square root property and solve.\nThe solutions are and .\nTRY IT #7 Solve by completing the square:\nUsing the Quadratic Formula\nThe fourth method of solving aquadratic equationis by using thequadratic formula, a formula that will solve all"
	},
	{
		"chunk_id" : "00000435",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to\nmake errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses\nwhen inserting a negative number.\nWe can derive the quadratic formula bycompleting the square. We will assume that the leading coefficient is positive; if\nit is negative, we can multiply the equation by and obtain a positivea. Given we will\ncomplete the square as follows:"
	},
	{
		"chunk_id" : "00000436",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "complete the square as follows:\n1. First, move the constant term to the right side of the equal sign:\n2. As we want the leading coefficient to equal 1, divide through bya:\n3. Then, find of the middle term, and add to both sides of the equal sign:\nAccess for free at openstax.org\n2.5  Quadratic Equations 143\n4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single\nfraction:\n5. Now, use the square root property, which gives"
	},
	{
		"chunk_id" : "00000437",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. Now, use the square root property, which gives\n6. Finally, add to both sides of the equation and combine the terms on the right side. Thus,\nThe Quadratic Formula\nWritten in standard form, any quadratic equation can be solved using thequadratic formula:\nwherea,b, andcare real numbers and\n...\nHOW TO\nGiven a quadratic equation, solve it using the quadratic formula\n1. Make sure the equation is in standard form:\n2. Make note of the values of the coefficients and constant term, and"
	},
	{
		"chunk_id" : "00000438",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses\naround each number input into the formula.\n4. Calculate and solve.\nEXAMPLE9\nSolve the Quadratic Equation Using the Quadratic Formula\nSolve the quadratic equation:\nSolution\nIdentify the coefficients: Then use the quadratic formula.\nEXAMPLE10\nSolving a Quadratic Equation with the Quadratic Formula\nUse the quadratic formula to solve\n144 2  Equations and Inequalities\nSolution"
	},
	{
		"chunk_id" : "00000439",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "144 2  Equations and Inequalities\nSolution\nFirst, we identify the coefficients: and\nSubstitute these values into the quadratic formula.\nThe solutions to the equation are and\nTRY IT #8 Solve the quadratic equation using the quadratic formula:\nThe Discriminant\nThequadratic formulanot only generates the solutions to a quadratic equation, it tells us about the nature of the\nsolutions when we consider thediscriminant, or the expression under the radical, The discriminant tells us"
	},
	{
		"chunk_id" : "00000440",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect.Table 1\nrelates the value of the discriminant to the solutions of a quadratic equation.\nValue of Discriminant Results\nOne rational solution (double solution)\nperfect square Two rational solutions\nnot a perfect square Two irrational solutions\nTwo complex solutions\nTable1\nThe Discriminant\nFor , where , , and are real numbers, thediscriminantis the expression under the radical in"
	},
	{
		"chunk_id" : "00000441",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the quadratic formula: It tells us whether the solutions are real numbers or complex numbers and how\nmany solutions of each type to expect.\nEXAMPLE11\nUsing the Discriminant to Find the Nature of the Solutions to a Quadratic Equation\nUse the discriminant to find the nature of the solutions to the following quadratic equations:\n   \nSolution\nCalculate the discriminant for each equation and state the expected type of solutions.\nAccess for free at openstax.org\n2.5  Quadratic Equations 145\n"
	},
	{
		"chunk_id" : "00000442",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.5  Quadratic Equations 145\n\nThere will be one rational double solution.\n\nAs is a perfect square, there will be two rational solutions.\n\nAs is a perfect square, there will be two rational solutions.\n\nThere will be two complex solutions.\nUsing the Pythagorean Theorem\nOne of the most famous formulas in mathematics is thePythagorean Theorem. It is based on a right triangle, and\nstates the relationship among the lengths of the sides as where and refer to the legs of a right triangle"
	},
	{
		"chunk_id" : "00000443",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "adjacent to the angle, and refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the\nsciences, geometry, trigonometry, and algebra, and in everyday applications.\nWe use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other\ntwo. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a"
	},
	{
		"chunk_id" : "00000444",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for\nthe missing side.\nThe Pythagorean Theorem is given as\nwhere and refer to the legs of a right triangle adjacent to the angle, and refers to the hypotenuse, as shown in\nFigure 4.\nFigure4\nEXAMPLE12\nFinding the Length of the Missing Side of a Right Triangle\nFind the length of the missing side of the right triangle inFigure 5.\nFigure5\n146 2  Equations and Inequalities\nSolution"
	},
	{
		"chunk_id" : "00000445",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "146 2  Equations and Inequalities\nSolution\nAs we have measurements for sideband the hypotenuse, the missing side isa.\nTRY IT #9 Use the Pythagorean Theorem to solve the right triangle problem: Legameasures 4 units, legb\nmeasures 3 units. Find the length of the hypotenuse.\nMEDIA\nAccess these online resources for additional instruction and practice with quadratic equations.\nSolving Quadratic Equations by Factoring(http://openstax.org/l/quadreqfactor)"
	},
	{
		"chunk_id" : "00000446",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Zero-Product Property(http://openstax.org/l/zeroprodprop)\nCompleting the Square(http://openstax.org/l/complthesqr)\nQuadratic Formula with Two Rational Solutions(http://openstax.org/l/quadrformrat)\nLength of a leg of a right triangle(http://openstax.org/l/leglengthtri)\n2.5 SECTION EXERCISES\nVerbal\n1. How do we recognize when 2. When we solve a quadratic 3. When we solve a quadratic\nan equation is quadratic? equation, how many equation by factoring, why"
	},
	{
		"chunk_id" : "00000447",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions should we always do we move all terms to one\nstart out seeking? Explain side, having zero on the\nwhy when solving a other side?\nquadratic equation in the\nform we\nmay graph the equation\nand have\nno zeroes (x-intercepts).\n4. In the quadratic formula, 5. Describe two scenarios\nwhat is the name of the where using the square root\nexpression under the radical property to solve a\nsign and how does quadratic equation would\nit determine the number of be the most efficient"
	},
	{
		"chunk_id" : "00000448",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "it determine the number of be the most efficient\nand nature of our solutions? method.\nAlgebraic\nFor the following exercises, solve the quadratic equation by factoring.\n6. 7. 8.\nAccess for free at openstax.org\n2.5  Quadratic Equations 147\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n18.\nFor the following exercises, solve the quadratic equation by using the square root property.\n19. 20. 21.\n22. 23. 24.\nFor the following exercises, solve the quadratic equation by completing the square. Show each step.\n25. 26. 27."
	},
	{
		"chunk_id" : "00000449",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "25. 26. 27.\n28. 29. 30.\n31.\nFor the following exercises, determine the discriminant, and then state how many solutions there are and the nature of\nthe solutions. Do not solve.\n32. 33. 34.\n35. 36. 37.\nFor the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real,\nstateNo Real Solution.\n38. 39. 40.\n41. 42. 43.\nTechnology\nFor the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the"
	},
	{
		"chunk_id" : "00000450",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-intercepts) by using2ndCALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the\nzero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between\nthe bounds near the zero, enter). Round your answers to the nearest thousandth.\n44. 45. 46.\n148 2  Equations and Inequalities\n47. To solve the quadratic 48. To solve the quadratic\nequation equation\nwe can graph these two we can\nequations graph these two equations"
	},
	{
		"chunk_id" : "00000451",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations graph these two equations\nand find the points of and find the points of\nintersection. Recall 2nd intersection. Recall 2nd\nCALC 5:intersection. Do this CALC 5:intersection. Do this\nand find the solutions to and find the solutions to\nthe nearest tenth. the nearest tenth.\nExtensions\n49. Beginning with the general 50. Show that the sum of the 51. A person has a garden that\nform of a quadratic two solutions to the has a length 10 feet longer\nequation, quadratic equation is . than the width. Set up a"
	},
	{
		"chunk_id" : "00000452",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solve for quadratic equation to find\nxby using the completing the dimensions of the\nthe square method, thus garden if its area is 119 ft.2.\nderiving the quadratic Solve the quadratic\nformula. equation to find the length\nand width.\n52. Abercrombie and Fitch 53. Suppose that an equation\nstock had a price given as is given\nwhere is the time in where represents the\nmonths from 1999 to 2001. number of items sold at an\n( is January 1999). auction and is the profit\nFind the two months in made by the business that"
	},
	{
		"chunk_id" : "00000453",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the two months in made by the business that\nwhich the price of the stock ran the auction. How many\nwas $30. items sold would make this\nprofit a maximum? Solve\nthis by graphing the\nexpression in your\ngraphing utility and finding\nthe maximum using 2nd\nCALC maximum. To obtain\na good window for the\ncurve, set [0,200] and\n[0,10000].\nAccess for free at openstax.org\n2.6  Other Types of Equations 149\nReal-World Applications"
	},
	{
		"chunk_id" : "00000454",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Real-World Applications\n54. A formula for the normal 55. The cost function for a 56. A falling object travels a\nsystolic blood pressure for a certain company is distance given by the\nman age measured in and the formula ft,\nmmHg, is given as revenue is given by where is measured in\nRecall seconds. How long will it\nFind the age to the nearest that profit is revenue take for the object to travel\nyear of a man whose normal minus cost. Set up a 74 ft?\nblood pressure measures quadratic equation and"
	},
	{
		"chunk_id" : "00000455",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "blood pressure measures quadratic equation and\n125 mmHg. find two values ofx\n(production level) that will\ncreate a profit of $300.\n57. A vacant lot is being converted 58. An epidemiological study\ninto a community garden. The of the spread of a certain\ngarden and the walkway around influenza strain that hit a\nits perimeter have an area of 378 small school population\nft2. Find the width of the walkway found that the total\nif the garden is 12 ft. wide by 15 number of students, ,"
	},
	{
		"chunk_id" : "00000456",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ft. long. who contracted the flu\ndays after it broke out is\ngiven by the model\nwhere Find the\nday that 160 students had\nthe flu. Recall that the\nrestriction on is at most 6.\n2.6 Other Types of Equations\nLearning Objectives\nIn this section, you will:\nSolve equations involving rational exponents.\nSolve equations using factoring.\nSolve radical equations.\nSolve absolute value equations.\nSolve other types of equations."
	},
	{
		"chunk_id" : "00000457",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve other types of equations.\nWe have solved linear equations, rational equations, and quadratic equations using several methods. However, there are\nmany other types of equations, and we will investigate a few more types in this section. We will look at equations\ninvolving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic\nform, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the"
	},
	{
		"chunk_id" : "00000458",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never\nchanges.\nSolving Equations Involving Rational Exponents\nRational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For\nexample, is another way of writing is another way of writing The ability to work with rational\nexponents is a useful skill, as it is highly applicable in calculus."
	},
	{
		"chunk_id" : "00000459",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the\nreciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to\neliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,\n150 2  Equations and Inequalities\nand so on.\nRational Exponents"
	},
	{
		"chunk_id" : "00000460",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and so on.\nRational Exponents\nA rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of\nwriting an expression, a variable, or a number with a rational exponent:\nEXAMPLE1\nEvaluating a Number Raised to a Rational Exponent\nEvaluate\nSolution\nWhether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite\nas\nTRY IT #1 Evaluate\nEXAMPLE2"
	},
	{
		"chunk_id" : "00000461",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as\nTRY IT #1 Evaluate\nEXAMPLE2\nSolve the Equation Including a Variable Raised to a Rational Exponent\nSolve the equation in which a variable is raised to a rational exponent:\nSolution\nThe way to remove the exponent onxis by raising both sides of the equation to a power that is the reciprocal of\nwhich is\nTRY IT #2 Solve the equation\nEXAMPLE3\nSolving an Equation Involving Rational Exponents and Factoring\nSolve\nAccess for free at openstax.org\n2.6  Other Types of Equations 151\nSolution"
	},
	{
		"chunk_id" : "00000462",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2.6  Other Types of Equations 151\nSolution\nThis equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time.\nFirst, put the variable terms on one side of the equal sign and set the equation equal to zero.\nNow, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest\nexponent. Rewrite as Then, factor out from both terms on the left."
	},
	{
		"chunk_id" : "00000463",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Where did come from? Remember, when we multiply two numbers with the same base, we add the exponents.\nTherefore, if we multiply back in using the distributive property, we get the expression we had before the factoring,\nwhich is what should happen. We need an exponent such that when added to equals Thus, the exponent onxin\nthe parentheses is\nLet us continue. Now we have two factors and can use the zero factor theorem.\nThe two solutions are and\nTRY IT #3 Solve:\nSolving Equations Using Factoring"
	},
	{
		"chunk_id" : "00000464",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Equations Using Factoring\nWe have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial\nequations, which are equations that contain a string of terms including numerical coefficients and variables. When we\nare faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.\nPolynomial Equations\nA polynomial of degreenis an expression of the type\nwherenis a positive integer and are real numbers and"
	},
	{
		"chunk_id" : "00000465",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "152 2  Equations and Inequalities\nSetting the polynomial equal to zero gives apolynomial equation. The total number of solutions (real and complex)\nto a polynomial equation is equal to the highest exponentn.\nEXAMPLE4\nSolving a Polynomial by Factoring\nSolve the polynomial by factoring:\nSolution\nFirst, set the equation equal to zero. Then factor out what is common to both terms, the GCF.\nNotice that we have the difference of squares in the factor which we will continue to factor and obtain two"
	},
	{
		"chunk_id" : "00000466",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions. The first term, generates, technically, two solutions as the exponent is 2, but they are the same solution.\nThe solutions are and\nAnalysis\nWe can see the solutions on the graph inFigure 1. Thex-coordinates of the points where the graph crosses thex-axis are\nthe solutionsthex-intercepts. Notice on the graph that at the solution the graph touches thex-axis and bounces\nback. It does not cross thex-axis. This is typical of double solutions.\nFigure1\nTRY IT #4 Solve by factoring:"
	},
	{
		"chunk_id" : "00000467",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nTRY IT #4 Solve by factoring:\nAccess for free at openstax.org\n2.6  Other Types of Equations 153\nEXAMPLE5\nSolve a Polynomial by Grouping\nSolve a polynomial by grouping:\nSolution\nThis polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two\nterms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process\nand solve, unless more factoring is suggested."
	},
	{
		"chunk_id" : "00000468",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and solve, unless more factoring is suggested.\nThe grouping process ends here, as we can factor using the difference of squares formula.\nThe solutions are and Note that the highest exponent is 3 and we obtained 3 solutions. We can see the\nsolutions, thex-intercepts, on the graph inFigure 2.\nFigure2\nAnalysis\nWe looked at solving quadratic equations by factoring when the leading coefficient is 1. When the leading coefficient is"
	},
	{
		"chunk_id" : "00000469",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "not 1, we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term of quadratic\nequations. We can also use grouping for some polynomials of degree higher than 2, as we saw here, since there were\nalready four terms.\nSolving Radical Equations\nRadical equationsare equations that contain variables in theradicand(the expression under a radical symbol), such as"
	},
	{
		"chunk_id" : "00000470",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We\nhave to be careful when solving radical equations, as it is not unusual to findextraneous solutions, roots that are not, in\nfact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the\nprocess of raising both sides of an equation to a power. However, checking each answer in the original equation will\nconfirm the true solutions."
	},
	{
		"chunk_id" : "00000471",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "confirm the true solutions.\n154 2  Equations and Inequalities\nRadical Equations\nAn equation containing terms with a variable in the radicand is called aradical equation.\n...\nHOW TO\nGiven a radical equation, solve it.\n1. Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.\n2. If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of"
	},
	{
		"chunk_id" : "00000472",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equation to the third power. In other words, for annth root radical, raise both sides to thenth power. Doing\nso eliminates the radical symbol.\n3. Solve the remaining equation.\n4. If a radical term still remains, repeat steps 12.\n5. Confirm solutions by substituting them into the original equation.\nEXAMPLE6\nSolving an Equation with One Radical\nSolve\nSolution\nThe radical is already isolated on the left side of the equal side, so proceed to square both sides."
	},
	{
		"chunk_id" : "00000473",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We see that the remaining equation is a quadratic. Set it equal to zero and solve.\nThe proposed solutions are and Let us check each solution back in the original equation. First, check\nThis is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not\nsatisfy the original equation.\nCheck\nThe solution is\nAccess for free at openstax.org\n2.6  Other Types of Equations 155\nTRY IT #5 Solve the radical equation:\nEXAMPLE7"
	},
	{
		"chunk_id" : "00000474",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Solve the radical equation:\nEXAMPLE7\nSolving a Radical Equation Containing Two Radicals\nSolve\nSolution\nAs this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.\nUse the perfect square formula to expand the right side:\nNow that both radicals have been eliminated, set the quadratic equal to zero and solve.\nThe proposed solutions are and Check each solution in the original equation.\nOne solution is\nCheck"
	},
	{
		"chunk_id" : "00000475",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "One solution is\nCheck\nThe only solution is We see that is an extraneous solution.\nTRY IT #6 Solve the equation with two radicals:\n156 2  Equations and Inequalities\nSolving an Absolute Value Equation\nNext, we will learn how to solve anabsolute value equation. To solve an equation such as we notice that\nthe absolute value will be equal to 8 if the quantity inside the absolute value bars is or This leads to two different\nequations we can solve independently."
	},
	{
		"chunk_id" : "00000476",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations we can solve independently.\nKnowing how to solve problems involving absolute value functions is useful. For example, we may need to identify\nnumbers or points on a line that are at a specified distance from a given reference point.\nAbsolute Value Equations\nThe absolute value ofxis written as It has the following properties:\nFor real numbers and an equation of the form with will have solutions when or\nIf the equation has no solution."
	},
	{
		"chunk_id" : "00000477",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If the equation has no solution.\nAnabsolute value equationin the form has the following properties:\n...\nHOW TO\nGiven an absolute value equation, solve it.\n1. Isolate the absolute value expression on one side of the equal sign.\n2. If write and solve two equations: and\nEXAMPLE8\nSolving Absolute Value Equations\nSolve the following absolute value equations:\n   \nSolution\n\nWrite two equations and solve each:\n\nThe two solutions are and There is no solution as an absolute value cannot be negative."
	},
	{
		"chunk_id" : "00000478",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n2.6  Other Types of Equations 157\n\nIsolate the absolute value expression and then write two equations.\nThere are two solutions: and\n\n(d)\nThe equation is set equal to zero, so we have to write only one equation.\nThere is one solution:\nTRY IT #7 Solve the absolute value equation:\nSolving Other Types of Equations\nThere are many other types of equations in addition to the ones we have discussed so far. We will see more of them"
	},
	{
		"chunk_id" : "00000479",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a\nquadratic.\nSolving Equations in Quadratic Form\nEquations in quadratic formare equations with three terms. The first term has a power other than 2. The middle term\nhas an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations\nin this form as if they were quadratic. A few examples of these equations include"
	},
	{
		"chunk_id" : "00000480",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and In each one, doubling the exponent of the middle term equals the exponent on the leading\nterm. We can solve these equations by substituting a variable for the middle term.\nQuadratic Form\nIf the exponent on the middle term is one-half of the exponent on the leading term, we have anequation in\nquadratic form, which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve\nequations in quadratic form.\n...\nHOW TO\nGiven an equation quadratic in form, solve it."
	},
	{
		"chunk_id" : "00000481",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given an equation quadratic in form, solve it.\n1. Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.\n2. If it is, substitute a variable, such asu, for the variable portion of the middle term.\n3. Rewrite the equation so that it takes on the standard form of a quadratic.\n4. Solve using one of the usual methods for solving a quadratic.\n5. Replace the substitution variable with the original term.\n158 2  Equations and Inequalities"
	},
	{
		"chunk_id" : "00000482",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "158 2  Equations and Inequalities\n6. Solve the remaining equation.\nEXAMPLE9\nSolving a Fourth-degree Equation in Quadratic Form\nSolve this fourth-degree equation:\nSolution\nThis equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we\nwill make a substitution for the variable term in the middle. Let Rewrite the equation inu.\nNow solve the quadratic.\nSolve each factor and replace the original term foru.\nThe solutions are and"
	},
	{
		"chunk_id" : "00000483",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solutions are and\nTRY IT #8 Solve using substitution:\nEXAMPLE10\nSolving an Equation in Quadratic Form Containing a Binomial\nSolve the equation in quadratic form:\nSolution\nThis equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However,\nrecognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make\na substitution, letting Then rewrite the equation inu."
	},
	{
		"chunk_id" : "00000484",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve using the zero-factor property and then replaceuwith the original expression.\nAccess for free at openstax.org\n2.6  Other Types of Equations 159\nThe second factor results in\nWe have two solutions: and\nTRY IT #9 Solve:\nSolving Rational Equations Resulting in a Quadratic\nEarlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens,"
	},
	{
		"chunk_id" : "00000485",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that\nthere is no solution.\nEXAMPLE11\nSolving a Rational Equation Leading to a Quadratic\nSolve the following rational equation:\nSolution\nWe want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further.\nHowever, Then, the LCD is Next, we multiply the whole equation by the LCD."
	},
	{
		"chunk_id" : "00000486",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.\nTRY IT #10 Solve\nMEDIA\nAccess these online resources for additional instruction and practice with different types of equations.\nRational Equation with no Solution(http://openstax.org/l/rateqnosoln)\nSolving equations with rational exponents using reciprocal powers(http://openstax.org/l/ratexprecpexp)\nSolving radical equations part 1 of 2(http://openstax.org/l/radeqsolvepart1)"
	},
	{
		"chunk_id" : "00000487",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving radical equations part 2 of 2(http://openstax.org/l/radeqsolvepart2)\n160 2  Equations and Inequalities\n2.6 SECTION EXERCISES\nVerbal\n1. In a radical equation, what 2. Explain why possible 3. Your friend tries to calculate\ndoes it mean if a number is solutionsmustbe checked\nthe value and keeps\nan extraneous solution? in radical equations.\ngetting an ERROR message.\nWhat mistake are they\nprobably making?\n4. Explain why 5. Explain how to change a\nhas no solutions. rational exponent into the"
	},
	{
		"chunk_id" : "00000488",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has no solutions. rational exponent into the\ncorrect radical expression.\nAlgebraic\nFor the following exercises, solve the rational exponent equation. Use factoring where necessary.\n6. 7. 8.\n9. 10. 11.\n12.\nFor the following exercises, solve the following polynomial equations by grouping and factoring.\n13. 14. 15.\n16. 17. 18.\n19.\nFor the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.\n20. 21. 22.\n23. 24. 25.\n26. 27. 28."
	},
	{
		"chunk_id" : "00000489",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "20. 21. 22.\n23. 24. 25.\n26. 27. 28.\nFor the following exercises, solve the equation involving absolute value.\n29. 30. 31.\n32. 33. 34.\nAccess for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 161\n35. 36.\nFor the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all\nreal solutions by factoring.\n37. 38. 39.\n40. 41.\nExtensions\nFor the following exercises, solve for the unknown variable.\n42. 43. 44.\n45."
	},
	{
		"chunk_id" : "00000490",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "42. 43. 44.\n45.\nReal-World Applications\nFor the following exercises, use the model for the period of a pendulum, such that where the length of\nthe pendulum isLand the acceleration due to gravity is\n46. If the acceleration due to 47. If the gravity is 32 ft/s2and\ngravity is 9.8 m/s2and the the period equals 1 s, find\nperiod equals 1 s, find the the length to the nearest\nlength to the nearest cm in. (12 in. = 1 ft). Round\n(100 cm = 1 m). your answer to the nearest\nin."
	},
	{
		"chunk_id" : "00000491",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(100 cm = 1 m). your answer to the nearest\nin.\nFor the following exercises, use a model for body surface area, BSA, such that wherew= weight in kg\nandh= height in cm.\n48. Find the height of a 72-kg 49. Find the weight of a\nfemale to the nearest cm 177-cm male to the nearest\nwhose kg whose\n2.7 Linear Inequalities and Absolute Value Inequalities\nLearning Objectives\nIn this section, you will:\nUse interval notation\nUse properties of inequalities.\nSolve inequalities in one variable algebraically."
	},
	{
		"chunk_id" : "00000492",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve inequalities in one variable algebraically.\nSolve absolute value inequalities.\n162 2  Equations and Inequalities\nFigure1\nIt is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at\nleast 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be\nexpressed mathematically? In this section, we will explore various ways to express different sets of numbers,"
	},
	{
		"chunk_id" : "00000493",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inequalities, and absolute value inequalities.\nUsing Interval Notation\nIndicating the solution to an inequality such as can be achieved in several ways.\nWe can use a number line as shown inFigure 2.The blue ray begins at and, as indicated by the arrowhead,\ncontinues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\nFigure2\nWe can useset-builder notation: which translates to all real numbersxsuch thatxis greater than or equal"
	},
	{
		"chunk_id" : "00000494",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to 4. Notice that braces are used to indicate a set.\nThe third method isinterval notation, in which solution sets are indicated with parentheses or brackets. The solutions\nto are represented as  This is perhaps the most useful method, as it applies to concepts studied later in\nthis course and to other higher-level math courses.\nThe main concept to remember is that parentheses represent solutions greater or less than the number, and brackets"
	},
	{
		"chunk_id" : "00000495",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent\ninfinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and,\ntherefore, cannot be equaled. A few examples of aninterval, or a set of numbers in which a solution falls, are\nor all numbers between and including but not including all real numbers between, but not including"
	},
	{
		"chunk_id" : "00000496",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and and  all real numbers less than and including Table 1outlines the possibilities.\nSet Indicated Set-Builder Notation Interval Notation\nAll real numbers betweenaandb, but not includingaorb\nAll real numbers greater thana, but not includinga \nAll real numbers less thanb, but not includingb \nAll real numbers greater thana, includinga \nAll real numbers less thanb, includingb \nAll real numbers betweenaandb, includinga\nAll real numbers betweenaandb, includingb\nTable1\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000497",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nAccess for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 163\nSet Indicated Set-Builder Notation Interval Notation\nAll real numbers betweenaandb, includingaandb\nAll real numbers less thanaor greater thanb  \nAll real numbers  \nTable1\nEXAMPLE1\nUsing Interval Notation to Express All Real Numbers Greater Than or Equal toa\nUse interval notation to indicate all real numbers greater than or equal to\nSolution"
	},
	{
		"chunk_id" : "00000498",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUse a bracket on the left of and parentheses after infinity:  The bracket indicates that is included in the\nset with all real numbers greater than to infinity.\nTRY IT #1 Use interval notation to indicate all real numbers between and including and\nEXAMPLE2\nUsing Interval Notation to Express All Real Numbers Less Than or Equal toaor Greater Than or Equal tob\nWrite the interval expressing all real numbers less than or equal to or greater than or equal to\nSolution"
	},
	{
		"chunk_id" : "00000499",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1.\nSo, this interval begins at and ends at which is written as \nThe second interval must show all real numbers greater than or equal to which is written as  However, we\nwant to combine these two sets. We accomplish this by inserting the union symbol, between the two intervals.\n \nTRY IT #2 Express all real numbers less than or greater than or equal to 3 in interval notation."
	},
	{
		"chunk_id" : "00000500",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using the Properties of Inequalities\nWhen we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use\ntheaddition propertyand themultiplication propertyto help us solve them. The one exception is when we multiply or\ndivide by a negative number; doing so reverses the inequality symbol.\nProperties of Inequalities\nThese properties also apply to and\n164 2  Equations and Inequalities\nEXAMPLE3\nDemonstrating the Addition Property"
	},
	{
		"chunk_id" : "00000501",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nDemonstrating the Addition Property\nIllustrate the addition property for inequalities by solving each of the following:\n1. \n2. \n3. \nSolution\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both\nsides does not change the inequality.\n \n\nTRY IT #3 Solve:\nEXAMPLE4\nDemonstrating the Multiplication Property\nIllustrate the multiplication property for inequalities by solving each of the following:\n1. \n2. \n3. \nSolution\n\n\n"
	},
	{
		"chunk_id" : "00000502",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. \n2. \n3. \nSolution\n\n\n\nTRY IT #4 Solve:\nSolving Inequalities in One Variable Algebraically\nAs the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with\nequations; we combine like terms and perform operations. To solve, we isolate the variable.\nAccess for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 165\nEXAMPLE5\nSolving an Inequality Algebraically\nSolve the inequality:\nSolution"
	},
	{
		"chunk_id" : "00000503",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the inequality:\nSolution\nSolving this inequality is similar to solving an equation up until the last step.\nThe solution set is given by the interval  or all real numbers less than and including 1.\nTRY IT #5 Solve the inequality and write the answer using interval notation:\nEXAMPLE6\nSolving an Inequality with Fractions\nSolve the following inequality and write the answer in interval notation:\nSolution\nWe begin solving in the same way we do when solving an equation.\nThe solution set is the interval "
	},
	{
		"chunk_id" : "00000504",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solution set is the interval \nTRY IT #6 Solve the inequality and write the answer in interval notation:\nUnderstanding Compound Inequalities\nAcompound inequalityincludes two inequalities in one statement. A statement such as means and\nThere are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the\ncompound inequality intact and performing operations on all three parts at the same time. We will illustrate both\nmethods.\nEXAMPLE7"
	},
	{
		"chunk_id" : "00000505",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "methods.\nEXAMPLE7\nSolving a Compound Inequality\nSolve the compound inequality:\nSolution\nThe first method is to write two separate inequalities: and We solve them independently.\n166 2  Equations and Inequalities\nThen, we can rewrite the solution as a compound inequality, the same way the problem began.\nIn interval notation, the solution is written as\nThe second method is to leave the compound inequality intact, and perform solving procedures on the three parts at the\nsame time.\nWe get the same solution:"
	},
	{
		"chunk_id" : "00000506",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same time.\nWe get the same solution:\nTRY IT #7 Solve the compound inequality:\nEXAMPLE8\nSolving a Compound Inequality with the Variable in All Three Parts\nSolve the compound inequality with variables in all three parts:\nSolution\nLet's try the first method. Write two inequalities:\nThe solution set is or in interval notation Notice that when we write the solution in interval\nnotation, the smaller number comes first. We read intervals from left to right, as they appear on a number line. See\nFigure 3.\nFigure3"
	},
	{
		"chunk_id" : "00000507",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 3.\nFigure3\nTRY IT #8 Solve the compound inequality:\nSolving Absolute Value Inequalities\nAs we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at has\nan absolute value of as it isxunits away. Consider absolute value as the distance from one point to another point.\nRegardless of direction, positive or negative, the distance between the two points is represented as a positive number or\nzero.\nAnabsolute value inequalityis an equation of the form"
	},
	{
		"chunk_id" : "00000508",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 167\nWhereA, and sometimesB, represents an algebraic expression dependent on a variablex.Solving the inequality means\nfinding the set of all -values that satisfy the problem. Usually this set will be an interval or the union of two intervals\nand will include a range of values.\nThere are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the"
	},
	{
		"chunk_id" : "00000509",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the\nalgebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.\nSuppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of\n$600. We can solve algebraically for the set ofx-values such that the distance between and 600 is less than or equal to"
	},
	{
		"chunk_id" : "00000510",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "200. We represent the distance between and 600 as and therefore, or\nThis means our returns would be between $400 and $800.\nTo solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them\nindependently.\nAbsolute Value Inequalities\nFor an algebraic expressionX,and anabsolute value inequalityis an inequality of the form\nThese statements also apply to and\nEXAMPLE9\nDetermining a Number within a Prescribed Distance"
	},
	{
		"chunk_id" : "00000511",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determining a Number within a Prescribed Distance\nDescribe all values within a distance of 4 from the number 5.\nSolution\nWe want the distance between and 5 to be less than or equal to 4. We can draw a number line, such as inFigure 4,to\nrepresent the condition to be satisfied.\nFigure4\nThe distance from to 5 can be represented using an absolute value symbol, Write the values of that satisfy\nthe condition as an absolute value inequality."
	},
	{
		"chunk_id" : "00000512",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the condition as an absolute value inequality.\nWe need to write two inequalities as there are always two solutions to an absolute value equation.\nIf the solution set is and then the solution set is an interval including all real numbers between and\nincluding 1 and 9.\nSo is equivalent to in interval notation.\nTRY IT #9 Describe allx-values within a distance of 3 from the number 2.\n168 2  Equations and Inequalities\nEXAMPLE10\nSolving an Absolute Value Inequality\nSolve .\nSolution\nEXAMPLE11"
	},
	{
		"chunk_id" : "00000513",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve .\nSolution\nEXAMPLE11\nUsing a Graphical Approach to Solve Absolute Value Inequalities\nGiven the equation determine thex-values for which they-values are negative.\nSolution\nWe are trying to determine where which is when We begin by isolating the absolute value.\nNext, we solve for the equality\nNow, we can examine the graph to observe where they-values are negative. We observe where the branches are below"
	},
	{
		"chunk_id" : "00000514",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thex-axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the\nhorizontal axis at and and that the graph opens downward. SeeFigure 5.\nFigure5\nTRY IT #10 Solve\nMEDIA\nAccess these online resources for additional instruction and practice with linear inequalities and absolute value\ninequalities.\nAccess for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 169\nInterval notation(http://openstax.org/l/intervalnotn)"
	},
	{
		"chunk_id" : "00000515",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "How to solve linear inequalities(http://openstax.org/l/solvelinineq)\nHow to solve an inequality(http://openstax.org/l/solveineq)\nAbsolute value equations(http://openstax.org/l/absvaleq)\nCompound inequalities(http://openstax.org/l/compndineqs)\nAbsolute value inequalities(http://openstax.org/l/absvalineqs)\n2.7 SECTION EXERCISES\nVerbal\n1. When solving an inequality, 2. When solving an inequality, 3. When writing our solution in\nexplain what happened we arrive at: interval notation, how do we"
	},
	{
		"chunk_id" : "00000516",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from Step 1 to Step 2: represent all the real\nnumbers?\nExplain what our solution\nset is.\n4. When solving an inequality, 5. Describe how to graph\nwe arrive at:\nExplain what our solution\nset is.\nAlgebraic\nFor the following exercises, solve the inequality. Write your final answer in interval notation.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\nFor the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23."
	},
	{
		"chunk_id" : "00000517",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "15. 16. 17.\n18. 19. 20.\n21. 22. 23.\nFor the following exercises, describe all thex-values within or including a distance of the given values.\n24. Distance of 5 units from 25. Distance of 3 units from 26. Distance of 10 units from\nthe number 7 the number 9 the number 4\n170 2  Equations and Inequalities\n27. Distance of 11 units from\nthe number 1\nFor the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write\nyour answer using interval notation."
	},
	{
		"chunk_id" : "00000518",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "your answer using interval notation.\n28. 29. 30.\n31. 32.\nGraphical\nFor the following exercises, graph the function. Observe the points of intersection and shade thex-axis representing the\nsolution set to the inequality. Show your graph and write your final answer in interval notation.\n33. 34. 35.\n36. 37.\nFor the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same"
	},
	{
		"chunk_id" : "00000519",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "axes. Find the point of intersection and solve the inequality by observing where it is true comparing they-values of the\nlines.\n38. 39. 40.\n41. 42.\nNumeric\nFor the following exercises, write the set in interval notation.\n43. 44. 45.\n46.\nFor the following exercises, write the interval in set-builder notation.\n47.  48.  49.\n50. \nFor the following exercises, write the set of numbers represented on the number line in interval notation.\n51. 52. 53.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000520",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "51. 52. 53.\nAccess for free at openstax.org\n2.7  Linear Inequalities and Absolute Value Inequalities 171\nTechnology\nFor the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the\nright-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of\nintersection, recall (2ndCALC 5:intersection, 1stcurve, enter, 2ndcurve, enter, guess, enter). Copy a sketch of the graph"
	},
	{
		"chunk_id" : "00000521",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and shade thex-axis for your solution set to the inequality. Write final answers in interval notation.\n54. 55. 56.\n57. 58.\nExtensions\n59. Solve 60. Solve 61.\n62. is\na profit formula for a small\nbusiness. Find the set of\nx-values that will keep this\nprofit positive.\nReal-World Applications\n63. In chemistry the volume 64. A basic cellular package\nfor a certain gas is given by costs $20/mo. for 60 min of\nwhereVis calling, with an additional\nmeasured in cc andTis charge of $.30/min beyond"
	},
	{
		"chunk_id" : "00000522",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "measured in cc andTis charge of $.30/min beyond\ntemperature in C. If the that time.. The cost\ntemperature varies formula would be\nbetween 80C and 120C, If\nfind the set of volume you have to keep your bill\nvalues. no greater than $50, what\nis the maximum calling\nminutes you can use?\n172 2  Chapter Review\nChapter Review\nKey Terms\nabsolute value equation an equation in which the variable appears in absolute value bars, typically with two solutions,"
	},
	{
		"chunk_id" : "00000523",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one accounting for the positive expression and one for the negative expression\narea in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular\nregion:\nCartesian coordinate system a grid system designed with perpendicular axes invented by Ren Descartes\ncompleting the square a process for solving quadratic equations in which terms are added to or subtracted from both\nsides of the equation in order to make one side a perfect square"
	},
	{
		"chunk_id" : "00000524",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "complex conjugate a complex number containing the same terms as another complex number, but with the opposite\noperator. Multiplying a complex number by its conjugate yields a real number.\ncomplex number the sum of a real number and an imaginary number; the standard form is whereais the real\npart and is the complex part.\ncomplex plane the coordinate plane in which the horizontal axis represents the real component of a complex number,\nand the vertical axis represents the imaginary component, labeledi."
	},
	{
		"chunk_id" : "00000525",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compound inequality a problem or a statement that includes two inequalities\nconditional equation an equation that is true for some values of the variable\ndiscriminant the expression under the radical in the quadratic formula that indicates the nature of the solutions, real\nor complex, rational or irrational, single or double roots.\ndistance formula a formula that can be used to find the length of a line segment if the endpoints are known"
	},
	{
		"chunk_id" : "00000526",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation in two variables a mathematical statement, typically written inxandy, in which two expressions are equal\nequations in quadratic form equations with a power other than 2 but with a middle term with an exponent that is\none-half the exponent of the leading term\nextraneous solutions any solutions obtained that are not valid in the original equation\ngraph in two variables the graph of an equation in two variables, which is always shown in two variables in the two-\ndimensional plane"
	},
	{
		"chunk_id" : "00000527",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "dimensional plane\nidentity equation an equation that is true for all values of the variable\nimaginary number the square root of :\ninconsistent equation an equation producing a false result\nintercepts the points at which the graph of an equation crosses thex-axis and they-axis\ninterval an interval describes a set of numbers within which a solution falls\ninterval notation a mathematical statement that describes a solution set and uses parentheses or brackets to indicate\nwhere an interval begins and ends"
	},
	{
		"chunk_id" : "00000528",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where an interval begins and ends\nlinear equation an algebraic equation in which each term is either a constant or the product of a constant and the first\npower of a variable\nlinear inequality similar to a linear equation except that the solutions will include sets of numbers\nmidpoint formula a formula to find the point that divides a line segment into two parts of equal length\nordered pair a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also"
	},
	{
		"chunk_id" : "00000529",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "known as a coordinate pair,\norigin the point where the two axes cross in the center of the plane, described by the ordered pair\nperimeter in linear units, the perimeter formula is used to find the linear measurement, or outside length and width,\naround a two-dimensional regular object; for a rectangle:\npolynomial equation an equation containing a string of terms including numerical coefficients and variables raised to\nwhole-number exponents"
	},
	{
		"chunk_id" : "00000530",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "whole-number exponents\nPythagorean Theorem a theorem that states the relationship among the lengths of the sides of a right triangle, used\nto solve right triangle problems\nquadrant one quarter of the coordinate plane, created when the axes divide the plane into four sections\nquadratic equation an equation containing a second-degree polynomial; can be solved using multiple methods\nquadratic formula a formula that will solve all quadratic equations"
	},
	{
		"chunk_id" : "00000531",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radical equation an equation containing at least one radical term where the variable is part of the radicand\nrational equation an equation consisting of a fraction of polynomials\nslope the change iny-values over the change inx-values\nsolution set the set of all solutions to an equation\nsquare root property one of the methods used to solve a quadratic equation, in which the term is isolated so that\nthe square root of both sides of the equation can be taken to solve forx"
	},
	{
		"chunk_id" : "00000532",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "volume in cubic units, the volume measurement includes length, width, and depth:\nx-axis the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right\nAccess for free at openstax.org\n2  Chapter Review 173\nx-coordinate the first coordinate of an ordered pair, representing the horizontal displacement and direction from the\norigin\nx-intercept the point where a graph intersects thex-axis; an ordered pair with ay-coordinate of zero"
	},
	{
		"chunk_id" : "00000533",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-axis the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top\ny-coordinate the second coordinate of an ordered pair, representing the vertical displacement and direction from the\norigin\ny-intercept a point where a graph intercepts they-axis; an ordered pair with anx-coordinate of zero\nzero-product property the property that formally states that multiplication by zero is zero, so that each factor of a"
	},
	{
		"chunk_id" : "00000534",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadratic equation can be set equal to zero to solve equations\nKey Equations\nquadratic formula\nKey Concepts\n2.1The Rectangular Coordinate Systems and Graphs\n We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as\ndisplacement from thex-axis and displacement from they-axis. SeeExample 1.\n An equation can be graphed in the plane by creating a table of values and plotting points. SeeExample 2."
	},
	{
		"chunk_id" : "00000535",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations\nusually have to be entered in the formy=_____. SeeExample 3.\n Finding thex-andy-intercepts can define the graph of a line. These are the points where the graph crosses the\naxes. SeeExample 4.\n The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See\nExample 5andExample 6."
	},
	{
		"chunk_id" : "00000536",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 5andExample 6.\n The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the\nx-coordinates and the sum of they-coordinates of the endpoints by 2. SeeExample 7andExample 8.\n2.2Linear Equations in One Variable\n We can solve linear equations in one variable in the form using standard algebraic properties. See\nExample 1andExample 2.\n A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See"
	},
	{
		"chunk_id" : "00000537",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 3andExample 4.\n All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or\nzero in the denominator. SeeExample 5andExample 6andExample 7.\n Given two points, we can find the slope of a line using the slope formula. SeeExample 8.\n We can identify the slope andy-intercept of an equation in slope-intercept form. SeeExample 9.\n We can find the equation of a line given the slope and a point. SeeExample 10."
	},
	{
		"chunk_id" : "00000538",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See\nExample 11.\n The standard form of a line has no fractions. SeeExample 12.\n Horizontal lines have a slope of zero and are defined as wherecis a constant.\n Vertical lines have an undefined slope (zero in the denominator), and are defined as wherecis a constant.\nSeeExample 13.\n Parallel lines have the same slope and differenty-intercepts. SeeExample 14andExample 15."
	},
	{
		"chunk_id" : "00000539",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is\nvertical. SeeExample 16.\n2.3Models and Applications\n A linear equation can be used to solve for an unknown in a number problem. SeeExample 1.\n Applications can be written as mathematical problems by identifying known quantities and assigning a variable to\nunknown quantities. SeeExample 2."
	},
	{
		"chunk_id" : "00000540",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unknown quantities. SeeExample 2.\n There are many known formulas that can be used to solve applications. Distance problems, for example, are solved\nusing the formula. SeeExample 3.\n Many geometry problems are solved using the perimeter formula the area formula or the\n174 2  Chapter Review\nvolume formula SeeExample 4,Example 5, andExample 6.\n2.4Complex Numbers\n The square root of any negative number can be written as a multiple of SeeExample 1."
	},
	{
		"chunk_id" : "00000541",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the\nreal axis, and the vertical axis is the imaginary axis. SeeExample 2.\n Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.\nSeeExample 3.\n Complex numbers can be multiplied and divided.\n To multiply complex numbers, distribute just as with polynomials. SeeExample 4andExample 5."
	},
	{
		"chunk_id" : "00000542",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the\ndenominator to eliminate the complex number from the denominator. SeeExample 6andExample 7.\n The powers of are cyclic, repeating every fourth one. SeeExample 8.\n2.5Quadratic Equations\n Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the\nequation is a difference of squares. The zero-product property is then used to find solutions. SeeExample 1,"
	},
	{
		"chunk_id" : "00000543",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 2, andExample 3.\n Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping\nmethod. SeeExample 4andExample 5.\n Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared\nterm and take the square root of both sides of the equation. The solution will yield a positive and negative solution.\nSeeExample 6andExample 7."
	},
	{
		"chunk_id" : "00000544",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 6andExample 7.\n Completing the square is a method of solving quadratic equations when the equation cannot be factored. See\nExample 8.\n A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and\nthe constant term in the equation. SeeExample 9andExample 10.\n The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex,\nrational or irrational, and how many of each. SeeExample 11."
	},
	{
		"chunk_id" : "00000545",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems\nand has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a\nquadratic equation. SeeExample 12.\n2.6Other Types of Equations\n Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve,"
	},
	{
		"chunk_id" : "00000546",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See\nExample 1,Example 2, andExample 3.\n Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See\nExample 4andExample 5.\n We can solve radical equations by isolating the radical and raising both sides of the equation to a power that\nmatches the index. SeeExample 6andExample 7."
	},
	{
		"chunk_id" : "00000547",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "matches the index. SeeExample 6andExample 7.\n To solve absolute value equations, we need to write two equations, one for the positive value and one for the\nnegative value. SeeExample 8.\n Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the\nsecond term and the third term is a constant. We may also see a binomial in place of the single variable. We use\nsubstitution to solve. SeeExample 9andExample 10."
	},
	{
		"chunk_id" : "00000548",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitution to solve. SeeExample 9andExample 10.\n Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. SeeExample 11.\n2.7Linear Inequalities and Absolute Value Inequalities\n Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system\nof parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are\nincluded as well. SeeTable 1andExample 2."
	},
	{
		"chunk_id" : "00000549",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "included as well. SeeTable 1andExample 2.\n Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or\ndividing by a negative number reverses the inequality. SeeExample 3,Example 4,Example 5, andExample 6.\n Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are\ngiven by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to"
	},
	{
		"chunk_id" : "00000550",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the two inequalities. SeeExample 7andExample 8.\nAccess for free at openstax.org\n2  Exercises 175\n Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing\ntwo equations: one equal to a positive value and one equal to a negative value. SeeExample 9andExample 10.\n Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by"
	},
	{
		"chunk_id" : "00000551",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graphing, as we cannot depend on a visual for a precise solution. SeeExample 11.\nExercises\nReview Exercises\nThe Rectangular Coordinate Systems and Graphs\nFor the following exercises, find thex-intercept and they-intercept without graphing.\n1. 2.\nFor the following exercises, solve foryin terms ofx, putting the equation in slopeintercept form.\n3. 4.\nFor the following exercises, find the distance between the two points.\n5. 6. 7. Find the distance between\nthe two points\nand using your"
	},
	{
		"chunk_id" : "00000552",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the two points\nand using your\ncalculator, and round your\nanswer to the nearest\nthousandth.\nFor the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.\n8. and 9. and\nFor the following exercises, construct a table and graph the equation by plotting at least three points.\n10. 11.\nLinear Equations in One Variable\nFor the following exercises, solve for\n12. 13. 14.\n15. 16."
	},
	{
		"chunk_id" : "00000553",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16.\nFor the following exercises, solve for State allx-values that are excluded from the solution set.\n17. 18.\n176 2  Exercises\nFor the following exercises, find the equation of the line using the point-slope formula.\n19. Passes through these two 20. Passes through the point 21. Passes through the point\npoints: and has a slope of and is parallel to\nthe graph\n22. Passes through these two\npoints:\nModels and Applications"
	},
	{
		"chunk_id" : "00000554",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "points:\nModels and Applications\nFor the following exercises, write and solve an equation to answer each question.\n23. The number of male fish in 24. A landscaper has 72 ft. of 25. A truck rental is $25 plus\nthe tank is five more than fencing to put around a $.30/mi. Find out how\nthree times the number of rectangular garden. If the many miles Ken traveled if\nfemales. If the total length is 3 times the width, his bill was $50.20.\nnumber of fish is 73, how find the dimensions of the"
	},
	{
		"chunk_id" : "00000555",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "many of each sex are in the garden.\ntank?\nComplex Numbers\nFor the following exercises, use the quadratic equation to solve.\n26. 27.\nFor the following exercises, name the horizontal component and the vertical component.\n28. 29.\nFor the following exercises, perform the operations indicated.\n30. 31. 32.\n33. 34. 35.\n36. 37. 38.\n39.\nQuadratic Equations\nFor the following exercises, solve the quadratic equation by factoring.\n40. 41. 42.\n43.\nAccess for free at openstax.org\n2  Exercises 177"
	},
	{
		"chunk_id" : "00000556",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n2  Exercises 177\nFor the following exercises, solve the quadratic equation by using the square-root property.\n44. 45.\nFor the following exercises, solve the quadratic equation by completing the square.\n46. 47.\nFor the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real,\nstateNo real solution.\n48. 49.\nFor the following exercises, solve the quadratic equation by the method of your choice.\n50. 51."
	},
	{
		"chunk_id" : "00000557",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "50. 51.\nOther Types of Equations\nFor the following exercises, solve the equations.\n52. 53. 54.\n55. 56. 57.\n58. 59.\nLinear Inequalities and Absolute Value Inequalities\nFor the following exercises, solve the inequality. Write your final answer in interval notation.\n60. 61. 62.\n63. 64. 65.\nFor the following exercises, solve the compound inequality. Write your answer in interval notation.\n66. 67.\n178 2  Exercises\nFor the following exercises, graph as described."
	},
	{
		"chunk_id" : "00000558",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, graph as described.\n68. Graph the absolute value 69. Graph both straight lines\nfunction and graph the (left-hand side being y1\nconstant function. Observe and right-hand side being\nthe points of intersection y2) on the same axes. Find\nand shade thex-axis the point of intersection\nrepresenting the solution and solve the inequality by\nset to the inequality. Show observing where it is true\nyour graph and write your comparing they-values of"
	},
	{
		"chunk_id" : "00000559",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "final answer in interval the lines. See the interval\nnotation. where the inequality is\ntrue.\nPractice Test\n1. Graph the following: 2. Find thex-andy-intercepts 3. Find thex-andy-intercepts\nfor the following: of this equation, and sketch\nthe graph of the line using\njust the intercepts plotted.\n4. Find the exact distance 5. Write the interval notation 6. Solve forx:\nbetween and for the set of numbers\nFind the represented by\ncoordinates of the midpoint\nof the line segment joining\nthe two points."
	},
	{
		"chunk_id" : "00000560",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the line segment joining\nthe two points.\n7. Solve for : 8. Solve forx: 9. Solve forx:\n10. The perimeter of a triangle 11. Solve forx. Write the 12. Solve:\nis 30 in. The longest side is answer in simplest radical\n2 less than 3 times the form.\nshortest side and the other\nside is 2 more than twice\nthe shortest side. Find the\nlength of each side.\n13. Solve: 14. Solve:\nFor the following exercises, find the equation of the line with the given information."
	},
	{
		"chunk_id" : "00000561",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "15. Passes through the points 16. Has an undefined slope 17. Passes through the point\nand and passes through the and is perpendicular\npoint to\nAccess for free at openstax.org\n2  Exercises 179\n18. Add these complex 19. Simplify: 20. Multiply:\nnumbers:\n21. Divide: 22. Solve this quadratic 23. Solve:\nequation and write the two\ncomplex roots in\nform:\n24. Solve: 25. Solve: 26. Solve:\n27. Solve: 28. Solve:\nFor the following exercises, find the real solutions of each equation by factoring.\n29. 30."
	},
	{
		"chunk_id" : "00000562",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "29. 30.\n180 2  Exercises\nAccess for free at openstax.org\n3  Introduction 181\n3 FUNCTIONS\nStandard and Poors Index with dividends reinvested (credit \"bull\"\": modification of work by Prayitno Hadinata; credit"
	},
	{
		"chunk_id" : "00000563",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.7Inverse Functions\nIntroduction to Functions\nToward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically.\nAs a result, the Standard and Poors stock market average rose as well. The graph above tracks the value of that initial\ninvestment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about"
	},
	{
		"chunk_id" : "00000564",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the dot-com\nbubble because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually\nburst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline\nrepresented on the graph beginning at the end of 2000."
	},
	{
		"chunk_id" : "00000565",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For\nany year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will\nexplore these kinds of relationships and their properties.\n3.1 Functions and Function Notation\nLearning Objectives\nIn this section, you will:\nDetermine whether a relation represents a function.\nFind the value of a function.\nDetermine whether a function is one-to-one."
	},
	{
		"chunk_id" : "00000566",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determine whether a function is one-to-one.\nUse the vertical line test to identify functions.\nGraph the functions listed in the library of functions.\nA jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child\n182 3  Functions\nincreases with time. In each case, one quantity depends on another. There is a relationship between the two quantities"
	},
	{
		"chunk_id" : "00000567",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.\nDetermining Whether a Relation Represents a Function\nArelationis a set of ordered pairs. The set of the first components of eachordered pairis called thedomainand the set\nof the second components of each ordered pair is called therange. Consider the following set of ordered pairs. The first"
	},
	{
		"chunk_id" : "00000568",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.\nThe domain is The range is\nNote that each value in the domain is also known as aninputvalue, orindependent variable, and is often labeled with\nthe lowercase letter Each value in the range is also known as anoutputvalue, ordependent variable, and is often\nlabeled lowercase letter"
	},
	{
		"chunk_id" : "00000569",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "labeled lowercase letter\nA function is a relation that assigns a single value in the range to each value in the domain.In other words, nox-values\nare repeated. For our example that relates the first fivenatural numbersto numbers double their values, this relation is\na function because each element in the domain, is paired with exactly one element in the range,\nNow lets consider the set of ordered pairs that relates the terms even and odd to the first five natural numbers. It\nwould appear as"
	},
	{
		"chunk_id" : "00000570",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would appear as\nNotice that each element in the domain, isnotpaired with exactly one element in the range,\nFor example, the term odd corresponds to three values from the range, and the term\neven corresponds to two values from the range, This violates the definition of a function, so this relation is not\na function.\nFigure 1compares relations that are functions and not functions.\nFigure1 (a) This relationship is a function because each input is associated with a single output. Note that input and"
	},
	{
		"chunk_id" : "00000571",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both give output (b) This relationship is also a function. In this case, each input is associated with a single output. (c)\nThis relationship is not a function because input is associated with two different outputs.\nFunction\nAfunctionis a relation in which each possible input value leads to exactly one output value. We say the output is a\nfunction of the input.\nTheinputvalues make up thedomain, and theoutputvalues make up therange.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000572",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a relationship between two quantities, determine whether the relationship is a function.\n1. Identify the input values.\n2. Identify the output values.\n3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads\nAccess for free at openstax.org\n3.1  Functions and Function Notation 183\nto two or more outputs, do not classify the relationship as a function.\nEXAMPLE1\nDetermining If Menu Price Lists Are Functions"
	},
	{
		"chunk_id" : "00000573",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determining If Menu Price Lists Are Functions\nThe coffee shop menu, shown below, consists of items and their prices.\n Is price a function of the item?\n Is the item a function of the price?\nSolution\n Lets begin by considering the input as the items on the menu. The output values are then the prices.\nEach item on the menu has only one price, so the price is a function of the item.\n Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the"
	},
	{
		"chunk_id" : "00000574",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output, then the same input value could have more than one output associated with it. See the image below.\nTherefore, the item is a not a function of price.\nEXAMPLE2\nDetermining If Class Grade Rules Are Functions\nIn a particular math class, the overall percent grade corresponds to a grade point average. Is grade point average a\nfunction of the percent grade? Is the percent grade a function of the grade point average?Table 1shows a possible rule\nfor assigning grade points.\n184 3  Functions"
	},
	{
		"chunk_id" : "00000575",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for assigning grade points.\n184 3  Functions\nPercent grade 056 5761 6266 6771 7277 7886 8791 92100\nGrade point average 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0\nTable1\nSolution\nFor any percent grade earned, there is an associated grade point average, so the grade point average is a function of the\npercent grade. In other words, if we input the percent grade, the output is a specific grade point average."
	},
	{
		"chunk_id" : "00000576",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the grading system given, there is a range of percent grades that correspond to the same grade point average. For\nexample, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all\nthe way to 86. Thus, percent grade is not a function of grade point average.\nTRY IT #1 Table 21 lists the five greatest baseball players of all time in order of rank.\nPlayer Rank\nBabe Ruth 1\nWillie Mays 2\nTy Cobb 3\nWalter Johnson 4\nHank Aaron 5\nTable2"
	},
	{
		"chunk_id" : "00000577",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Ty Cobb 3\nWalter Johnson 4\nHank Aaron 5\nTable2\n Is the rank a function of the player name?  Is the player name a function of the rank?\nUsing Function Notation\nOnce we determine that a relationship is a function, we need to display and define the functional relationships so that\nwe can understand and use them, and sometimes also so that we can program them into computers. There are various\nways of representing functions. A standardfunction notationis one representation that facilitates working with"
	},
	{
		"chunk_id" : "00000578",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions.\nTo represent height is a function of age, we start by identifying the descriptive variables for height and for age. The\nletters and are often used to represent functions just as we use and to represent numbers and\nand to represent sets.\nRemember, we can use any letter to name the function; the notation shows us that depends on The value\nmust be put into the function to get a result. The parentheses indicate that age is input into the function; they do not\nindicate multiplication."
	},
	{
		"chunk_id" : "00000579",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "indicate multiplication.\nWe can also give an algebraic expression as the input to a function. For example means first addaandb, and\nthe result is the input for the functionf. The operations must be performed in this order to obtain the correct result.\n1 http://www.baseball-almanac.com/legendary/lisn100.shtml. Accessed 3/24/2014.\nAccess for free at openstax.org\n3.1  Functions and Function Notation 185\nFunction Notation"
	},
	{
		"chunk_id" : "00000580",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Function Notation\nThe notation defines a function named This is read as is a function of The letter represents the\ninput value, or independent variable. The letter or represents the output value, or dependent variable.\nEXAMPLE3\nUsing Function Notation for Days in a Month\nUse function notation to represent a function whose input is the name of a month and output is the number of days in\nthat month. Assume that the domain does not include leap years.\nSolution"
	},
	{
		"chunk_id" : "00000581",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe number of days in a month is a function of the name of the month, so if we name the function we write\nor The name of the month is the input to a rule that associates a specific number (the\noutput) with each input.\nFigure2\nFor example, because March has 31 days. The notation reminds us that the number of days,\n(the output), is dependent on the name of the month, (the input).\nAnalysis"
	},
	{
		"chunk_id" : "00000582",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNote that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of\ngeometric objects, or any other element that determines some kind of output. However, most of the functions we will\nwork with in this book will have numbers as inputs and outputs.\nEXAMPLE4\nInterpreting Function Notation\nA function gives the number of police officers, in a town in year What does represent?\nSolution"
	},
	{
		"chunk_id" : "00000583",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWhen we read we see that the input year is 2005. The value for the output, the number of police officers\nis 300. Remember, The statement tells us that in the year 2005 there were 300 police\nofficers in the town.\nTRY IT #2 Use function notation to express the weight of a pig in pounds as a function of its age in days\nQ&A Instead of a notation such as could we use the same symbol for the output as for the\nfunction, such as meaning yis a function ofx?"
	},
	{
		"chunk_id" : "00000584",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, such as meaning yis a function ofx?\nYes, this is often done, especially in applied subjects that use higher math, such as physics and\nengineering. However, in exploring math itself we like to maintain a distinction between a function such\nas which is a rule or procedure, and the output we get by applying to a particular input This is\nwhy we usually use notation such as and so on.\nRepresenting Functions Using Tables"
	},
	{
		"chunk_id" : "00000585",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Representing Functions Using Tables\nA common method of representing functions is in the form of a table. The table rows or columns display the\ncorresponding input and output values.In some cases, these values represent all we know about the relationship; other\ntimes, the table provides a few select examples from a more complete relationship.\n186 3  Functions\nTable 3lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number"
	},
	{
		"chunk_id" : "00000586",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of days in that month. This information represents all we know about the months and days for a given year (that is not a\nleap year). Note that, in this table, we define a days-in-a-month function where identifies months by an\ninteger rather than by name.\nMonth number, (input) 1 2 3 4 5 6 7 8 9 10 11 12\nDays in month, (output) 31 28 31 30 31 30 31 31 30 31 30 31\nTable3\nTable 4defines a function Remember, this notation tells us that is the name of the function that takes the\ninput and gives the output"
	},
	{
		"chunk_id" : "00000587",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input and gives the output\n1 2 3 4 5\n8 6 7 6 8\nTable4\nTable 5displays the age of children in years and their corresponding heights. This table displays just some of the data\navailable for the heights and ages of children. We can see right away that this table does not represent a function\nbecause the same input value, 5 years, has two different output values, 40 in. and 42 in.\nAge in years, (input) 5 5 6 7 8 9 10\nHeight in inches, (output) 40 42 44 47 50 52 54\nTable5\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000588",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table5\n...\nHOW TO\nGiven a table of input and output values, determine whether the table represents a function.\n1. Identify the input and output values.\n2. Check to see if each input value is paired with only one output value. If so, the table represents a function.\nEXAMPLE5\nIdentifying Tables that Represent Functions\nWhich table,Table 6,Table 7, orTable 8, represents a function (if any)?\nInput Output\n2 1\n5 3\n8 6\nTable6\nAccess for free at openstax.org\n3.1  Functions and Function Notation 187\nInput Output"
	},
	{
		"chunk_id" : "00000589",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Input Output\n3 5\n0 1\n4 5\nTable7\nInput Output\n1 0\n5 2\n5 4\nTable8\nSolution\nTable 6andTable 7define functions. In both, each input value corresponds to exactly one output value.Table 8does not\ndefine a function because the input value of 5 corresponds to two different output values.\nWhen a table represents a function, corresponding input and output values can also be specified using function\nnotation.\nThe function represented byTable 6can be represented by writing\nSimilarly, the statements"
	},
	{
		"chunk_id" : "00000590",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, the statements\nrepresent the function inTable 7.\nTable 8cannot be expressed in a similar way because it does not represent a function.\nTRY IT #3 DoesTable 9represent a function?\nInput Output\n1 10\n2 100\n3 1000\nTable9\nFinding Input and Output Values of a Function\nWhen we know an input value and want to determine the corresponding output value for a function, weevaluatethe\nfunction. Evaluating will always produce one result because each input value of a function corresponds to exactly one"
	},
	{
		"chunk_id" : "00000591",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output value.\nWhen we know an output value and want to determine the input values that would produce that output value, we set\n188 3  Functions\nthe output equal to the functions formula andsolvefor the input. Solving can produce more than one solution because\ndifferent input values can produce the same output value.\nEvaluation of Functions in Algebraic Forms\nWhen we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the"
	},
	{
		"chunk_id" : "00000592",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function can be evaluated by squaring the input value, multiplying by 3, and then subtracting the\nproduct from 5.\n...\nHOW TO\nGiven the formula for a function, evaluate.\n1. Substitute the input variable in the formula with the value provided.\n2. Calculate the result.\nEXAMPLE6\nEvaluating Functions at Specific Values\nEvaluate at:\n    Now evaluate\nSolution\nReplace the in the function with each specified value.\n Because the input value is a number, 2, we can use simple algebra to simplify."
	},
	{
		"chunk_id" : "00000593",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " In this case, the input value is a letter so we cannot simplify the answer any further.\nWith an input value of we must use the distributive property.\n In this case, we apply the input values to the function more than once, and then perform algebraic operations on\nthe result. We already found that\nand we know that\nNow we combine the results and simplify.\nAccess for free at openstax.org\n3.1  Functions and Function Notation 189\nEXAMPLE7\nEvaluating Functions\nGiven the function evaluate\nSolution"
	},
	{
		"chunk_id" : "00000594",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given the function evaluate\nSolution\nTo evaluate we substitute the value 4 for the input variable in the given function.\nTherefore, for an input of 4, we have an output of 24.\nTRY IT #4 Given the function evaluate\nEXAMPLE8\nSolving Functions\nGiven the function solve for\nSolution\nIf either or (or both of them equal 0). We will set each factor equal to 0 and\nsolve for in each case.\nThis gives us two solutions. The output when the input is either or We can also verify by graphing"
	},
	{
		"chunk_id" : "00000595",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as inFigure 3. The graph verifies that and\nFigure3\nTRY IT #5 Given the function solve\nEvaluating Functions Expressed in Formulas\nSome functions are defined by mathematical rules or procedures expressed inequationform. If it is possible to express\nthe function output with aformulainvolving the input quantity, then we can define a function in algebraic form. For\nexample, the equation expresses a functional relationship between and We can rewrite it to decide if\n190 3  Functions\nis a function of\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000596",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "190 3  Functions\nis a function of\n...\nHOW TO\nGiven a function in equation form, write its algebraic formula.\n1. Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an\nexpression that involvesonlythe input variable.\n2. Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or\nfrom both sides, or multiplying or dividing both sides of the equation by the same quantity.\nEXAMPLE9"
	},
	{
		"chunk_id" : "00000597",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nFinding an Equation of a Function\nExpress the relationship as a function if possible.\nSolution\nTo express the relationship in this form, we need to be able to write the relationship where is a function of which\nmeans writing it as\nTherefore, as a function of is written as\nAnalysis\nIt is important to note that not every relationship expressed by an equation can also be expressed as a function with a\nformula.\nEXAMPLE10\nExpressing the Equation of a Circle as a Function"
	},
	{
		"chunk_id" : "00000598",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Expressing the Equation of a Circle as a Function\nDoes the equation represent a function with as input and as output? If so, express the relationship as a\nfunction\nSolution\nFirst we subtract from both sides.\nWe now try to solve for in this equation.\nWe get two outputs corresponding to the same input, so this relationship cannot be represented as a single function\nTRY IT #6 If express as a function of\nAccess for free at openstax.org\n3.1  Functions and Function Notation 191"
	},
	{
		"chunk_id" : "00000599",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.1  Functions and Function Notation 191\nQ&A Are there relationships expressed by an equation that do represent a function but which still\ncannot be represented by an algebraic formula?\nYes, this can happen. For example, given the equation if we want to express as a function of\nthere is no simple algebraic formula involving only that equals However, each does determine a\nunique value for and there are mathematical procedures by which can be found to any desired"
	},
	{
		"chunk_id" : "00000600",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "accuracy. In this case, we say that the equation gives an implicit (implied) rule for as a function of\neven though the formula cannot be written explicitly.\nEvaluating a Function Given in Tabular Form\nAs we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions,\nand we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share"
	},
	{
		"chunk_id" : "00000601",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can\nremember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppys memory span is no\nlonger than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory\nspan lasts for 16 hours.\nThe function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a\ntable. SeeTable 10.2"
	},
	{
		"chunk_id" : "00000602",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "table. SeeTable 10.2\nPet Memory span in hours\nPuppy 0.008\nAdult dog 0.083\nCat 16\nGoldfish 2160\nBeta fish 3600\nTable10\nAt times, evaluating a function in table form may be more useful than using equations.Here let us call the function\nThedomainof the function is the type of pet and the range is a real number representing the number of hours the pets\nmemory span lasts. We can evaluate the function at the input value of goldfish. We would write"
	},
	{
		"chunk_id" : "00000603",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from\nthe pertinent row of the table. The tabular form for function seems ideally suited to this function, more so than writing\nit in paragraph or function form.\n...\nHOW TO\nGiven a function represented by a table, identify specific output and input values.\n1. Find the given input in the row (or column) of input values.\n2. Identify the corresponding output value paired with that input value."
	},
	{
		"chunk_id" : "00000604",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Find the given output values in the row (or column) of output values, noting every time that output value\nappears.\n4. Identify the input value(s) corresponding to the given output value.\n2 http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.\n192 3  Functions\nEXAMPLE11\nEvaluating and Solving a Tabular Function\nUsingTable 11,\n Evaluate\n Solve\n1 2 3 4 5\n8 6 7 6 8\nTable11\nSolution"
	},
	{
		"chunk_id" : "00000605",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Solve\n1 2 3 4 5\n8 6 7 6 8\nTable11\nSolution\n Evaluating means determining the output value of the function for the input value of The table output\nvalue corresponding to is 7, so\n Solving means identifying the input values, that produce an output value of 6. The table below shows\ntwo solutions: and\n1 2 3 4 5\n8 6 7 6 8\nWhen we input 2 into the function our output is 6. When we input 4 into the function our output is also 6."
	},
	{
		"chunk_id" : "00000606",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Using the table fromEvaluating and Solving a Tabular Functionabove, evaluate\nFinding Function Values from a Graph\nEvaluating a function using a graph also requires finding the corresponding output value for a given input value, only in\nthis case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all\ninstances of the given output value on the graph and observing the corresponding input value(s).\nEXAMPLE12"
	},
	{
		"chunk_id" : "00000607",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE12\nReading Function Values from a Graph\nGiven the graph inFigure 4,\n Evaluate\n Solve\nFigure4\nAccess for free at openstax.org\n3.1  Functions and Function Notation 193\nSolution\n To evaluate locate the point on the curve where then read they-coordinate of that point. The point has\ncoordinates so SeeFigure 5.\nFigure5\n To solve we find the output value on the vertical axis. Moving horizontally along the line we"
	},
	{
		"chunk_id" : "00000608",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "locate two points of the curve with output value and These points represent the two solutions to\nor This means and or when the input is or the output is SeeFigure 6.\nFigure6\nTRY IT #8 UsingFigure 4, solve\nDetermining Whether a Function is One-to-One\nSome functions have a given output value that corresponds to two or more input values. For example, in the stock chart\nshown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that"
	},
	{
		"chunk_id" : "00000609",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "there were five different input values that all resulted in the same output value of $1000.\nHowever, some functions have only one input value for each output value, as well as having only one output for each\ninput. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and\ndecimal equivalents, as listed inTable 12.\nLetter grade Grade point average\nA 4.0\nB 3.0\nC 2.0\nTable12\n194 3  Functions\nLetter grade Grade point average\nD 1.0\nTable12"
	},
	{
		"chunk_id" : "00000610",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Letter grade Grade point average\nD 1.0\nTable12\nThis grading system represents a one-to-one function, because each letter input yields one particular grade point\naverage output and each grade point average corresponds to one input letter.\nTo visualize this concept, lets look again at the two simple functions sketched inFigure 1(a)andFigure 1(b). The\nfunction in part (a) shows a relationship that is not a one-to-one function because inputs and both give output The"
	},
	{
		"chunk_id" : "00000611",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single\noutput.\nOne-to-One Function\nAone-to-one functionis a function in which each output value corresponds to exactly one input value.\nEXAMPLE13\nDetermining Whether a Relationship Is a One-to-One Function\nIs the area of a circle a function of its radius? If yes, is the function one-to-one?\nSolution"
	},
	{
		"chunk_id" : "00000612",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nA circle of radius has a unique area measure given by so for any input, there is only one output, The area\nis a function of radius\nIf the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area\nmeasure is given by the formula Because areas and radii are positive numbers, there is exactly one solution:\nSo the area of a circle is a one-to-one function of the circles radius.\nTRY IT #9  Is a balance a function of the bank account number?"
	},
	{
		"chunk_id" : "00000613",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Is a bank account number a function of the balance?\n Is a balance a one-to-one function of the bank account number?\nTRY IT #10 Evaluate the following:\n If each percent grade earned in a course translates to one letter grade, is the letter grade a\nfunction of the percent grade?\n If so, is the function one-to-one?\nUsing the Vertical Line Test\nAs we have seen in some examples above, we can represent a function using a graph. Graphs display a great many"
	},
	{
		"chunk_id" : "00000614",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input-output pairs in a small space. The visual information they provide often makes relationships easier to understand.\nBy convention, graphs are typically constructed with the input values along the horizontal axis and the output values\nalong the vertical axis.\nThe most common graphs name the input value and the output value and we say is a function of or\nwhen the function is named The graph of the function is the set of all points in the plane that satisfies the"
	},
	{
		"chunk_id" : "00000615",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation If the function is defined for only a few input values, then the graph of the function is only a few\npoints, where thex-coordinate of each point is an input value and they-coordinate of each point is the corresponding\noutput value. For example, the black dots on the graph inFigure 7tell us that and However, the set\nof all points satisfying is a curve. The curve shown includes and because the curve passes\nthrough those points.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000616",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.1  Functions and Function Notation 195\nFigure7\nThevertical line testcan be used to determine whether a graph represents a function. If we can draw any vertical line\nthat intersects a graph more than once, then the graph doesnotdefine a function because a function has only one\noutput value for each input value. SeeFigure 8.\nFigure8\n...\nHOW TO\nGiven a graph, use the vertical line test to determine if the graph represents a function."
	},
	{
		"chunk_id" : "00000617",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Inspect the graph to see if any vertical line drawn would intersect the curve more than once.\n2. If there is any such line, determine that the graph does not represent a function.\nEXAMPLE14\nApplying the Vertical Line Test\nWhich of the graphs inFigure 9represent(s) a function\nFigure9\n196 3  Functions\nSolution\nIf any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Notice that"
	},
	{
		"chunk_id" : "00000618",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) ofFigure 9. From this we\ncan conclude that these two graphs represent functions. The third graph does not represent a function because, at most\nx-values, a vertical line would intersect the graph at more than one point, as shown inFigure 10.\nFigure10\nTRY IT #11 Does the graph inFigure 11represent a function?\nFigure11\nUsing the Horizontal Line Test"
	},
	{
		"chunk_id" : "00000619",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure11\nUsing the Horizontal Line Test\nOnce we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to\nuse thehorizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more\nthan once, then the graph does not represent a one-to-one function.\n...\nHOW TO\nGiven a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one\nfunction."
	},
	{
		"chunk_id" : "00000620",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\n1. Inspect the graph to see if any horizontal line drawn would intersect the curve more than once.\n2. If there is any such line, determine that the function is not one-to-one.\nAccess for free at openstax.org\n3.1  Functions and Function Notation 197\nEXAMPLE15\nApplying the Horizontal Line Test\nConsider the functions shown inFigure 9(a)andFigure 9(b). Are either of the functions one-to-one?\nSolution"
	},
	{
		"chunk_id" : "00000621",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe function inFigure 9(a)is not one-to-one. The horizontal line shown inFigure 12intersects the graph of the function\nat two points (and we can even find horizontal lines that intersect it at three points.)\nFigure12\nThe function inFigure 9(b)is one-to-one. Any horizontal line will intersect a diagonal line at most once.\nTRY IT #12 Is the graph shown inFigure 9one-to-one?\nIdentifying Basic Toolkit Functions"
	},
	{
		"chunk_id" : "00000622",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Basic Toolkit Functions\nIn this text, we will be exploring functionsthe shapes of their graphs, their unique characteristics, their algebraic\nformulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do\narithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-\nblock elements. We call these our toolkit functions, which form a set of basic named functions for which we know the"
	},
	{
		"chunk_id" : "00000623",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph, formula, and special properties. Some of these functions are programmed to individual buttons on many\ncalculators. For these definitions we will use as the input variable and as the output variable.\nWe will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently\nthroughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by"
	},
	{
		"chunk_id" : "00000624",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "name, formula, graph, and basic table properties. The graphs and sample table values are included with each function\nshown inTable 13.\nToolkit Functions\nName Function Graph\nConstant where is a constant\nTable13\n198 3  Functions\nToolkit Functions\nName Function Graph\nIdentity\nAbsolute value\nQuadratic\nCubic\nReciprocal\nTable13\nAccess for free at openstax.org\n3.1  Functions and Function Notation 199\nToolkit Functions\nName Function Graph\nReciprocal squared\nSquare root\nCube root\nTable13\nMEDIA"
	},
	{
		"chunk_id" : "00000625",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Square root\nCube root\nTable13\nMEDIA\nAccess the following online resources for additional instruction and practice with functions.\nDetermine if a Relation is a Function(http://openstax.org/l/relationfunction)\nVertical Line Test(http://openstax.org/l/vertlinetest)\nIntroduction to Functions(http://openstax.org/l/introtofunction)\nVertical Line Test on Graph(http://openstax.org/l/vertlinegraph)\nOne-to-one Functions(http://openstax.org/l/onetoone)"
	},
	{
		"chunk_id" : "00000626",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphs as One-to-one Functions(http://openstax.org/l/graphonetoone)\n3.1 SECTION EXERCISES\nVerbal\n1. What is the difference 2. What is the difference 3. Why does the vertical line\nbetween a relation and a between the input and the test tell us whether the\nfunction? output of a function? graph of a relation\nrepresents a function?\n200 3  Functions\n4. How can you determine if a 5. Why does the horizontal line\nrelation is a one-to-one test tell us whether the\nfunction? graph of a function is one-\nto-one?"
	},
	{
		"chunk_id" : "00000627",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function? graph of a function is one-\nto-one?\nAlgebraic\nFor the following exercises, determine whether the relation represents a function.\n6. 7.\nFor the following exercises, determine whether the relation represents as a function of\n8. 9. 10.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23. 24. 25.\n26.\nFor the following exercises, evaluate\n27. 28. 29.\n30. 31. 32. Given the function\nevaluate\n33. Given the function 34. Given the function 35. Given the function\nevaluate\n Evaluate  Evaluate"
	},
	{
		"chunk_id" : "00000628",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "evaluate\n Evaluate  Evaluate\n Solve  Solve\n36. Given the function 37. Given the function 38. Given the function\n Evaluate  Evaluate  Evaluate\n Solve  Solve  Solve\nAccess for free at openstax.org\n3.1  Functions and Function Notation 201\n39. Consider the relationship\n Write the relationship\nas a function\n Evaluate\n Solve\nGraphical\nFor the following exercises, use the vertical line test to determine which graphs show relations that are functions.\n40. 41. 42.\n43. 44. 45.\n46. 47. 48."
	},
	{
		"chunk_id" : "00000629",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "40. 41. 42.\n43. 44. 45.\n46. 47. 48.\n202 3  Functions\n49. 50. 51.\n52. Given the following graph, 53. Given the following graph, 54. Given the following graph,\n Evaluate  Evaluate  Evaluate\n Solve for  Solve for  Solve for\nFor the following exercises, determine if the given graph is a one-to-one function.\n55. 56. 57.\nAccess for free at openstax.org\n3.1  Functions and Function Notation 203\n58. 59.\nNumeric\nFor the following exercises, determine whether the relation represents a function.\n60. 61. 62."
	},
	{
		"chunk_id" : "00000630",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "60. 61. 62.\nFor the following exercises, determine if the relation represented in table form represents as a function of\n63. 64. 65.\n5 10 15 5 10 15 5 10 10\n3 8 14 3 8 8 3 8 14\nFor the following exercises, use the function represented in the table below.\n0 1 2 3 4 5 6 7 8 9\n74 28 1 53 56 3 36 45 14 47\nTable14\n66. Evaluate 67. Solve\nFor the following exercises, evaluate the function at the values and\n68. 69. 70.\n71. 72. 73.\nFor the following exercises, evaluate the expressions, given functions and\n74. 75."
	},
	{
		"chunk_id" : "00000631",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "74. 75.\n204 3  Functions\nTechnology\nFor the following exercises, graph on the given domain. Determine the corresponding range. Show each graph.\n76. 77. 78.\nFor the following exercises, graph on the given domain. Determine the corresponding range. Show each graph.\n79. 80.\n81.\nFor the following exercises, graph on the given domain. Determine the corresponding range. Show each graph.\n82. 83.\n84.\nFor the following exercises, graph on the given domain. Determine the corresponding range. Show each graph."
	},
	{
		"chunk_id" : "00000632",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "85. 86.\n87.\nReal-World Applications\n88. The amount of garbage, 89. The number of cubic yards 90. Let be the number of\nproduced by a city with of dirt, needed to cover ducks in a lake years after\npopulation is given by a garden with area 1990. Explain the meaning\nis measured square feet is given by of each statement:\nin tons per week, and is\n\nmeasured in thousands of\npeople.  A garden with area \n5000 ft2requires 50 yd3of\n The town of Tola has a dirt. Express this"
	},
	{
		"chunk_id" : "00000633",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The town of Tola has a dirt. Express this\npopulation of 40,000 and information in terms of the\nproduces 13 tons of function\ngarbage each week.  Explain the meaning of\nExpress this information in the statement\nterms of the function\n Explain the meaning of\nthe statement\n91. Let be the height 92. Show that the function\nabove ground, in feet, of a is\nrocket seconds after notone-to-one.\nlaunching. Explain the\nmeaning of each\nstatement:\n\n\nAccess for free at openstax.org\n3.2  Domain and Range 205"
	},
	{
		"chunk_id" : "00000634",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.2  Domain and Range 205\n3.2 Domain and Range\nLearning Objectives\nIn this section, you will:\nFind the domain of a function defined by an equation.\nGraph piecewise-defined functions.\nHorror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting\nlocations, and special effects are included, however, studios count on even more viewership to be successful. Consider"
	},
	{
		"chunk_id" : "00000635",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "five major thriller/horror entries from the early 2000sI am Legend,Hannibal,The Ring,The Grudge, andThe Conjuring.\nFigure 1shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales\nfor horror movies in general by year. Notice that we can use the data to create a function of the amount each movie\nearned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify"
	},
	{
		"chunk_id" : "00000636",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "different independent and dependent variables, and we can analyze the data and the functions to determine thedomain\nand range. In this section, we will investigate methods for determining the domain and range of functions such as these.\nFigure1 Based on data compiled by www.the-numbers.com.3\nFinding the Domain of a Function Defined by an Equation\nInFunctions and Function Notation, we were introduced to the concepts ofdomain and range. In this section, we will"
	},
	{
		"chunk_id" : "00000637",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges,\nwe need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in\nthe horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot\ninclude any input value that leads us to take an even root of a negative number if the domain and range consist of real"
	},
	{
		"chunk_id" : "00000638",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to\ndivide by 0.\nWe can visualize the domain as a holding area that contains raw materials for a function machine and the range\nas another holding area for the machines products. SeeFigure 2.\nFigure2\nWe can write thedomain and rangeininterval notation, which uses values within brackets to describe a set of"
	},
	{
		"chunk_id" : "00000639",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to\nindicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to\nspend, they would need to express the interval that is more than 0 and less than or equal to 100 and write We\nwill discuss interval notation in greater detail later."
	},
	{
		"chunk_id" : "00000640",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lets turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain\n3 The Numbers: Where Data and the Movie Business Meet. Box Office History for Horror Movies. http://www.the-numbers.com/market/\ngenre/Horror. Accessed 3/24/2014\n206 3  Functions\nof such functions involves remembering three different forms. First, if the function has no denominator or an odd root,"
	},
	{
		"chunk_id" : "00000641",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "consider whether the domain could be all real numbers. Second, if there is a denominator in the functions equation,\nexclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding\nvalues that would make the radicand negative.\nBefore we begin, let us review the conventions of interval notation:\n The smallest number from the interval is written first.\n The largest number in the interval is written second, following a comma."
	},
	{
		"chunk_id" : "00000642",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.\n Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.\nSeeFigure 3for a summary of interval notation.\nFigure3\nEXAMPLE1\nFinding the Domain of a Function as a Set of Ordered Pairs\nFind thedomainof the following function: .\nAccess for free at openstax.org\n3.2  Domain and Range 207\nSolution"
	},
	{
		"chunk_id" : "00000643",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.2  Domain and Range 207\nSolution\nFirst identify the input values. The input value is the first coordinate in anordered pair. There are no restrictions, as the\nordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.\nTRY IT #1 Find the domain of the function:\n...\nHOW TO\nGiven a function written in equation form, find the domain.\n1. Identify the input values.\n2. Identify any restrictions on the input and exclude those values from the domain."
	},
	{
		"chunk_id" : "00000644",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Write the domain in interval form, if possible.\nEXAMPLE2\nFinding the Domain of a Function\nFind the domain of the function\nSolution\nThe input value, shown by the variable in the equation, is squared and then the result is lowered by one. Any real\nnumber may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The\ndomain is the set of real numbers.\nIn interval form, the domain of is  \nTRY IT #2 Find the domain of the function:\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000645",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a function written in an equation form that includes a fraction, find the domain.\n1. Identify the input values.\n2. Identify any restrictions on the input. If there is a denominator in the functions formula, set the denominator\nequal to zero and solve for . If the functions formula contains an even root, set the radicand greater than or\nequal to 0, and then solve.\n3. Write the domain in interval form, making sure to exclude any restricted values from the domain.\nEXAMPLE3"
	},
	{
		"chunk_id" : "00000646",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nFinding the Domain of a Function Involving a Denominator\nFind thedomainof the function\nSolution\nWhen there is a denominator, we want to include only values of the input that do not force the denominator to be zero.\nSo, we will set the denominator equal to 0 and solve for\n208 3  Functions\nNow, we will exclude 2 from the domain. The answers are all real numbers where or as shown inFigure 4.\nWe can use a symbol known as the union, to combine the two sets. In interval notation, we write the solution:"
	},
	{
		"chunk_id" : "00000647",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "\nFigure4\nTRY IT #3 Find the domain of the function:\n...\nHOW TO\nGiven a function written in equation form including an even root, find the domain.\n1. Identify the input values.\n2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the\nradicand greater than or equal to zero and solve for\n3. The solution(s) are the domain of the function. If possible, write the answer in interval form.\nEXAMPLE4\nFinding the Domain of a Function with an Even Root"
	},
	{
		"chunk_id" : "00000648",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find thedomainof the function\nSolution\nWhen there is an even root in the formula, we exclude any real numbers that result in a negative number in the\nradicand.\nSet the radicand greater than or equal to zero and solve for\nNow, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal\nto or \nTRY IT #4 Find the domain of the function\nQ&A Can there be functions in which the domain and range do not intersect at all?"
	},
	{
		"chunk_id" : "00000649",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Yes. For example, the function has the set of all positive real numbers as its domain but the\nset of all negative real numbers as its range. As a more extreme example, a functions inputs and outputs\nAccess for free at openstax.org\n3.2  Domain and Range 209\ncan be completely different categories (for example, names of weekdays as inputs and numbers as\noutputs, as on an attendance chart), in such cases the domain and range have no elements in common.\nUsing Notations to Specify Domain and Range"
	},
	{
		"chunk_id" : "00000650",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Notations to Specify Domain and Range\nIn the previous examples, we used inequalities and lists to describe the domain of functions. We can also use\ninequalities, or other statements that might define sets of values or data, to describe the behavior of the variable inset-\nbuilder notation. For example, describes the behavior of in set-builder notation. The braces are\nread as the set of, and the vertical bar | is read as such that, so we would read as the set of"
	},
	{
		"chunk_id" : "00000651",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-values such that 10 is less than or equal to and is less than 30.\nFigure 5compares inequality notation, set-builder notation, and interval notation.\nFigure5\nTo combine two intervals using inequality notation or set-builder notation, we use the word or. As we saw in earlier\nexamples, we use the union symbol, to combine two unconnected intervals. For example, the union of the sets\nand is the set It is the set of all elements that belong to oneorthe other (or both) of the"
	},
	{
		"chunk_id" : "00000652",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending\norder of numerical value. If the original two sets have some elements in common, those elements should be listed only\nonce in the union set. For sets of real numbers on intervals, another example of a union is\n \nSet-Builder Notation and Interval Notation\nSet-builder notationis a method of specifying a set of elements that satisfy a certain condition. It takes the form"
	},
	{
		"chunk_id" : "00000653",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "which is read as, the set of all such that the statement about is true. For example,\n210 3  Functions\nInterval notationis a way of describing sets that include all real numbers between a lower limit that may or may not\nbe included and an upper limit that may or may not be included. The endpoint values are listed between brackets or\nparentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For\nexample,\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000654",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "example,\n...\nHOW TO\nGiven a line graph, describe the set of values using interval notation.\n1. Identify the intervals to be included in the set by determining where the heavy line overlays the real line.\n2. At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each\nexcluded end value (open dot).\n3. At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each\nexcluded end value (open dot)."
	},
	{
		"chunk_id" : "00000655",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "excluded end value (open dot).\n4. Use the union symbol to combine all intervals into one set.\nEXAMPLE5\nDescribing Sets on the Real-Number Line\nDescribe the intervals of values shown inFigure 6using inequality notation, set-builder notation, and interval notation.\nFigure6\nSolution\nTo describe the values, included in the intervals shown, we would say,  is a real number greater than or equal to 1\nand less than or equal to 3, or a real number greater than 5.\nInequality\nSet-builder notation"
	},
	{
		"chunk_id" : "00000656",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Inequality\nSet-builder notation\nInterval notation \nRemember that, when writing or reading interval notation, using a square bracket means the boundary is included in the\nset. Using a parenthesis means the boundary is not included in the set.\nTRY IT #5 GivenFigure 7, specify the graphed set in\n words  set-builder notation  interval notation\nFigure7\nFinding Domain and Range from Graphs\nAnother way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of"
	},
	{
		"chunk_id" : "00000657",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "possible input values, the domain of a graph consists of all the input values shown on thex-axis. The range is the set of\npossible output values, which are shown on they-axis. Keep in mind that if the graph continues beyond the portion of\nthe graph we can see, the domain and range may be greater than the visible values. SeeFigure 8.\nAccess for free at openstax.org\n3.2  Domain and Range 211\nFigure8\nWe can observe that the graph extends horizontally from to the right without bound, so the domain is  The"
	},
	{
		"chunk_id" : "00000658",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertical extent of the graph is all range values and below, so the range is Note that the domain and range are\nalways written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the\ntop of the graph for range.\nEXAMPLE6\nFinding Domain and Range from a Graph\nFind the domain and range of the function whose graph is shown inFigure 9.\nFigure9\nSolution\nWe can observe that the horizontal extent of the graph is 3 to 1, so the domain of is"
	},
	{
		"chunk_id" : "00000659",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The vertical extent of the graph is 0 to 4, so the range is SeeFigure 10.\nFigure10\n212 3  Functions\nEXAMPLE7\nFinding Domain and Range from a Graph of Oil Production\nFind the domain and range of the function whose graph is shown inFigure 11.\nFigure11 (credit: modification of work by the U.S. Energy Information Administration)4\nSolution\nThe input quantity along the horizontal axis is years, which we represent with the variable for time. The output"
	},
	{
		"chunk_id" : "00000660",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quantity is thousands of barrels of oil per day, which we represent with the variable for barrels. The graph may\ncontinue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can\ndetermine the domain as and the range as approximately\nIn interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we\napproximate the smallest and largest values since they do not fall exactly on the grid lines."
	},
	{
		"chunk_id" : "00000661",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #6 GivenFigure 12, identify the domain and range using interval notation.\nFigure12\nQ&A Can a functions domain and range be the same?\nYes. For example, the domain and range of the cube root function are both the set of all real numbers.\nFinding Domains and Ranges of the Toolkit Functions\nWe will now return to our set of toolkit functions to determine the domain and range of each.\n4 http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000662",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.2  Domain and Range 213\nFigure13 For theconstant function the domain consists of all real numbers; there are no restrictions on the\ninput. The only output value is the constant so the range is the set that contains this single element. In interval\nnotation, this is written as the interval that both begins and ends with\nFigure14 For theidentity function there is no restriction on Both the domain and range are the set of all\nreal numbers."
	},
	{
		"chunk_id" : "00000663",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "real numbers.\nFigure15 For theabsolute value function there is no restriction on However, because absolute value is\ndefined as a distance from 0, the output can only be greater than or equal to 0.\nFigure16 For thequadratic function the domain is all real numbers since the horizontal extent of the\ngraph is the whole real number line. Because the graph does not include any negative values for the range, the range is\nonly nonnegative real numbers.\n214 3  Functions"
	},
	{
		"chunk_id" : "00000664",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "only nonnegative real numbers.\n214 3  Functions\nFigure17 For thecubic function the domain is all real numbers because the horizontal extent of the graph\nis the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all\nreal numbers.\nFigure18 For thereciprocal function we cannot divide by 0, so we must exclude 0 from the domain.\nFurther, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also"
	},
	{
		"chunk_id" : "00000665",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "write the set of all real numbers that are not zero.\nFigure19 For thereciprocal squared function we cannot divide by so we must exclude from the\ndomain. There is also no that can give an output of 0, so 0 is excluded from the range as well. Note that the output of\nthis function is always positive due to the square in the denominator, so the range includes only positive numbers.\nFigure20 For thesquare root function we cannot take the square root of a negative real number, so the"
	},
	{
		"chunk_id" : "00000666",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number\nis defined to be positive, even though the square of the negative number also gives us\nAccess for free at openstax.org\n3.2  Domain and Range 215\nFigure21 For thecube root function the domain and range include all real numbers. Note that there is\nno problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is\nan odd function).\n..."
	},
	{
		"chunk_id" : "00000667",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "an odd function).\n...\nHOW TO\nGiven the formula for a function, determine the domain and range.\n1. Exclude from the domain any input values that result in division by zero.\n2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.\n3. Use the valid input values to determine the range of the output values.\n4. Look at the function graph and table values to confirm the actual function behavior.\nEXAMPLE8\nFinding the Domain and Range Using Toolkit Functions"
	},
	{
		"chunk_id" : "00000668",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain and range of\nSolution\nThere are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.\nThe domain is   and the range is also  \nEXAMPLE9\nFinding the Domain and Range\nFind the domain and range of\nSolution\nWe cannot evaluate the function at because division by zero is undefined. The domain is  \nBecause the function is never zero, we exclude 0 from the range. The range is  \nEXAMPLE10\nFinding the Domain and Range\nFind the domain and range of"
	},
	{
		"chunk_id" : "00000669",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain and range of\nSolution\nWe cannot take the square root of a negative number, so the value inside the radical must be nonnegative.\nThe domain of is \n216 3  Functions\nWe then find the range. We know that and the function value increases as increases without any upper\nlimit. We conclude that the range of is \nAnalysis\nFigure 22represents the function\nFigure22\nTRY IT #7 Find the domain and range of\nGraphing Piecewise-Defined Functions"
	},
	{
		"chunk_id" : "00000670",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Piecewise-Defined Functions\nSometimes, we come across a function that requires more than one formula in order to obtain the given output. For\nexample, in the toolkit functions, we introduced the absolute value function With a domain of all real\nnumbers and a range of values greater than or equal to 0,absolute valuecan be defined as themagnitude, ormodulus,\nof a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the"
	},
	{
		"chunk_id" : "00000671",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output to be greater than or equal to 0.\nIf we input 0, or a positive value, the output is the same as the input.\nIf we input a negative value, the output is the opposite of the input.\nBecause this requires two different processes or pieces, the absolute value function is an example of a piecewise\nfunction. Apiecewise functionis a function in which more than one formula is used to define the output over different\npieces of the domain."
	},
	{
		"chunk_id" : "00000672",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "pieces of the domain.\nWe use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses\ncertain boundaries. For example, we often encounter situations in business for which the cost per piece of a certain\nitem is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of\npiecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and"
	},
	{
		"chunk_id" : "00000673",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "any additional income is taxed at 20%. The tax on a total income would be if and\nif\nPiecewise Function\nApiecewise functionis a function in which more than one formula is used to define the output. Each formula has its\nown domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:\nIn piecewise notation, the absolute value function is\nAccess for free at openstax.org\n3.2  Domain and Range 217\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000674",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.2  Domain and Range 217\n...\nHOW TO\nGiven a piecewise function, write the formula and identify the domain for each interval.\n1. Identify the intervals for which different rules apply.\n2. Determine formulas that describe how to calculate an output from an input in each interval.\n3. Use braces and if-statements to write the function.\nEXAMPLE11\nWriting a Piecewise Function\nA museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or"
	},
	{
		"chunk_id" : "00000675",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "more people. Write afunctionrelating the number of people, to the cost,\nSolution\nTwo different formulas will be needed. Forn-values under 10, For values of that are 10 or greater,\nAnalysis\nThe function is represented inFigure 23. The graph is a diagonal line from to and a constant after that. In\nthis example, the two formulas agree at the meeting point where but not all piecewise functions have this\nproperty.\nFigure23\nEXAMPLE12\nWorking with a Piecewise Function"
	},
	{
		"chunk_id" : "00000676",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE12\nWorking with a Piecewise Function\nA cell phone company uses the function below to determine the cost, in dollars for gigabytes of data transfer.\nFind the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.\nSolution\nTo find the cost of using 1.5 gigabytes of data, we first look to see which part of the domain our input falls in.\nBecause 1.5 is less than 2, we use the first formula."
	},
	{
		"chunk_id" : "00000677",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To find the cost of using 4 gigabytes of data, we see that our input of 4 is greater than 2, so we use the second\nformula.\n218 3  Functions\nAnalysis\nThe function is represented inFigure 24. We can see where the function changes from a constant to a shifted and\nstretched identity at We plot the graphs for the different formulas on a common set of axes, making sure each\nformula is applied on its proper domain.\nFigure24\n...\nHOW TO\nGiven a piecewise function, sketch a graph."
	},
	{
		"chunk_id" : "00000678",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given a piecewise function, sketch a graph.\n1. Indicate on thex-axis the boundaries defined by the intervals on each piece of the domain.\n2. For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece.\nDo not graph two functions over one interval because it would violate the criteria of a function.\nEXAMPLE13\nGraphing a Piecewise Function\nSketch a graph of the function.\nSolution"
	},
	{
		"chunk_id" : "00000679",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketch a graph of the function.\nSolution\nEach of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine\ngraphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw\nopen circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a\nclosed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality."
	},
	{
		"chunk_id" : "00000680",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 25shows the three components of the piecewise function graphed on separate coordinate systems.\nAccess for free at openstax.org\n3.2  Domain and Range 219\nFigure25 (a) (b) (c)\nNow that we have sketched each piece individually, we combine them in the same coordinate plane. SeeFigure 26.\nFigure26\nAnalysis\nNote that the graph does pass the vertical line test even at and because the points and are not\npart of the graph of the function, though and are.\nTRY IT #8 Graph the following piecewise function."
	},
	{
		"chunk_id" : "00000681",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #8 Graph the following piecewise function.\nQ&A Can more than one formula from a piecewise function be applied to a value in the domain?\nNo. Each value corresponds to one equation in a piecewise formula.\nMEDIA\nAccess these online resources for additional instruction and practice with domain and range.\nDomain and Range of Square Root Functions(http://openstax.org/l/domainsqroot)\nDetermining Domain and Range(http://openstax.org/l/determinedomain)"
	},
	{
		"chunk_id" : "00000682",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find Domain and Range Given the Graph(http://openstax.org/l/drgraph)\nFind Domain and Range Given a Table(http://openstax.org/l/drtable)\nFind Domain and Range Given Points on a Coordinate Plane(http://openstax.org/l/drcoordinate)\n220 3  Functions\n3.2 SECTION EXERCISES\nVerbal\n1. Why does the domain differ 2. How do we determine the 3. Explain why the domain of\nfor different functions? domain of a function is different from\ndefined by an equation? the domain of"
	},
	{
		"chunk_id" : "00000683",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "defined by an equation? the domain of\n4. When describing sets of 5. How do you graph a\nnumbers using interval piecewise function?\nnotation, when do you use a\nparenthesis and when do\nyou use a bracket?\nAlgebraic\nFor the following exercises, find the domain of each function using interval notation.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23.\n24. 25. 26. Find the domain of the\nfunction\nby:\n using algebra.\n graphing the function\nin the radicand and\ndetermining intervals on"
	},
	{
		"chunk_id" : "00000684",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "in the radicand and\ndetermining intervals on\nthex-axis for which the\nradicand is nonnegative.\nAccess for free at openstax.org\n3.2  Domain and Range 221\nGraphical\nFor the following exercises, write the domain and range of each function using interval notation.\n27. 28. 29.\n30. 31. 32.\n33. 34. 35.\n36. 37.\n222 3  Functions\nFor the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.\n38. 39. 40.\n41. 42. 43.\n44. 45.\nNumeric"
	},
	{
		"chunk_id" : "00000685",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "38. 39. 40.\n41. 42. 43.\n44. 45.\nNumeric\nFor the following exercises, given each function evaluate and\n46. 47. 48.\nFor the following exercises, given each function evaluate and\n49. 50. 51.\nFor the following exercises, write the domain for the piecewise function in interval notation.\n52. 53.\n54.\nTechnology\n55. Graph on the viewing window 56. Graph on the viewing window\nand Determine the and Determine the corresponding\ncorresponding range for the viewing window. range for the viewing window. Show the graphs."
	},
	{
		"chunk_id" : "00000686",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Show the graphs.\nExtension\n57. Suppose the range of a 58. Create a function in which 59. Create a function in which\nfunction is What the range is all the domain is\nis the range of nonnegative real numbers.\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 223\nReal-World Applications\n60. The height of a projectile 61. The cost in dollars of\nis a function of the time it making items is given by\nis in the air. The height in the function\nfeet for seconds is given\nby the function"
	},
	{
		"chunk_id" : "00000687",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "feet for seconds is given\nby the function\n The fixed cost is\nWhat is\ndetermined when zero\nthe domain of the\nitems are produced. Find\nfunction? What does the\nthe fixed cost for this item.\ndomain mean in the\n What is the cost of\ncontext of the problem?\nmaking 25 items?\n Suppose the maximum\ncost allowed is $1500. What\nare the domain and range\nof the cost function,\n3.3 Rates of Change and Behavior of Graphs\nLearning Objectives\nIn this section, you will:\nFind the average rate of change of a function."
	},
	{
		"chunk_id" : "00000688",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the average rate of change of a function.\nUse a graph to determine where a function is increasing, decreasing, or constant.\nUse a graph to locate local maxima and local minima.\nUse a graph to locate the absolute maximum and absolute minimum.\nGasoline costs have experienced some wild fluctuations over the last several decades.Table 15 lists the average cost, in\ndollars, of a gallon of gasoline for the years 20052012. The cost of gasoline can be considered as a function of year."
	},
	{
		"chunk_id" : "00000689",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2005 2006 2007 2008 2009 2010 2011 2012\n2.31 2.62 2.84 3.30 2.41 2.84 3.58 3.68\nTable1\nIf we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost\nper gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to\nlook at how much the price changedper year. In this section, we will investigate changes such as these.\nFinding the Average Rate of Change of a Function"
	},
	{
		"chunk_id" : "00000690",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Average Rate of Change of a Function\nThe price change per year is arate of changebecause it describes how an output quantity changes relative to the\nchange in the input quantity. We can see that the price of gasoline inTable 1did not change by the same amount each\nyear, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the\naverage rate of changeover the specified period of time. To find the average rate of change, we divide the change in"
	},
	{
		"chunk_id" : "00000691",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the output value by the change in the input value.\nThe Greek letter (delta) signifies the change in a quantity; we read the ratio as delta-yover delta-x or the change in\ndivided by the change in  Occasionally we write instead of which still represents the change in the functions\n5 http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014.\n224 3  Functions\noutput value resulting from a change to its input value. It does not mean we are changing the function into some other"
	},
	{
		"chunk_id" : "00000692",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\nIn our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was\nOn average, the price of gas increased by about 19.6 each year.\nOther examples of rates of change include:\n A population of rats increasing by 40 rats per week\n A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)\n A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)"
	},
	{
		"chunk_id" : "00000693",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage\n The amount of money in a college account decreasing by $4,000 per quarter\nRate of Change\nA rate of change describes how an output quantity changes relative to the change in the input quantity. The units on\na rate of change are output units per input units.\nThe average rate of change between two input values is the total change of the function values (output values)"
	},
	{
		"chunk_id" : "00000694",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "divided by the change in the input values.\n...\nHOW TO\nGiven the value of a function at different points, calculate the average rate of change of a function for the\ninterval between two values and\n1. Calculate the difference\n2. Calculate the difference\n3. Find the ratio\nEXAMPLE1\nComputing an Average Rate of Change\nUsing the data inTable 1, find the average rate of change of the price of gasoline between 2007 and 2009.\nSolution"
	},
	{
		"chunk_id" : "00000695",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nIn 2007, the price of gasoline was $2.84. In 2009, the cost was $2.41. The average rate of change is\nAnalysis\nNote that a decrease is expressed by a negative change or negative increase. A rate of change is negative when the\noutput decreases as the input increases or when the output increases as the input decreases.\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 225\nTRY IT #1 Using the data inTable 1, find the average rate of change between 2005 and 2010.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00000696",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nComputing Average Rate of Change from a Graph\nGiven the function shown inFigure 1, find the average rate of change on the interval\nFigure1\nSolution\nAt Figure 2shows At the graph shows\nFigure2\nThe horizontal change is shown by the red arrow, and the vertical change is shown by the turquoise\narrow. The average rate of change is shown by the slope of the orange line segment. The output changes by 3 while the\ninput changes by 3, giving an average rate of change of\nAnalysis"
	},
	{
		"chunk_id" : "00000697",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNote that the order we choose is very important. If, for example, we use we will not get the correct answer.\nDecide which point will be 1 and which point will be 2, and keep the coordinates fixed as and\nEXAMPLE3\nComputing Average Rate of Change from a Table\nAfter picking up a friend who lives 10 miles away and leaving on a trip, Anna records her distance from home over time.\nThe values are shown inTable 2. Find her average speed over the first 6 hours.\n226 3  Functions\nt(hours) 0 1 2 3 4 5 6 7"
	},
	{
		"chunk_id" : "00000698",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "226 3  Functions\nt(hours) 0 1 2 3 4 5 6 7\nD(t) (miles) 10 55 90 153 214 240 292 300\nTable2\nSolution\nHere, the average speed is the average rate of change. She traveled 282 miles in 6 hours.\nThe average speed is 47 miles per hour.\nAnalysis\nBecause the speed is not constant, the average speed depends on the interval chosen. For the interval [2,3], the average\nspeed is 63 miles per hour.\nEXAMPLE4\nComputing Average Rate of Change for a Function Expressed as a Formula"
	},
	{
		"chunk_id" : "00000699",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Compute the average rate of change of on the interval\nSolution\nWe can start by computing the function values at eachendpointof the interval.\nNow we compute the average rate of change.\nTRY IT #2 Find the average rate of change of on the interval\nEXAMPLE5\nFinding the Average Rate of Change of a Force\nTheelectrostatic force measured in newtons, between two charged particles can be related to the distance between"
	},
	{
		"chunk_id" : "00000700",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the particles in centimeters, by the formula Find the average rate of change of force if the distance\nbetween the particles is increased from 2 cm to 6 cm.\nSolution\nWe are computing the average rate of change of on the interval\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 227\nThe average rate of change is newton per centimeter.\nEXAMPLE6\nFinding an Average Rate of Change as an Expression"
	},
	{
		"chunk_id" : "00000701",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the average rate of change of on the interval The answer will be an expression involving\nin simplest form.\nSolution\nWe use the average rate of change formula.\nThis result tells us the average rate of change in terms of between and any other point For example, on the\ninterval the average rate of change would be\nTRY IT #3 Find the average rate of change of on the interval in simplest forms in\nterms\nof\nUsing a Graph to Determine Where a Function is Increasing, Decreasing, or\nConstant"
	},
	{
		"chunk_id" : "00000702",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Constant\nAs part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.\nWe say that a function is increasing on an interval if the function values increase as the input values increase within that\ninterval. Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over\nthat interval. The average rate of change of an increasing function is positive, and the average rate of change of a"
	},
	{
		"chunk_id" : "00000703",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decreasing function is negative.Figure 3shows examples of increasing and decreasing intervals on a function.\n228 3  Functions\nFigure3 The function is increasing on   and is decreasing on\nWhile some functions are increasing (or decreasing) over their entire domain, many others are not. A value of the input\nwhere a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable"
	},
	{
		"chunk_id" : "00000704",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increases) is the location of alocal maximum. The function value at that point is the local maximum. If a function has\nmore than one, we say it has local maxima. Similarly, a value of the input where a function changes from decreasing to\nincreasing as the input variable increases is the location of alocal minimum. The function value at that point is the local\nminimum. The plural form is local minima. Together, local maxima and minima are calledlocal extrema, or local"
	},
	{
		"chunk_id" : "00000705",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "extreme values, of the function. (The singular form is extremum.) Often, the termlocalis replaced by the termrelative.\nIn this text, we will use the termlocal.\nClearly, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither\nincreasing nor decreasing at extrema. Note that we have to speak oflocalextrema, because any given local extremum as\ndefined here is not necessarily the highest maximum or lowest minimum in the functions entire domain."
	},
	{
		"chunk_id" : "00000706",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the function whose graph is shown inFigure 4, the local maximum is 16, and it occurs at The local minimum\nis and it occurs at\nFigure4\nTo locate the local maxima and minima from a graph, we need to observe the graph to determine where the graph\nattains its highest and lowest points, respectively, within an open interval. Like the summit of a roller coaster, the graph\nof a function is higher at a local maximum than at nearby points on both sides. The graph will also be lower at a local"
	},
	{
		"chunk_id" : "00000707",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minimum than at neighboring points.Figure 5illustrates these ideas for a local maximum.\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 229\nFigure5 Definition of a local maximum\nThese observations lead us to a formal definition of local extrema.\nLocal Minima and Local Maxima\nA function is anincreasing functionon an open interval if for any two input values and in the\ngiven interval where"
	},
	{
		"chunk_id" : "00000708",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given interval where\nA function is adecreasing functionon an open interval if for any two input values and in the given\ninterval where\nA function has a local maximum at if there exists an interval with such that, for any in the\ninterval Likewise, has a local minimum at if there exists an interval with\nsuch that, for any in the interval\nEXAMPLE7\nFinding Increasing and Decreasing Intervals on a Graph\nGiven the function inFigure 6, identify the intervals on which the function appears to be increasing.\nFigure6"
	},
	{
		"chunk_id" : "00000709",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nSolution\nWe see that the function is not constant on any interval. The function is increasing where it slants upward as we move to\nthe right and decreasing where it slants downward as we move to the right. The function appears to be increasing from\nto and from on.\nIninterval notation, we would say the function appears to be increasing on the interval (1,3) and the interval \nAnalysis"
	},
	{
		"chunk_id" : "00000710",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNotice in this example that we used open intervals (intervals that do not include the endpoints), because the function is\nneither increasing nor decreasing at , , and . These points are the local extrema (two minima and a\n230 3  Functions\nmaximum).\nEXAMPLE8\nFinding Local Extrema from a Graph\nGraph the function Then use the graph to estimate the local extrema of the function and to determine\nthe intervals on which the function is increasing.\nSolution"
	},
	{
		"chunk_id" : "00000711",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUsing technology, we find that the graph of the function looks like that inFigure 7. It appears there is a low point, or\nlocal minimum, between and and a mirror-image high point, or local maximum, somewhere between\nand\nFigure7\nAnalysis\nMost graphing calculators and graphing utilities can estimate the location of maxima and minima.Figure 8provides\nscreen images from two different technologies, showing the estimate for the local maximum and minimum.\nFigure8"
	},
	{
		"chunk_id" : "00000712",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nBased on these estimates, the function is increasing on the interval  and  Notice that, while we\nexpect the extrema to be symmetric, the two different technologies agree only up to four decimals due to the differing\napproximation algorithms used by each. (The exact location of the extrema is at but determining this requires\ncalculus.)\nTRY IT #4 Graph the function to estimate the local extrema of the function.\nUse these to determine the intervals on which the function is increasing and decreasing."
	},
	{
		"chunk_id" : "00000713",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 231\nEXAMPLE9\nFinding Local Maxima and Minima from a Graph\nFor the function whose graph is shown inFigure 9, find all local maxima and minima.\nFigure9\nSolution\nObserve the graph of The graph attains a local maximum at because it is the highest point in an open interval\naround The local maximum is the -coordinate at which is\nThe graph attains a local minimum at because it is the lowest point in an open interval around The local"
	},
	{
		"chunk_id" : "00000714",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minimum is they-coordinate at which is\nAnalyzing the Toolkit Functions for Increasing or Decreasing Intervals\nWe will now return to our toolkit functions and discuss their graphical behavior inFigure 10,Figure 11, andFigure 12.\nFigure10\n232 3  Functions\nFigure11\nFigure12\nUse A Graph to Locate the Absolute Maximum and Absolute Minimum\nThere is a difference between locating the highest and lowest points on a graph in a region around an open interval"
	},
	{
		"chunk_id" : "00000715",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(locally) and locating the highest and lowest points on the graph for the entire domain. The coordinates (output) at\nthe highest and lowest points are called theabsolute maximumandabsolute minimum, respectively.\nTo locate absolute maxima and minima from a graph, we need to observe the graph to determine where the graph\nattains it highest and lowest points on the domain of the function. SeeFigure 13.\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 233\nFigure13"
	},
	{
		"chunk_id" : "00000716",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure13\nNot every function has an absolute maximum or minimum value. The toolkit function is one such function.\nAbsolute Maxima and Minima\nTheabsolute maximumof at is where for all in the domain of\nTheabsolute minimumof at is where for all in the domain of\nEXAMPLE10\nFinding Absolute Maxima and Minima from a Graph\nFor the function shown inFigure 14, find all absolute maxima and minima.\nFigure14\nSolution\nObserve the graph of The graph attains an absolute maximum in two locations, and because at these"
	},
	{
		"chunk_id" : "00000717",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "locations, the graph attains its highest point on the domain of the function. The absolute maximum is they-coordinate\nat and which is\nThe graph attains an absolute minimum at because it is the lowest point on the domain of the functions graph.\nThe absolute minimum is they-coordinate at which is\nMEDIA\nAccess this online resource for additional instruction and practice with rates of change.\n234 3  Functions\nAverage Rate of Change(http://openstax.org/l/aroc)\n3.3 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000718",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.3 SECTION EXERCISES\nVerbal\n1. Can the average rate of 2. If a function is increasing 3. How are the absolute\nchange of a function be on and decreasing on maximum and minimum\nconstant? then what can be said similar to and different from\nabout the local extremum of the local extrema?\non\n4. How does the graph of the\nabsolute value function\ncompare to the graph of the\nquadratic function,\nin terms of increasing and\ndecreasing intervals?\nAlgebraic"
	},
	{
		"chunk_id" : "00000719",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decreasing intervals?\nAlgebraic\nFor the following exercises, find the average rate of change of each function on the interval specified for real numbers\nor in simplest form.\n5. on 6. on 7. on\n8. on 9. on 10. on\n11. on 12. on 13. on\n14. on 15. given\non\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 235\nGraphical\nFor the following exercises, consider the graph of shown inFigure 15.\nFigure15\n16. Estimate the average rate 17. Estimate the average rate"
	},
	{
		"chunk_id" : "00000720",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of change from to of change from to\nFor the following exercises, use the graph of each function to estimate the intervals on which the function is increasing\nor decreasing.\n18. 19. 20.\n21.\n236 3  Functions\nFor the following exercises, consider the graph shown inFigure 16.\nFigure16\n22. Estimate the intervals 23. Estimate the point(s) at\nwhere the function is which the graph of has a\nincreasing or decreasing. local maximum or a local\nminimum.\nFor the following exercises, consider the graph inFigure 17."
	},
	{
		"chunk_id" : "00000721",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure17\n24. If the complete graph of 25. If the complete graph of\nthe function is shown, the function is shown,\nestimate the intervals estimate the absolute\nwhere the function is maximum and absolute\nincreasing or decreasing. minimum.\nAccess for free at openstax.org\n3.3  Rates of Change and Behavior of Graphs 237\nNumeric\n26. Table 3gives the annual 27. Table 4gives the population of a town (in\nsales (in millions of dollars) thousands) from 2000 to 2008. What was the"
	},
	{
		"chunk_id" : "00000722",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of a product from 1998 to average rate of change of population (a) between\n2006. What was the 2002 and 2004, and (b) between 2002 and 2006?\naverage rate of change of\nannual sales (a) between Population\nYear\n2001 and 2002, and (b) (thousands)\nbetween 2001 and 2004?\n2000 87\nSales\nYear (millions of 2001 84\ndollars)\n2002 83\n1998 201\n2003 80\n1999 219\n2004 77\n2000 233\n2005 76\n2001 243\n2006 78\n2002 249\n2007 81\n2003 251\n2008 85\n2004 249\nTable4\n2005 243\n2006 233\nTable3"
	},
	{
		"chunk_id" : "00000723",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2008 85\n2004 249\nTable4\n2005 243\n2006 233\nTable3\nFor the following exercises, find the average rate of change of each function on the interval specified.\n28. on 29. on 30. on\n31. on 32. on 33. on\n34. on\nTechnology\nFor the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the\nintervals on which the function is increasing and decreasing.\n35. 36.\n238 3  Functions\n37. 38.\n39. 40.\nExtension"
	},
	{
		"chunk_id" : "00000724",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "238 3  Functions\n37. 38.\n39. 40.\nExtension\n41. The graph of the function is shown inFigure 18. 42. Let Find a\nnumber such that the\naverage rate of change of\nthe function on the\ninterval is\nFigure18\nBased on the calculator screen shot, the point\nis which of the following?\n a relative (local) maximum of the function\n the vertex of the function\n the absolute maximum of the function\n a zero of the function\n43. Let . Find the\nnumber such that the\naverage rate of change of\non the interval is"
	},
	{
		"chunk_id" : "00000725",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "average rate of change of\non the interval is\nReal-World Applications\n44. At the start of a trip, the 45. A driver of a car stopped at 46. Near the surface of the\nodometer on a car read a gas station to fill up their moon, the distance that an\n21,395. At the end of the gas tank. They looked at object falls is a function of\ntrip, 13.5 hours later, the their watch, and the time time. It is given by\nodometer read 22,125. read exactly 3:40 p.m. At where is"
	},
	{
		"chunk_id" : "00000726",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Assume the scale on the this time, they started in seconds and is in\nodometer is in miles. What pumping gas into the tank. feet. If an object is dropped\nis the average speed the At exactly 3:44, the tank from a certain height, find\ncar traveled during this was full and the driver the average velocity of the\ntrip? noticed that they had object from to\npumped 10.7 gallons. What\nis the average rate of flow\nof the gasoline into the gas\ntank?\nAccess for free at openstax.org\n3.4  Composition of Functions 239"
	},
	{
		"chunk_id" : "00000727",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.4  Composition of Functions 239\n47. The graph inFigure 19illustrates the decay of a\nradioactive substance over days.\nFigure19\nUse the graph to estimate the average decay rate\nfrom to\n3.4 Composition of Functions\nLearning Objectives\nIn this section, you will:\nCombine functions using algebraic operations.\nCreate a new function by composition of functions.\nEvaluate composite functions.\nFind the domain of a composite function.\nDecompose a composite function into its component functions."
	},
	{
		"chunk_id" : "00000728",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house\nwill depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day\nof the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the\ntemperature depends on the day.\nUsing descriptive variables, we can notate these two functions. The function gives the cost of heating a house"
	},
	{
		"chunk_id" : "00000729",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for a given average daily temperature in degrees Celsius. The function gives the average daily temperature on\nday of the year. For any given day, means that the cost depends on the temperature, which in turns\ndepends on the day of the year. Thus, we can evaluate the cost function at the temperature For example, we could\nevaluate to determine the average daily temperature on the 5th day of the year. Then, we could evaluate thecost\nfunctionat that temperature. We would write"
	},
	{
		"chunk_id" : "00000730",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functionat that temperature. We would write\nBy combining these two relationships into one function, we have performed function composition, which is the focus of\nthis section.\nCombining Functions Using Algebraic Operations\nFunction composition is only one way to combine existing functions. Another way is to carry out the usual algebraic\noperations on functions, such as addition, subtraction, multiplication and division. We do this by performing the"
	},
	{
		"chunk_id" : "00000731",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "operations with the function outputs, defining the result as the output of our new function.\nSuppose we need to add two columns of numbers that represent a husband and wifes separate annual incomes over a\nperiod of years, with the result being their total household income. We want to do this for every year, adding only that\n240 3  Functions\nyears incomes and then collecting all the data in a new column. If is the wifes income and is the husbands"
	},
	{
		"chunk_id" : "00000732",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "income in year and we want to represent the total income, then we can define a new function.\nIf this holds true for every year, then we can focus on the relation between the functions without reference to a year and\nwrite\nJust as for this sum of two functions, we can define difference, product, and ratio functions for any pair of functions that\nhave the same kinds of inputs (not necessarily numbers) and also the same kinds of outputs (which do have to be"
	},
	{
		"chunk_id" : "00000733",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers so that the usual operations of algebra can apply to them, and which also must have the same units or no units\nwhen we add and subtract). In this way, we can think of adding, subtracting, multiplying, and dividing functions.\nFor two functions and with real number outputs, we define new functions and by the\nrelations\nEXAMPLE1\nPerforming Algebraic Operations on Functions\nFind and simplify the functions and given and Are they the same\nfunction?\nSolution"
	},
	{
		"chunk_id" : "00000734",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function?\nSolution\nBegin by writing the general form, and then substitute the given functions.\nNo, the functions are not the same.\nNote: For the condition is necessary because when the denominator is equal to 0, which makes\nthe function undefined.\nTRY IT #1 Find and simplify the functions and\nAre they the same function?\nAccess for free at openstax.org\n3.4  Composition of Functions 241\nCreate a Function by Composition of Functions"
	},
	{
		"chunk_id" : "00000735",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Create a Function by Composition of Functions\nPerforming algebraic operations on functions combines them into a new function, but we can also create functions by\ncomposing functions. When we wanted to compute a heating cost from a day of the year, we created a new function that\ntakes a day as input and yields a cost as output. The process ofcombining functionsso that the output of one function\nbecomes the input of another is known as acomposition of functions.The resulting function is known as acomposite"
	},
	{
		"chunk_id" : "00000736",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function. We represent this combination by the following notation:\nWe read the left-hand side as composed with at and the right-hand side as of of The two sides of the\nequation have the same mathematical meaning and are equal. The open circle symbol is called the composition\noperator. We use this operator mainly when we wish to emphasize the relationship between the functions themselves\nwithout referring to any particular input value. Composition is a binary operation that takes two functions and forms a"
	},
	{
		"chunk_id" : "00000737",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "new function, much as addition or multiplication takes two numbers and gives a new number. However, it is important\nnot to confuse function composition with multiplication because, as we learned above, in most cases\nIt is also important to understand the order of operations in evaluating a composite function. We follow the usual\nconvention with parentheses by starting with the innermost parentheses first, and then working to the outside. In the"
	},
	{
		"chunk_id" : "00000738",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation above, the function takes the input first and yields an output Then the function takes as an\ninput and yields an output\nIn general, and are different functions. In other words, in many cases for all We will also\nsee that sometimes two functions can be composed only in one specific order.\nFor example, if and then\nbut\nThese expressions are not equal for all values of so the two functions are not equal. It is irrelevant that the\nexpressions happen to be equal for the single input value"
	},
	{
		"chunk_id" : "00000739",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the\noutside function. Less formally, the composition has to make sense in terms of inputs and outputs.\nComposition of Functions\nWhen the output of one function is used as the input of another, we call the entire operation a composition of\nfunctions. For any input and functions and this action defines acomposite function, which we write as\nsuch that"
	},
	{
		"chunk_id" : "00000740",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "such that\nThe domain of the composite function is all such that is in the domain of and is in the domain of\nIt is important to realize that the product of functions is not the same as the function composition\nbecause, in general,\n242 3  Functions\nEXAMPLE2\nDetermining whether Composition of Functions is Commutative\nUsing the functions provided, find and Determine whether the composition of the functions is\ncommutative.\nSolution\nLets begin by substituting into\nNow we can substitute into"
	},
	{
		"chunk_id" : "00000741",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now we can substitute into\nWe find that so the operation of function composition is not commutative.\nEXAMPLE3\nInterpreting Composite Functions\nThe function gives the number of calories burned completing sit-ups, and gives the number of sit-ups a person\ncan complete in minutes. Interpret\nSolution\nThe inside expression in the composition is Because the input to thes-function is time, represents 3 minutes,\nand is the number of sit-ups completed in 3 minutes."
	},
	{
		"chunk_id" : "00000742",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using as the input to the function gives us the number of calories burned during the number of sit-ups that can\nbe completed in 3 minutes, or simply the number of calories burned in 3 minutes (by doing sit-ups).\nEXAMPLE4\nInvestigating the Order of Function Composition\nSuppose gives miles that can be driven in hours and gives the gallons of gas used in driving miles. Which of\nthese expressions is meaningful: or\nSolution"
	},
	{
		"chunk_id" : "00000743",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "these expressions is meaningful: or\nSolution\nThe function is a function whose output is the number of miles driven corresponding to the number of hours\ndriven.\nThe function is a function whose output is the number of gallons used corresponding to the number of miles\ndriven. This means:\nThe expression takes miles as the input and a number of gallons as the output. The function requires a\nnumber of hours as the input. Trying to input a number of gallons does not make sense. The expression is\nmeaningless."
	},
	{
		"chunk_id" : "00000744",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "meaningless.\nThe expression takes hours as input and a number of miles driven as the output. The function requires a\nnumber of miles as the input. Using (miles driven) as an input value for where gallons of gas depends on\nmiles driven, does make sense. The expression makes sense, and will yield the number of gallons of gas used,\ndriving a certain number of miles, in hours.\nAccess for free at openstax.org\n3.4  Composition of Functions 243"
	},
	{
		"chunk_id" : "00000745",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.4  Composition of Functions 243\nQ&A Are there any situations where and would both be meaningful or useful expressions?\nYes. For many pure mathematical functions, both compositions make sense, even though they usually\nproduce different new functions. In real-world problems, functions whose inputs and outputs have the\nsame units also may give compositions that are meaningful in either order.\nTRY IT #2 The gravitational force on a planet a distancerfrom the sun is given by the function The"
	},
	{
		"chunk_id" : "00000746",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "acceleration of a planet subjected to any force is given by the function Form a meaningful\ncomposition of these two functions, and explain what it means.\nEvaluating Composite Functions\nOnce we compose a new function from two existing functions, we need to be able to evaluate it for any input in its\ndomain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with"
	},
	{
		"chunk_id" : "00000747",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variables as inputs to functions expressed as formulas. In each case, we evaluate the inner function using the starting\ninput and then use the inner functions output as the input for the outer function.\nEvaluating Composite Functions Using Tables\nWhen working with functions given as tables, we read input and output values from the table entries and always work\nfrom the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the"
	},
	{
		"chunk_id" : "00000748",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input to the outside function.\nEXAMPLE5\nUsing a Table to Evaluate a Composite Function\nUsingTable 1, evaluate and\n1 6 3\n2 8 5\n3 3 2\n4 1 7\nTable1\nSolution\nTo evaluate we start from the inside with the input value 3. We then evaluate the inside expression using\nthe table that defines the function We can then use that result as the input to the function so is\nreplaced by 2 and we get Then, using the table that defines the function we find that"
	},
	{
		"chunk_id" : "00000749",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To evaluate we first evaluate the inside expression using the first table: Then, using the table for\nwe can evaluate\nTable 2shows the composite functions and as tables.\nTable2\n244 3  Functions\n3 2 8 3 2\nTable2\nTRY IT #3 UsingTable 1, evaluate and\nEvaluating Composite Functions Using Graphs\nWhen we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the"
	},
	{
		"chunk_id" : "00000750",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "process we use for evaluating tables. We read the input and output values, but this time, from the and axes of the\ngraphs.\n...\nHOW TO\nGiven a composite function and graphs of its individual functions, evaluate it using the information provided\nby the graphs.\n1. Locate the given input to the inner function on the axis of its graph.\n2. Read off the output of the inner function from the axis of its graph.\n3. Locate the inner function output on the axis of the graph of the outer function."
	},
	{
		"chunk_id" : "00000751",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Read the output of the outer function from the axis of its graph. This is the output of the composite function.\nEXAMPLE6\nUsing a Graph to Evaluate a Composite Function\nUsingFigure 1, evaluate\nFigure1\nSolution\nTo evaluate we start with the inside evaluation. SeeFigure 2.\nAccess for free at openstax.org\n3.4  Composition of Functions 245\nFigure2\nWe evaluate using the graph of finding the input of 1 on the axis and finding the output value of the graph"
	},
	{
		"chunk_id" : "00000752",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "at that input. Here, We use this value as the input to the function\nWe can then evaluate the composite function by looking to the graph of finding the input of 3 on the axis and\nreading the output value of the graph at this input. Here, so\nAnalysis\nFigure 3shows how we can mark the graphs with arrows to trace the path from the input value to the output value.\nFigure3\nTRY IT #4 UsingFigure 1, evaluate\nEvaluating Composite Functions Using Formulas"
	},
	{
		"chunk_id" : "00000753",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Evaluating Composite Functions Using Formulas\nWhen evaluating a composite function where we have either created or been given formulas, the rule of working from\nthe inside out remains the same. The input value to the outer function will be the output of the inner function, which\nmay be a numerical value, a variable name, or a more complicated expression.\nWhile we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that"
	},
	{
		"chunk_id" : "00000754",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will calculate the result of a composition To do this, we will extend our idea of function evaluation. Recall that,\nwhen we evaluate a function like we substitute the value inside the parentheses into the formula wherever\n246 3  Functions\nwe see the input variable.\n...\nHOW TO\nGiven a formula for a composite function, evaluate the function.\n1. Evaluate the inside function using the input value or variable provided.\n2. Use the resulting output as the input to the outside function.\nEXAMPLE7"
	},
	{
		"chunk_id" : "00000755",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nEvaluating a Composition of Functions Expressed as Formulas with a Numerical Input\nGiven and evaluate\nSolution\nBecause the inside expression is we start by evaluating at 1.\nThen so we evaluate at an input of 5.\nAnalysis\nIt makes no difference what the input variables and were called in this problem because we evaluated for specific\nnumerical values.\nTRY IT #5 Given and evaluate\n \nFinding the Domain of a Composite Function"
	},
	{
		"chunk_id" : "00000756",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nFinding the Domain of a Composite Function\nAs we discussed previously, thedomain of a composite functionsuch as is dependent on the domain of and the\ndomain of It is important to know when we can apply a composite function and when we cannot, that is, to know the\ndomain of a function such as Let us assume we know the domains of the functions and separately. If we write\nthe composite function for an input as we can see right away that must be a member of the domain of in"
	},
	{
		"chunk_id" : "00000757",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation.\nHowever, we also see that must be a member of the domain of otherwise the second function evaluation in\ncannot be completed, and the expression is still undefined. Thus the domain of consists of only those\ninputs in the domain of that produce outputs from belonging to the domain of Note that the domain of\ncomposed with is the set of all such that is in the domain of and is in the domain of"
	},
	{
		"chunk_id" : "00000758",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Domain of a Composite Function\nThe domain of a composite function is the set of those inputs in the domain of for which is in the\ndomain of\n...\nHOW TO\nGiven a function composition determine its domain.\nAccess for free at openstax.org\n3.4  Composition of Functions 247\n1. Find the domain of\n2. Find the domain of\n3. Find those inputs in the domain of for which is in the domain of That is, exclude those inputs from\nthe domain of for which is not in the domain of The resulting set is the domain of\nEXAMPLE8"
	},
	{
		"chunk_id" : "00000759",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE8\nFinding the Domain of a Composite Function\nFind the domain of\nSolution\nThe domain of consists of all real numbers except since that input value would cause us to divide by 0.\nLikewise, the domain of consists of all real numbers except 1. So we need to exclude from the domain of that\nvalue of for which\nSo the domain of is the set of all real numbers except and This means that\nWe can write this in interval notation as\n \nEXAMPLE9\nFinding the Domain of a Composite Function Involving Radicals"
	},
	{
		"chunk_id" : "00000760",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain of\nSolution\nBecause we cannot take the square root of a negative number, the domain of is  Now we check the domain\nof the composite function\nFor since the radicand of a square root must be positive. Since square roots\nare positive, or, which gives a domain of .\nAnalysis\nThis example shows that knowledge of the range of functions (specifically the inner function) can also be helpful in"
	},
	{
		"chunk_id" : "00000761",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "finding the domain of a composite function. It also shows that the domain of can contain values that are not in the\ndomain of though they must be in the domain of\nTRY IT #6 Find the domain of\n248 3  Functions\nDecomposing a Composite Function into its Component Functions\nIn some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of\ntwo simpler functions. There may be more than one way todecompose a composite function, so we may choose the"
	},
	{
		"chunk_id" : "00000762",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decomposition that appears to be most expedient.\nEXAMPLE10\nDecomposing a Function\nWrite as the composition of two functions.\nSolution\nWe are looking for two functions, and so To do this, we look for a function inside a function in the\nformula for As one possibility, we might notice that the expression is the inside of the square root. We could\nthen decompose the function as\nWe can check our answer by recomposing the functions.\nTRY IT #7 Write as the composition of two functions.\nMEDIA"
	},
	{
		"chunk_id" : "00000763",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with composite functions.\nComposite Functions(http://openstax.org/l/compfunction)\nComposite Function Notation Application(http://openstax.org/l/compfuncnot)\nComposite Functions Using Graphs(http://openstax.org/l/compfuncgraph)\nDecompose Functions(http://openstax.org/l/decompfunction)\nComposite Function Values(http://openstax.org/l/compfuncvalue)\n3.4 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00000764",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.4 SECTION EXERCISES\nVerbal\n1. How does one find the 2. What is the composition of 3. If the order is reversed\ndomain of the quotient of two functions, when composing two\ntwo functions, functions, can the result\never be the same as the\nanswer in the original order\nof the composition? If yes,\ngive an example. If no,\nexplain why not.\n4. How do you find the domain\nfor the composition of two\nfunctions,\nAccess for free at openstax.org\n3.4  Composition of Functions 249\nAlgebraic"
	},
	{
		"chunk_id" : "00000765",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.4  Composition of Functions 249\nAlgebraic\nFor the following exercises, determine the domain for each function in interval notation.\n5. Given and find 6. Given and find\nand and\n7. Given and find 8. Given and find\nand and\n9. Given and find 10. Given and find\nand\n11. For the following exercise, find the indicated\nfunction given and\n  \n \nFor the following exercises, use each pair of functions to find and Simplify your answers.\n12. 13. 14.\n15. 16. 17."
	},
	{
		"chunk_id" : "00000766",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16. 17.\nFor the following exercises, use each set of functions to find Simplify your answers.\n18. 19. 20. Given and\nand and find the\nfollowing:\n\n the domain of\nin interval notation\n\n the domain of\n\n21. Given and 22. Given the functions 23. Given functions\nfind the\nand state\nfollowing: find the following: the domain of each of the\n   following functions using\ninterval notation:\n the domain of\nin interval  \nnotation\n\n250 3  Functions\n24. Given functions 25. For and\nwrite the"
	},
	{
		"chunk_id" : "00000767",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "24. Given functions 25. For and\nwrite the\nand state\ndomain of in\nthe domain of each of the\ninterval notation.\nfollowing functions using\ninterval notation.\n \n\nFor the following exercises, find functions and so the given function can be expressed as\n26. 27. 28.\n29. 30. 31.\n32. 33. 34.\n35. 36. 37.\n38. 39. 40.\n41.\nGraphical\nFor the following exercises, use the graphs of shown inFigure 4, and shown inFigure 5, to evaluate the expressions.\nFigure4\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000768",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\nAccess for free at openstax.org\n3.4  Composition of Functions 251\nFigure5\n42. 43. 44.\n45. 46. 47.\n48. 49.\nFor the following exercises, use graphs of shown inFigure 6, shown inFigure 7, and shown inFigure 8,\nto evaluate the expressions.\nFigure6\nFigure7\n252 3  Functions\nFigure8\n50. 51. 52.\n53. 54. 55.\n56. 57.\nNumeric\nFor the following exercises, use the function values for shown inTable 3to evaluate each expression.\n0 7 9\n1 6 5\n2 5 6\n3 8 2\n4 4 1\n5 0 8\n6 2 7\n7 1 3\n8 9 4\n9 3 0\nTable3\n58. 59. 60."
	},
	{
		"chunk_id" : "00000769",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5 0 8\n6 2 7\n7 1 3\n8 9 4\n9 3 0\nTable3\n58. 59. 60.\n61. 62. 63.\n64. 65.\nAccess for free at openstax.org\n3.4  Composition of Functions 253\nFor the following exercises, use the function values for shown inTable 4to evaluate the expressions.\n11\n9\n7 0\n0 5 1\n1 3 0\n2 1\n3\nTable4\n66. 67. 68.\n69. 70. 71.\nFor the following exercises, use each pair of functions to find and\n72. 73. 74.\n75.\nFor the following exercises, use the functions and to evaluate or find the composite\nfunction as indicated.\n76. 77. 78.\n79."
	},
	{
		"chunk_id" : "00000770",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function as indicated.\n76. 77. 78.\n79.\nExtensions\nFor the following exercises, use and\n80. Find and 81. Find and 82. What is the domain of\nCompare the\ntwo answers.\n254 3  Functions\n83. What is the domain of 84. Let\n Find\n Is for any\nfunction the same result\nas the answer to part (a)\nfor any function? Explain.\nFor the following exercises, let and\n85. True or False: 86. True or False:\nFor the following exercises, find the composition when for all and\n87. 88. 89.\nReal-World Applications"
	},
	{
		"chunk_id" : "00000771",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "87. 88. 89.\nReal-World Applications\n90. The function gives the 91. The function gives the 92. A store offers customers a\nnumber of items that will pain level on a scale of 0 to 30% discount on the price\nbe demanded when the 10 experienced by a patient of selected items. Then,\nprice is The production with milligrams of a pain- the store takes off an\ncost is the cost of reducing drug in her additional 15% at the cash\nproducing items. To system. The milligrams of register. Write a price"
	},
	{
		"chunk_id" : "00000772",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determine the cost of the drug in the patients function that\nproduction when the price system after minutes is computes the final price of\nis $6, you would do which modeled by Which of the item in terms of the\nof the following? the following would you do original price (Hint: Use\nin order to determine function composition to\n Evaluate\nwhen the patient will be at find your answer.)\n Evaluate a pain level of 4?\n Solve\n Evaluate\n Solve\n Evaluate\n Solve\n Solve"
	},
	{
		"chunk_id" : "00000773",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Evaluate\n Solve\n Evaluate\n Solve\n Solve\n93. A rain drop hitting a lake 94. A forest fire leaves behind 95. Use the function you found\nmakes a circular ripple. If an area of grass burned in in the previous exercise to\nthe radius, in inches, grows an expanding circular find the total area burned\nas a function of time in pattern. If the radius of the after 5 minutes.\nminutes according to circle of burning grass is\nfind the increasing with time\narea of the ripple as a according to the formula"
	},
	{
		"chunk_id" : "00000774",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "area of the ripple as a according to the formula\nfunction of time. Find the express the\narea of the ripple at area burned as a function\nof time, (minutes).\nAccess for free at openstax.org\n3.5  Transformation of Functions 255\n96. The radius in inches, of a spherical balloon is 97. The number of bacteria in a refrigerated food\nproduct is given by\nrelated to the volume, by Air is\nwhere is the temperature of the\npumped into the balloon, so the volume after\nfood. When the food is removed from the"
	},
	{
		"chunk_id" : "00000775",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "food. When the food is removed from the\nseconds is given by\nrefrigerator, the temperature is given by\n Find the composite function where is the time in hours.\n Find theexacttime when the radius reaches 10  Find the composite function\ninches.  Find the time (round to two decimal places)\nwhen the bacteria count reaches 6752.\n3.5 Transformation of Functions\nLearning Objectives\nIn this section, you will:\nGraph functions using vertical and horizontal shifts."
	},
	{
		"chunk_id" : "00000776",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph functions using reflections about the x-axis and the y-axis.\nDetermine whether a function is even, odd, or neither from its graph.\nGraph functions using compressions and stretches.\nCombine transformations.\nFigure1 (credit: \"Misko\"\"/Flickr)"
	},
	{
		"chunk_id" : "00000777",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or\nvertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing\nobjects or processes in the real world. In this section, we will take a look at several kinds of transformations.\nGraphing Functions Using Vertical and Horizontal Shifts"
	},
	{
		"chunk_id" : "00000778",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and\nequations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a\ngiven scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are\ntrying to solve.\nIdentifying Vertical Shifts"
	},
	{
		"chunk_id" : "00000779",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "trying to solve.\nIdentifying Vertical Shifts\nOne simple kind oftransformationinvolves shifting the entire graph of a function up, down, right, or left. The simplest\nshift is avertical shift, moving the graph up or down, because this transformation involves adding a positive or negative\nconstant to the function. In other words, we add the same constant to the output value of the function regardless of the\ninput. For a function the function is shifted vertically units. SeeFigure 2for an example."
	},
	{
		"chunk_id" : "00000780",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "256 3  Functions\nFigure2 Vertical shift by of the cube root function\nTo help you visualize the concept of a vertical shift, consider that Therefore, is equivalent to\nEvery unit of is replaced by so they-value increases or decreases depending on the value of The result is a\nshift upward or downward.\nVertical Shift\nGiven a function a new function where is a constant, is avertical shiftof the function"
	},
	{
		"chunk_id" : "00000781",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "All the output values change by units. If is positive, the graph will shift up. If is negative, the graph will shift\ndown.\nEXAMPLE1\nAdding a Constant to a Function\nTo regulate temperature in a green building, airflow vents near the roof open and close throughout the day.Figure 3\nshows the area of open vents (in square feet) throughout the day in hours after midnight, During the summer, the\nfacilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square"
	},
	{
		"chunk_id" : "00000782",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "feet throughout the day and night. Sketch a graph of this new function.\nFigure3\nSolution\nWe can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will\nhave the effect of shifting the graph vertically up, as shown inFigure 4.\nAccess for free at openstax.org\n3.5  Transformation of Functions 257\nFigure4\nNotice that inFigure 4, for each input value, the output value has increased by 20, so if we call the new function we\ncould write"
	},
	{
		"chunk_id" : "00000783",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "could write\nThis notation tells us that, for any value of can be found by evaluating the function at the same input and then\nadding 20 to the result. This defines as a transformation of the function in this case a vertical shift up 20 units.\nNotice that, with a vertical shift, the input values stay the same and only the output values change. SeeTable 1.\n0 8 10 17 19 24\n0 0 220 220 0 0\n20 20 240 240 20 20\nTable1\n...\nHOW TO\nGiven a tabular function, create a new row to represent a vertical shift."
	},
	{
		"chunk_id" : "00000784",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the output row or column.\n2. Determine themagnitudeof the shift.\n3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.\nEXAMPLE2\nShifting a Tabular Function Vertically\nA function is given inTable 2. Create a table for the function\n2 4 6 8\n1 3 7 11\nTable2\nSolution\nThe formula tells us that we can find the output values of by subtracting 3 from the output values of\n258 3  Functions\nFor example:"
	},
	{
		"chunk_id" : "00000785",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "258 3  Functions\nFor example:\nSubtracting 3 from each value, we can complete a table of values for as shown inTable 3.\n2 4 6 8\n1 3 7 11\n2 0 4 8\nTable3\nAnalysis\nAs with the earlier vertical shift, notice the input values stay the same and only the output values change.\nTRY IT #1 The function gives the height of a ball (in meters) thrown upward from the\nground after seconds. Suppose the ball was instead thrown from the top of a 10-m building.\nRelate this new height function to and then find a formula for"
	},
	{
		"chunk_id" : "00000786",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Horizontal Shifts\nWe just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes\nto input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the\ngraph of the function left or right in what is known as ahorizontal shift, shown inFigure 5.\nFigure5 Horizontal shift of the function Note that means , which shifts the graph to the left,\nthat is, towardsnegativevalues of"
	},
	{
		"chunk_id" : "00000787",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that is, towardsnegativevalues of\nFor example, if then is a new function. Each input is reduced by 2 prior to squaring the\nfunction. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by\n2 units to yield the same output value as given in\nHorizontal Shift\nGiven a function a new function where is a constant, is ahorizontal shiftof the function If\nis positive, the graph will shift right. If is negative, the graph will shift left."
	},
	{
		"chunk_id" : "00000788",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.5  Transformation of Functions 259\nEXAMPLE3\nAdding a Constant to an Input\nReturning to our building airflow example fromFigure 3, suppose that in autumn the facilities manager decides that the\noriginal venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the\nnew function.\nSolution\nWe can set to be the original program and to be the revised program."
	},
	{
		"chunk_id" : "00000789",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the new graph, at each time, the airflow is the same as the original function was 2 hours later. For example, in the\noriginal function the airflow starts to change at 8 a.m., whereas for the function the airflow starts to change at 6\na.m. The comparable function values are SeeFigure 6. Notice also that the vents first opened to at\n10 a.m. under the original plan, while under the new plan the vents reach at\n8 a.m., so"
	},
	{
		"chunk_id" : "00000790",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8 a.m., so\nIn both cases, we see that, because starts 2 hours sooner, That means that the same output values are\nreached when\nFigure6\nAnalysis\nNote that has the effect of shifting the graph to theleft.\nHorizontal changes or inside changes affect the domain of a function (the input) instead of the range and often seem\ncounterintuitive. The new function uses the same outputs as but matches those outputs to inputs 2 hours"
	},
	{
		"chunk_id" : "00000791",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "earlier than those of Said another way, we must add 2 hours to the input of to find the corresponding output for\n...\nHOW TO\nGiven a tabular function, create a new row to represent a horizontal shift.\n1. Identify the input row or column.\n2. Determine the magnitude of the shift.\n3. Add the shift to the value in each input cell.\n260 3  Functions\nEXAMPLE4\nShifting a Tabular Function Horizontally\nA function is given inTable 4. Create a table for the function\n2 4 6 8\n1 3 7 11\nTable4\nSolution"
	},
	{
		"chunk_id" : "00000792",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 4 6 8\n1 3 7 11\nTable4\nSolution\nThe formula tells us that the output values of are the same as the output value of when the input\nvalue is 3 less than the original value. For example, we know that To get the same output from the function\nwe will need an input value that is 3larger. We input a value that is 3 larger for because the function takes 3 away\nbefore evaluating the function\nWe continue with the other values to createTable 5.\n5 7 9 11\n2 4 6 8\n1 3 7 11\n1 3 7 11\nTable5"
	},
	{
		"chunk_id" : "00000793",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5 7 9 11\n2 4 6 8\n1 3 7 11\n1 3 7 11\nTable5\nThe result is that the function has been shifted to the right by 3. Notice the output values for remain the same\nas the output values for but the corresponding input values, have shifted to the right by 3. Specifically, 2 shifted\nto 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.\nAnalysis\nFigure 7represents both of the functions. We can see the horizontal shift in each point.\nAccess for free at openstax.org\n3.5  Transformation of Functions 261\nFigure7"
	},
	{
		"chunk_id" : "00000794",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.5  Transformation of Functions 261\nFigure7\nEXAMPLE5\nIdentifying a Horizontal Shift of a Toolkit Function\nFigure 8represents a transformation of the toolkit function Relate this new function to and then\nfind a formula for\nFigure8\nSolution\nNotice that the graph is identical in shape to the function, but thex-values are shifted to the right 2 units. The\nvertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the\nright, so"
	},
	{
		"chunk_id" : "00000795",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "right, so\nNotice how we must input the value to get the output value thex-values must be 2 units larger because of\nthe shift to the right by 2 units. We can then use the definition of the function to write a formula for by\nevaluating\nAnalysis\nTo determine whether the shift is or , consider a single reference point on the graph. For a quadratic, looking at\n262 3  Functions\nthe vertex point is convenient. In the original function, In our shifted function, To obtain the output"
	},
	{
		"chunk_id" : "00000796",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value of 0 from the function we need to decide whether a plus or a minus sign will work to satisfy\nFor this to work, we will need tosubtract2 units from our input values.\nEXAMPLE6\nInterpreting Horizontal versus Vertical Shifts\nThe function gives the number of gallons of gas required to drive miles. Interpret and\nSolution\ncan be interpreted as adding 10 to the output, gallons. This is the gas required to drive miles, plus another\n10 gallons of gas. The graph would indicate a vertical shift."
	},
	{
		"chunk_id" : "00000797",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive\n10 miles more than miles. The graph would indicate a horizontal shift.\nTRY IT #2 Given the function graph the original function and the transformation\non the same axes. Is this a horizontal or a vertical shift? Which way is the graph\nshifted and by how many units?\nCombining Vertical and Horizontal Shifts"
	},
	{
		"chunk_id" : "00000798",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Combining Vertical and Horizontal Shifts\nNow that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output\n(y-) values and shift the function up or down. Horizontal shifts are inside changes that affect the input (x-) values and\nshift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down\nandleft or right.\n...\nHOW TO\nGiven a function and both a vertical and a horizontal shift, sketch the graph."
	},
	{
		"chunk_id" : "00000799",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the vertical and horizontal shifts from the formula.\n2. The vertical shift results from a constant added to the output. Move the graph up for a positive constant and\ndown for a negative constant.\n3. The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and\nright for a negative constant.\n4. Apply the shifts to the graph in either order.\nEXAMPLE7\nGraphing Combined Vertical and Horizontal Shifts\nGiven sketch a graph of\nSolution"
	},
	{
		"chunk_id" : "00000800",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given sketch a graph of\nSolution\nThe function is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin.\nThe graph of has transformed in two ways: is a change on the inside of the function, giving a horizontal\nshift left by 1, and the subtraction by 3 in is a change to the outside of the function, giving a vertical shift\ndown by 3. The transformation of the graph is illustrated inFigure 9.\nLet us follow one point of the graph of"
	},
	{
		"chunk_id" : "00000801",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Let us follow one point of the graph of\n The point is transformed first by shifting left 1 unit:\n The point is transformed next by shifting down 3 units:\nAccess for free at openstax.org\n3.5  Transformation of Functions 263\nFigure9\nFigure 10shows the graph of\nFigure10\nTRY IT #3 Given sketch a graph of\nEXAMPLE8\nIdentifying Combined Vertical and Horizontal Shifts\nWrite a formula for the graph shown inFigure 11, which is a transformation of the toolkit square root function.\n264 3  Functions\nFigure11"
	},
	{
		"chunk_id" : "00000802",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "264 3  Functions\nFigure11\nSolution\nThe graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function\nnotation, we could write that as\nUsing the formula for the square root function, we can write\nAnalysis\nNote that this transformation has changed the domain and range of the function. This new graph has domain  and\nrange \nTRY IT #4 Write a formula for a transformation of the toolkit reciprocal function that shifts the"
	},
	{
		"chunk_id" : "00000803",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions graph one unit to the right and one unit up.\nGraphing Functions Using Reflections about the Axes\nAnother transformation that can be applied to a function is a reflection over thex- ory-axis. Avertical reflectionreflects\na graph vertically across thex-axis, while ahorizontal reflectionreflects a graph horizontally across they-axis. The\nreflections are shown inFigure 12.\nFigure12 Vertical and horizontal reflections of a function."
	},
	{
		"chunk_id" : "00000804",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the\nAccess for free at openstax.org\n3.5  Transformation of Functions 265\nx-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the\ny-axis.\nReflections\nGiven a function a new function is avertical reflectionof the function sometimes called a\nreflection about (or over, or through) thex-axis."
	},
	{
		"chunk_id" : "00000805",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reflection about (or over, or through) thex-axis.\nGiven a function a new function is ahorizontal reflectionof the function sometimes called\na reflection about they-axis.\n...\nHOW TO\nGiven a function, reflect the graph both vertically and horizontally.\n1. Multiply all outputs by 1 for a vertical reflection. The new graph is a reflection of the original graph about the\nx-axis.\n2. Multiply all inputs by 1 for a horizontal reflection. The new graph is a reflection of the original graph about the\ny-axis."
	},
	{
		"chunk_id" : "00000806",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-axis.\nEXAMPLE9\nReflecting a Graph Horizontally and Vertically\nReflect the graph of (a) vertically and (b) horizontally.\nSolution\n\nReflecting the graph vertically means that each output value will be reflected over the horizontalt-axis as shown in\nFigure 13.\nFigure13 Vertical reflection of the square root function\nBecause each output value is the opposite of the original output value, we can write"
	},
	{
		"chunk_id" : "00000807",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that this is an outside change, or vertical shift, that affects the output values, so the negative sign belongs\noutside of the function.\n266 3  Functions\n\nReflecting horizontally means that each input value will be reflected over the vertical axis as shown inFigure 14.\nFigure14 Horizontal reflection of the square root function\nBecause each input value is the opposite of the original input value, we can write"
	},
	{
		"chunk_id" : "00000808",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the\ninside of the function.\nNote that these transformations can affect the domain and range of the functions. While the original square root\nfunction has domain  and range  the vertical reflection gives the function the range  and\nthe horizontal reflection gives the function the domain \nTRY IT #5 Reflect the graph of (a) vertically and (b) horizontally.\nEXAMPLE10"
	},
	{
		"chunk_id" : "00000809",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE10\nReflecting a Tabular Function Horizontally and Vertically\nA function is given asTable 6. Create a table for the functions below.\n \n2 4 6 8\n1 3 7 11\nTable6\nAccess for free at openstax.org\n3.5  Transformation of Functions 267\nSolution\n\nFor the negative sign outside the function indicates a vertical reflection, so thex-values stay the same and each\noutput value will be the opposite of the original output value. SeeTable 7.\n2 4 6 8\n1 3 7 11\nTable7\n"
	},
	{
		"chunk_id" : "00000810",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 4 6 8\n1 3 7 11\nTable7\n\nFor the negative sign inside the function indicates a horizontal reflection, so each input value will be the\nopposite of the original input value and the values stay the same as the values. SeeTable 8.\n2 4 6 8\n1 3 7 11\nTable8\nTRY IT #6 A function is given asTable 9. Create a table for the functions below.\n \n2 0 2 4\n5 10 15 20\nTable9\nEXAMPLE11\nApplying a Learning Model Equation"
	},
	{
		"chunk_id" : "00000811",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE11\nApplying a Learning Model Equation\nA common model for learning has an equation similar to where is the percentage of mastery that can\nbe achieved after practice sessions. This is a transformation of the function shown inFigure 15. Sketch a\ngraph of\n268 3  Functions\nFigure15\nSolution\nThis equation combines three transformations into one equation.\n A horizontal reflection:\n A vertical reflection:\n A vertical shift:"
	},
	{
		"chunk_id" : "00000812",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A vertical reflection:\n A vertical shift:\nWe can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points\nthrough each of the three transformations. We will choose the points (0, 1) and (1, 2).\n1. First, we apply a horizontal reflection: (0, 1) (1, 2).\n2. Then, we apply a vertical reflection: (0, -1) (-1, 2)\n3. Finally, we apply a vertical shift: (0, 0) (-1, -1))."
	},
	{
		"chunk_id" : "00000813",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.\nInFigure 16, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third\nresults from a vertical shift up 1 unit.\nFigure16\nAnalysis\nAs a model for learning, this function would be limited to a domain of with corresponding range\nTRY IT #7 Given the toolkit function graph and Take note of any\nsurprising behavior for these functions."
	},
	{
		"chunk_id" : "00000814",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "surprising behavior for these functions.\nAccess for free at openstax.org\n3.5  Transformation of Functions 269\nDetermining Even and Odd Functions\nSome functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the\ntoolkit functions or will result in the original graph. We say that these types of graphs are\nsymmetric about they-axis. A function whose graph is symmetric about they-axis is called aneven function."
	},
	{
		"chunk_id" : "00000815",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If the graphs of or were reflected overbothaxes, the result would be the original graph, as shown\ninFigure 17.\nFigure17 (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical\nreflections reproduce the original cubic function.\nWe say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is\ncalled anodd function."
	},
	{
		"chunk_id" : "00000816",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "called anodd function.\nNote: A function can be neither even nor odd if it does not exhibit either symmetry. For example, is neither\neven nor odd. Also, the only function that is both even and odd is the constant function\nEven and Odd Functions\nA function is called aneven functionif for every input\nThe graph of an even function is symmetric about the axis.\nA function is called anodd functionif for every input\nThe graph of an odd function is symmetric about the origin.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000817",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the formula for a function, determine if the function is even, odd, or neither.\n1. Determine whether the function satisfies If it does, it is even.\n2. Determine whether the function satisfies If it does, it is odd.\n3. If the function does not satisfy either rule, it is neither even nor odd.\nEXAMPLE12\nDetermining whether a Function Is Even, Odd, or Neither\nIs the function even, odd, or neither?\n270 3  Functions\nSolution"
	},
	{
		"chunk_id" : "00000818",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "270 3  Functions\nSolution\nWithout looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections\nand determining if they return us to the original function. Lets begin with the rule for even functions.\nThis does not return us to the original function, so this function is not even. We can now test the rule for odd functions.\nBecause this is an odd function.\nAnalysis"
	},
	{
		"chunk_id" : "00000819",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because this is an odd function.\nAnalysis\nConsider the graph of inFigure 18. Notice that the graph is symmetric about the origin. For every point on the\ngraph, the corresponding point is also on the graph. For example, (1, 3) is on the graph of and the\ncorresponding point is also on the graph.\nFigure18\nTRY IT #8 Is the function even, odd, or neither?\nGraphing Functions Using Stretches and Compressions"
	},
	{
		"chunk_id" : "00000820",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it\ndid not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.\nWe can transform the inside (input values) of a function or we can transform the outside (output values) of a function.\nEach change has a specific effect that can be seen graphically.\nVertical Stretches and Compressions"
	},
	{
		"chunk_id" : "00000821",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Vertical Stretches and Compressions\nWhen we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically\nin relation to the graph of the original function. If the constant is greater than 1, we get avertical stretch; if the\nconstant is between 0 and 1, we get avertical compression.Figure 19shows a function multiplied by constant factors 2\nand 0.5 and the resulting vertical stretch and compression.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000822",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.5  Transformation of Functions 271\nFigure19 Vertical stretch and compression\nVertical Stretches and Compressions\nGiven a function a new function where is a constant, is avertical stretchorvertical\ncompressionof the function\n If then the graph will be stretched.\n If then the graph will be compressed.\n If then there will be combination of a vertical stretch or compression with a vertical reflection.\n...\nHOW TO\nGiven a function, graph its vertical stretch."
	},
	{
		"chunk_id" : "00000823",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given a function, graph its vertical stretch.\n1. Identify the value of\n2. Multiply all range values by\n3. If the graph is stretched by a factor of\nIf the graph is compressed by a factor of\nIf the graph is either stretched or compressed and also reflected about thex-axis.\nEXAMPLE13\nGraphing a Vertical Stretch\nA function models the population of fruit flies. The graph is shown inFigure 20.\n272 3  Functions\nFigure20"
	},
	{
		"chunk_id" : "00000824",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "272 3  Functions\nFigure20\nA scientist is comparing this population to another population, whose growth follows the same pattern, but is twice\nas large. Sketch a graph of this population.\nSolution\nBecause the population is always twice as large, the new populations output values are always twice the original\nfunctions output values. Graphically, this is shown inFigure 21.\nIf we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2."
	},
	{
		"chunk_id" : "00000825",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The following shows where the new points for the new graph will be located.\nFigure21\nSymbolically, the relationship is written as\nThis means that for any input the value of the function is twice the value of the function Notice that the effect on\nthe graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input\nvalues, stay the same while the output values are twice as large as before.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000826",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.5  Transformation of Functions 273\n...\nHOW TO\nGiven a tabular function and assuming that the transformation is a vertical stretch or compression, create a\ntable for a vertical compression.\n1. Determine the value of\n2. Multiply all of the output values by\nEXAMPLE14\nFinding a Vertical Compression of a Tabular Function\nA function is given asTable 10. Create a table for the function\n2 4 6 8\n1 3 7 11\nTable10\nSolution"
	},
	{
		"chunk_id" : "00000827",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 4 6 8\n1 3 7 11\nTable10\nSolution\nThe formula tells us that the output values of are half of the output values of with the same inputs. For\nexample, we know that Then\nWe do the same for the other values to produceTable 11.\nTable11\nAnalysis\nThe result is that the function has been compressed vertically by Each output value is divided in half, so the\ngraph is half the original height.\nTRY IT #9 A function is given asTable 12. Create a table for the function\n2 4 6 8\n12 16 20 0\nTable12\nEXAMPLE15"
	},
	{
		"chunk_id" : "00000828",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 4 6 8\n12 16 20 0\nTable12\nEXAMPLE15\nRecognizing a Vertical Stretch\nThe graph inFigure 22is a transformation of the toolkit function Relate this new function to and\nthen find a formula for\n274 3  Functions\nFigure22\nSolution\nWhen trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In\nthis graph, it appears that With the basic cubic function at the same input, Based on that, it"
	},
	{
		"chunk_id" : "00000829",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "appears that the outputs of are the outputs of the function because From this we can fairly safely\nconclude that\nWe can write a formula for by using the definition of the function\nTRY IT #10 Write the formula for the function that we get when we stretch the identity toolkit function by a\nfactor of 3, and then shift it down by 2 units.\nHorizontal Stretches and Compressions\nNow we consider changes to the inside of a function. When we multiply a functions input by a positive constant, we get"
	},
	{
		"chunk_id" : "00000830",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the\nconstant is between 0 and 1, we get ahorizontal stretch; if the constant is greater than 1, we get ahorizontal\ncompressionof the function.\nAccess for free at openstax.org\n3.5  Transformation of Functions 275\nFigure23\nGiven a function the form results in a horizontal stretch or compression. Consider the function"
	},
	{
		"chunk_id" : "00000831",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ObserveFigure 23. The graph of is a horizontal stretch of the graph of the function by a\nfactor of 2. The graph of is a horizontal compression of the graph of the function by a factor of .\nHorizontal Stretches and Compressions\nGiven a function a new function where is a constant, is ahorizontal stretchorhorizontal\ncompressionof the function\n If then the graph will be compressed by\n If then the graph will be stretched by"
	},
	{
		"chunk_id" : "00000832",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If then the graph will be stretched by\n If then there will be combination of a horizontal stretch or compression with a horizontal reflection.\n...\nHOW TO\nGiven a description of a function, sketch a horizontal compression or stretch.\n1. Write a formula to represent the function.\n2. Set where for a compression or for a stretch.\nEXAMPLE16\nGraphing a Horizontal Compression\nSuppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as"
	},
	{
		"chunk_id" : "00000833",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fast as the original population. In other words, this new population, will progress in 1 hour the same amount as the\noriginal population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours.\nSketch a graph of this population.\nSolution\nSymbolically, we could write\n276 3  Functions\nSeeFigure 24for a graphical comparison of the original population and the compressed population.\nFigure24 (a) Original population graph (b) Compressed population graph\nAnalysis"
	},
	{
		"chunk_id" : "00000834",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNote that the effect on the graph is a horizontal compression where all input values are half of their original distance\nfrom the vertical axis.\nEXAMPLE17\nFinding a Horizontal Stretch for a Tabular Function\nA function is given asTable 13. Create a table for the function\n2 4 6 8\n1 3 7 11\nTable13\nSolution\nThe formula tells us that the output values for are the same as the output values for the function at\nan input half the size. Notice that we do not have enough information to determine because"
	},
	{
		"chunk_id" : "00000835",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and we do not have a value for in our table. Our input values to will need to be twice as\nlarge to get inputs for that we can evaluate. For example, we can determine\nWe do the same for the other values to produceTable 14.\n4 8 12 16\nTable14\nAccess for free at openstax.org\n3.5  Transformation of Functions 277\n1 3 7 11\nTable14\nFigure 25shows the graphs of both of these sets of points.\nFigure25\nAnalysis"
	},
	{
		"chunk_id" : "00000836",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure25\nAnalysis\nBecause each input value has been doubled, the result is that the function has been stretched horizontally by a\nfactor of 2.\nEXAMPLE18\nRecognizing a Horizontal Compression on a Graph\nRelate the function to inFigure 26.\nFigure26\nSolution\nThe graph of looks like the graph of horizontally compressed. Because ends at and ends at\nwe can see that the values have been compressed by because We might also notice that\nand Either way, we can describe this relationship as This is a horizontal"
	},
	{
		"chunk_id" : "00000837",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compression by\nAnalysis\nNotice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression.\nSo to stretch the graph horizontally by a scale factor of 4, we need a coefficient of in our function: This means\nthat the input values must be four times larger to produce the same result, requiring the input to be larger, causing the\nhorizontal stretching.\n278 3  Functions"
	},
	{
		"chunk_id" : "00000838",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal stretching.\n278 3  Functions\nTRY IT #11 Write a formula for the toolkit square root function horizontally stretched by a factor of 3.\nPerforming a Sequence of Transformations\nWhen combining transformations, it is very important to consider the order of the transformations. For example,\nvertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and"
	},
	{
		"chunk_id" : "00000839",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only\nthe original function gets stretched when we stretch first.\nWhen we see an expression such as which transformation should we start with? The answer here follows\nnicely from the order of operations. Given the output value of we first multiply by 2, causing the vertical stretch,\nand then add 3, causing the vertical shift. In other words, multiplication before addition."
	},
	{
		"chunk_id" : "00000840",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Horizontal transformations are a little trickier to think about. When we write for example, we have to\nthink about how the inputs to the function relate to the inputs to the function Suppose we know What\ninput to would produce that output? In other words, what value of will allow We would need\nTo solve for we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a\nhorizontal compression."
	},
	{
		"chunk_id" : "00000841",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal compression.\nThis format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph\nbefore shifting. We can work around this by factoring inside the function.\nLets work through an example.\nWe can factor out a 2.\nNow we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way\nallows us to horizontally stretch first and then shift horizontally.\nCombining Transformations"
	},
	{
		"chunk_id" : "00000842",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Combining Transformations\nWhen combining vertical transformations written in the form first vertically stretch by and then vertically\nshift by\nWhen combining horizontal transformations written in the form first horizontally shift by and then\nhorizontally stretch by\nWhen combining horizontal transformations written in the form first horizontally stretch by and then\nhorizontally shift by\nHorizontal and vertical transformations are independent. It does not matter whether horizontal or vertical"
	},
	{
		"chunk_id" : "00000843",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "transformations are performed first.\nEXAMPLE19\nFinding a Triple Transformation of a Tabular Function\nGivenTable 15for the function create a table of values for the function\n6 12 18 24\n10 14 15 17\nTable15\nAccess for free at openstax.org\n3.5  Transformation of Functions 279\nSolution\nThere are three steps to this transformation, and we will work from the inside out. Starting with the horizontal\ntransformations, is a horizontal compression by which means we multiply each value by SeeTable 16.\n2 4 6 8"
	},
	{
		"chunk_id" : "00000844",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 4 6 8\n10 14 15 17\nTable16\nLooking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2.\nWe apply this to the previous transformation. SeeTable 17.\n2 4 6 8\n20 28 30 34\nTable17\nFinally, we can apply the vertical shift, which will add 1 to all the output values. SeeTable 18.\n2 4 6 8\n21 29 31 35\nTable18\nEXAMPLE20\nFinding a Triple Transformation of a Graph\nUse the graph of inFigure 27to sketch a graph of\nFigure27\nSolution"
	},
	{
		"chunk_id" : "00000845",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure27\nSolution\nTo simplify, lets start by factoring out the inside of the function.\n280 3  Functions\nBy factoring the inside, we can first horizontally stretch by 2, as indicated by the on the inside of the function.\nRemember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See\nFigure 28.\nFigure28\nNext, we horizontally shift left by 2 units, as indicated by SeeFigure 29.\nFigure29"
	},
	{
		"chunk_id" : "00000846",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure29\nLast, we vertically shift down by 3 to complete our sketch, as indicated by the on the outside of the function. See\nFigure 30.\nAccess for free at openstax.org\n3.5  Transformation of Functions 281\nFigure30\nMEDIA\nAccess this online resource for additional instruction and practice with transformation of functions.\nFunction Transformations(http://openstax.org/l/functrans)\n3.5 SECTION EXERCISES\nVerbal\n1. When examining the 2. When examining the 3. When examining the"
	},
	{
		"chunk_id" : "00000847",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "formula of a function that is formula of a function that is formula of a function that is\nthe result of multiple the result of multiple the result of multiple\ntransformations, how can transformations, how can transformations, how can\nyou tell a horizontal shift you tell a horizontal stretch you tell a horizontal\nfrom a vertical shift? from a vertical stretch? compression from a vertical\ncompression?\n4. When examining the 5. How can you determine\nformula of a function that is whether a function is odd or"
	},
	{
		"chunk_id" : "00000848",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the result of multiple even from the formula of\ntransformations, how can the function?\nyou tell a reflection with\nrespect to thex-axis from a\nreflection with respect to the\ny-axis?\nAlgebraic\nFor the following exercises, write a formula for the function obtained when the graph is shifted as described.\n6. is shifted up 1 7. is shifted down 3 8. is shifted down 4\nunit and to the left 2 units. units and to the right 1 unit. units and to the right 3\nunits.\n282 3  Functions\n9. is shifted up 2"
	},
	{
		"chunk_id" : "00000849",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "units.\n282 3  Functions\n9. is shifted up 2\nunits and to the left 4 units.\nFor the following exercises, describe how the graph of the function is a transformation of the graph of the original\nfunction\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19.\nFor the following exercises, determine the interval(s) on which the function is increasing and decreasing.\n20. 21. 22.\n23.\nGraphical\nFor the following exercises, use the graph of shown inFigure 31to sketch a graph of each transformation of\nFigure31\n24. 25. 26."
	},
	{
		"chunk_id" : "00000850",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure31\n24. 25. 26.\nFor the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit\nfunctions.\n27. 28. 29.\n30.\nAccess for free at openstax.org\n3.5  Transformation of Functions 283\nNumeric\n31. Tabular representations for the 32. Tabular representations for the functions\nfunctions and are given and are given below. Write and as\nbelow. Write and as transformations of\ntransformations of\n2 1 0 1 2\n2 1 0 1 2\n1 3 4 2 1\n2 1 3 1 2\n3 2 1 0 1\n1 0 1 2 3"
	},
	{
		"chunk_id" : "00000851",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 3 4 2 1\n2 1 3 1 2\n3 2 1 0 1\n1 0 1 2 3\n1 3 4 2 1\n2 1 3 1 2\n2 1 0 1 2\n2 1 0 1 2\n2 4 3 1 0\n1 0 2 2 3\nFor the following exercises, write an equation for each graphed function by using transformations of the graphs of one\nof the toolkit functions.\n33. 34. 35.\n284 3  Functions\n36. 37. 38.\n39. 40.\nFor the following exercises, use the graphs of transformations of the square root function to find a formula for each of\nthe functions.\n41. 42.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00000852",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "41. 42.\nAccess for free at openstax.org\n3.5  Transformation of Functions 285\nFor the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the\nresulting functions.\n43. 44. 45.\n46.\nFor the following exercises, determine whether the function is odd, even, or neither.\n47. 48. 49.\n50. 51. 52.\nFor the following exercises, describe how the graph of each function is a transformation of the graph of the original\nfunction\n53. 54. 55.\n56. 57. 58.\n59. 60. 61.\n62."
	},
	{
		"chunk_id" : "00000853",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function\n53. 54. 55.\n56. 57. 58.\n59. 60. 61.\n62.\n286 3  Functions\nFor the following exercises, write a formula for the function that results when the graph of a given toolkit function is\ntransformed as described.\n63. The graph of is 64. The graph of is 65. The graph of is\nreflected over the -axis reflected over the -axis vertically compressed by a\nand horizontally and horizontally stretched\nfactor of then shifted to\ncompressed by a factor of by a factor of 2.\nthe left 2 units and down 3\n.\nunits."
	},
	{
		"chunk_id" : "00000854",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the left 2 units and down 3\n.\nunits.\n66. The graph of is 67. The graph of is 68. The graph of is\nvertically stretched by a vertically compressed by a horizontally stretched by a\nfactor of 8, then shifted to factor of then shifted to factor of 3, then shifted to\nthe right 4 units and up 2 the right 5 units and up 1 the left 4 units and down 3\nunits. unit. units.\nFor the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of\nthe transformation."
	},
	{
		"chunk_id" : "00000855",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the transformation.\n69. 70. 71.\n72. 73. 74.\n75. 76. 77.\nFor the following exercises, use the graph inFigure 32to sketch the given transformations.\nFigure32\n78. 79. 80.\n81.\nAccess for free at openstax.org\n3.6  Absolute Value Functions 287\n3.6 Absolute Value Functions\nLearning Objectives\nIn this section, you will:\nGraph an absolute value function.\nSolve an absolute value equation.\nFigure1 Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of"
	},
	{
		"chunk_id" : "00000856",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "absolute values. (credit: \"s58y\"\"/Flickr)"
	},
	{
		"chunk_id" : "00000857",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value\nfunction. In this section, we will continue our investigation ofabsolute value functions.\nUnderstanding Absolute Value\nRecall that in its basic form the absolute value function is one of our toolkit functions. Theabsolute value\nfunction is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for"
	},
	{
		"chunk_id" : "00000858",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "whatever the input value is, the output is the value without regard to sign. Knowing this, we can use absolute value\nfunctions to solve some kinds of real-world problems.\nAbsolute Value Function\nThe absolute value function can be defined as a piecewise function\nEXAMPLE1\nUsing Absolute Value to Determine Resistance\nElectrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance,"
	},
	{
		"chunk_id" : "00000859",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat\nfrom piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to\nguarantee that the variations will stay within a specified range, often or\nSuppose we have a resistor rated at 680 ohms, Use the absolute value function to express the range of possible\nvalues of the actual resistance.\nSolution"
	},
	{
		"chunk_id" : "00000860",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values of the actual resistance.\nSolution\nWe can find that 5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal\nresistance should not exceed the stated variability, so, with the resistance in ohms,\n288 3  Functions\nTRY IT #1 Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using\nabsolute value notation.\nGraphing an Absolute Value Function"
	},
	{
		"chunk_id" : "00000861",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing an Absolute Value Function\nThe most significant feature of the absolute value graph is the corner point at which the graph changes direction. This\npoint is shown at theorigininFigure 2.\nFigure2\nFigure 3shows the graph of The graph of has been shifted right 3 units, vertically stretched by a\nfactor of 2, and shifted up 4 units. This means that the corner point is located at for this transformed function.\nFigure3\nEXAMPLE2\nWriting an Equation for an Absolute Value Function Given a Graph"
	},
	{
		"chunk_id" : "00000862",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write an equation for the function graphed inFigure 4.\nAccess for free at openstax.org\n3.6  Absolute Value Functions 289\nFigure4\nSolution\nThe basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and\ndown 2 units from the basic toolkit function. SeeFigure 5.\nFigure5\nWe also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not"
	},
	{
		"chunk_id" : "00000863",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value\nfunction. Instead, the width is equal to 1 times the vertical distance as shown inFigure 6.\n290 3  Functions\nFigure6\nFrom this information we can write the equation\nAnalysis\nNote that these equations are algebraically equivalentthe stretch for an absolute value function can be written"
	},
	{
		"chunk_id" : "00000864",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative,\nthere is also a reflection about the x-axis.\nQ&A If we couldnt observe the stretch of the function from the graphs, could we algebraically\ndetermine it?\nYes. If we are unable to determine the stretch based on the width of the graph, we can solve for the\nstretch factor by putting in a known pair of values for and\nNow substituting in the point(1, 2)"
	},
	{
		"chunk_id" : "00000865",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now substituting in the point(1, 2)\nTRY IT #2 Write the equation for the absolute value function that is horizontally shifted left 2 units, is\nvertically flipped, and vertically shifted up 3 units.\nQ&A Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?\nYes, they always intersect the vertical axis. The graph of an absolute value function will intersect the\nvertical axis when the input is zero."
	},
	{
		"chunk_id" : "00000866",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertical axis when the input is zero.\nNo, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal\naxis, depending on how the graph has been shifted and reflected. It is possible for the absolute value\nfunction to intersect the horizontal axis at zero, one, or two points (seeFigure 7).\nAccess for free at openstax.org\n3.6  Absolute Value Functions 291\nFigure7 (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function"
	},
	{
		"chunk_id" : "00000867",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.\nSolving an Absolute Value Equation\nInOther Type of Equations, we touched on the concepts of absolute value equations. Now that we understand a little\nmore about their graphs, we can take another look at these types of equations. Now that we can graph an absolute\nvalue function, we will learn how to solve an absolute value equation. To solve an equation such as we"
	},
	{
		"chunk_id" : "00000868",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two\ndifferent equations we can solve independently.\nKnowing how to solve problems involvingabsolute value functionsis useful. For example, we may need to identify\nnumbers or points on a line that are at a specified distance from a given reference point.\nAn absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,"
	},
	{
		"chunk_id" : "00000869",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solutions to Absolute Value Equations\nFor real numbers and , an equation of the form with will have solutions when or\nIf the equation has no solution.\n...\nHOW TO\nGiven the formula for an absolute value function, find the horizontal intercepts of its graph.\n1. Isolate the absolute value term.\n2. Use to write or assuming\n3. Solve for\nEXAMPLE3\nFinding the Zeros of an Absolute Value Function\nFor the function find the values of such that\n292 3  Functions\nSolution\nThe function outputs 0 when or SeeFigure 8."
	},
	{
		"chunk_id" : "00000870",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The function outputs 0 when or SeeFigure 8.\nFigure8\nTRY IT #3 For the function find the values of such that\nQ&A Should we always expect two answers when solving\nNo. We may find one, two, or even no answers. For example, there is no solution to\nMEDIA\nAccess these online resources for additional instruction and practice with absolute value.\nGraphing Absolute Value Functions(http://openstax.org/l/graphabsvalue)\nGraphing Absolute Value Functions 2(http://openstax.org/l/graphabsvalue2)"
	},
	{
		"chunk_id" : "00000871",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.6  Absolute Value Functions 293\n3.6 SECTION EXERCISES\nVerbal\n1. How do you solve an 2. How can you tell whether an 3. When solving an absolute\nabsolute value equation? absolute value function has value function, the isolated\ntwox-intercepts without absolute value term is equal\ngraphing the function? to a negative number. What\ndoes that tell you about the\ngraph of the absolute value\nfunction?\n4. How can you use the graph\nof an absolute value\nfunction to determine the"
	},
	{
		"chunk_id" : "00000872",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of an absolute value\nfunction to determine the\nx-values for which the\nfunction values are\nnegative?\nAlgebraic\n5. Describe all numbers that 6. Describe all numbers that 7. Describe the situation in\nare at a distance of 4 from are at a distance of from which the distance that\nthe number 8. Express this the number 4. Express this point is from 10 is at least\nset of numbers using set of numbers using 15 units. Express this set of\nabsolute value notation. absolute value notation. numbers using absolute"
	},
	{
		"chunk_id" : "00000873",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value notation.\n8. Find all function values\nsuch that the distance from\nto the value 8 is less\nthan 0.03 units. Express this\nset of numbers using\nabsolute value notation.\nFor the following exercises, find thex- andy-intercepts of the graphs of each function.\n9. 10. 11.\n12. 13. 14.\n15.\nGraphical\nFor the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.\n16. 17. 18.\n294 3  Functions\nFor the following exercises, graph the given functions by hand."
	},
	{
		"chunk_id" : "00000874",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31.\nTechnology\n32. Use a graphing utility to 33. Use a graphing utility to\ngraph on graph\nthe viewing window on\nIdentify the corresponding the viewing window\nrange. Show the graph. Identify the\ncorresponding range.\nShow the graph.\nFor the following exercises, graph each function using a graphing utility. Specify the viewing window.\n34. 35.\nExtensions\nFor the following exercises, solve the inequality."
	},
	{
		"chunk_id" : "00000875",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "36. If possible, find all values of such that there are 37. If possible, find all values of such that there are\nno intercepts for no -intercepts for\nReal-World Applications\n38. Cities A and B are on the 39. The true proportion of 40. Students who score within\nsame east-west line. people who give a 18 points of the number 82\nAssume that city A is favorable rating to will pass a particular test.\nlocated at the origin. If the Congress is 8% with a Write this statement using"
	},
	{
		"chunk_id" : "00000876",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distance from city A to city margin of error of 1.5%. absolute value notation\nB is at least 100 miles and Describe this statement and use the variable for\nrepresents the distance using an absolute value the score.\nfrom city B to city A, equation.\nexpress this using absolute\nvalue notation.\nAccess for free at openstax.org\n3.7  Inverse Functions 295\n41. A machinist must produce 42. The tolerance for a ball\na bearing that is within bearing is 0.01. If the true"
	},
	{
		"chunk_id" : "00000877",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0.01 inches of the correct diameter of the bearing is\ndiameter of 5.0 inches. to be 2.0 inches and the\nUsing as the diameter of measured value of the\nthe bearing, write this diameter is inches,\nstatement using absolute express the tolerance using\nvalue notation. absolute value notation.\n3.7 Inverse Functions\nLearning Objectives\nIn this section, you will:\nVerify inverse functions.\nDetermine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-\none."
	},
	{
		"chunk_id" : "00000878",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one.\nFind or evaluate the inverse of a function.\nUse the graph of a one-to-one function to graph its inverse function on the same axes.\nA reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in\none direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building\nfrom the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than"
	},
	{
		"chunk_id" : "00000879",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "conventional electrical resistance heating.\nIf some physical machines can run in two directions, we might ask whether some of the function machines we have\nbeen studying can also run backwards.Figure 1provides a visual representation of this question. In this section, we will\nconsider the reverse nature of functions.\nFigure1 Can a function machine operate in reverse?\nVerifying That Two Functions Are Inverse Functions"
	},
	{
		"chunk_id" : "00000880",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with\ntheCelsiusscale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees\nFahrenheitto degrees Celsius using the formula\nand substitutes 75 for to calculate\nKnowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, Betty gets the weeks weather forecast"
	},
	{
		"chunk_id" : "00000881",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fromFigure 2for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.\n296 3  Functions\nFigure2\nAt first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her\nalgebra, and can easily solve the equation for after substituting a value for For example, to convert 26 degrees\nCelsius, she could write\nAfter considering this option for a moment, however, she realizes that solving the equation for each of the temperatures"
	},
	{
		"chunk_id" : "00000882",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to\nhave a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.\nThe formula for which Betty is searching corresponds to the idea of aninverse function, which is a function for which\nthe input of the original function becomes the output of the inverse function and the output of the original function\nbecomes the input of the inverse function."
	},
	{
		"chunk_id" : "00000883",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "becomes the input of the inverse function.\nGiven a function we represent its inverse as read as inverse of The raised is part of the notation.\nIt is not an exponent; it does not imply a power of . In other words, doesnotmean because is the\nreciprocal of and not the inverse.\nThe exponent-like notation comes from an analogy between function composition and multiplication: just as\n(1 is the identity element for multiplication) for any nonzero number so equals the identity function, that is,"
	},
	{
		"chunk_id" : "00000884",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This holds for all in the domain of Informally, this means that inverse functions undo each other. However, just as\nzero does not have areciprocal, some functions do not have inverses.\nGiven a function we can verify whether some other function is the inverse of by checking whether either\nor is true. We can test whichever equation is more convenient to work with because they are\nlogically equivalent (that is, if one is true, then so is the other.)\nFor example, and are inverse functions.\nand"
	},
	{
		"chunk_id" : "00000885",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For example, and are inverse functions.\nand\nA few coordinate pairs from the graph of the function are (2, 8), (0, 0), and (2, 8). A few coordinate pairs from\nthe graph of the function are (8, 2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate\npair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.\nInverse Function\nFor anyone-to-one function a function is aninverse functionof if This can also be"
	},
	{
		"chunk_id" : "00000886",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "written as for all in the domain of It also follows that for all in the domain of\nAccess for free at openstax.org\n3.7  Inverse Functions 297\nif is the inverse of\nThe notation is read inverse. Like any other function, we can use any variable name as the input for so\nwe will often write which we read as inverse of Keep in mind that\nand not all functions have inverses.\nEXAMPLE1\nIdentifying an Inverse Function for a Given Input-Output Pair"
	},
	{
		"chunk_id" : "00000887",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If for a particular one-to-one function and what are the corresponding input and output values for\nthe inverse function?\nSolution\nThe inverse function reverses the input and output quantities, so if\nAlternatively, if we want to name the inverse function then and\nAnalysis\nNotice that if we show the coordinate pairs in a table form, the input and output are clearly reversed. SeeTable 1.\nTable1\nTRY IT #1 Given that what are the corresponding input and output values of the original\nfunction\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000888",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function\n...\nHOW TO\nGiven two functions and test whether the functions are inverses of each other.\n1. Determine whether or\n2. If either statement is true, then both are true, and and If either statement is false, then both\nare false, and and\nEXAMPLE2\nTesting Inverse Relationships Algebraically\nIf and is\n298 3  Functions\nSolution\nso\nThis is enough to answer yes to the question, but we can also verify the other formula.\nAnalysis"
	},
	{
		"chunk_id" : "00000889",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNotice the inverse operations are in reverse order of the operations from the original function.\nTRY IT #2 If and is\nEXAMPLE3\nDetermining Inverse Relationships for Power Functions\nIf (the cube function) and is\nSolution\nNo, the functions are not inverses.\nAnalysis\nThe correct inverse to the cube is, of course, the cube root that is, the one-third is an exponent, not a\nmultiplier.\nTRY IT #3 If is\nFinding Domain and Range of Inverse Functions"
	},
	{
		"chunk_id" : "00000890",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Domain and Range of Inverse Functions\nThe outputs of the function are the inputs to so the range of is also the domain of Likewise, because the\ninputs to are the outputs of the domain of is the range of We can visualize the situation as inFigure 3.\nFigure3 Domain and range of a function and its inverse\nWhen a function has no inverse function, it is possible to create a new function where that new function on a limited"
	},
	{
		"chunk_id" : "00000891",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "domain does have an inverse function. For example, the inverse of is because a square\nundoes a square root; but the square is only the inverse of the square root on the domain  since that is the\nAccess for free at openstax.org\n3.7  Inverse Functions 299\nrange of\nWe can look at this problem from the other side, starting with the square (toolkit quadratic) function If we\nwant to construct an inverse to this function, we run into a problem, because for every given output of the quadratic"
	},
	{
		"chunk_id" : "00000892",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic\nfunction corresponds to the inputs 3 and 3. But an output from a function is an input to its inverse; if this inverse input\ncorresponds to more than one inverse output (input of the original function), then the inverse is not a function at all!\nTo put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have"
	},
	{
		"chunk_id" : "00000893",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "an inverse function. In order for a function to have an inverse, it must be a one-to-one function.\nIn many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-\nto-one. For example, we can make a restricted version of the square function with its domain limited to\n which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root\nfunction).\nIf on  then the inverse function is"
	},
	{
		"chunk_id" : "00000894",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function).\nIf on  then the inverse function is\n The domain of = range of = \n The domain of = range of = \nQ&A Is it possible for a function to have more than one inverse?\nNo. If two supposedly different functions, say, and both meet the definition of being inverses of\nanother function then you can prove that We have just seen that some functions only have\ninverses if we restrict the domain of the original function. In these cases, there may be more than one way"
	},
	{
		"chunk_id" : "00000895",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to restrict the domain, leading to different inverses. However, on any one domain, the original function\nstill has only one unique inverse.\nDomain and Range of Inverse Functions\nThe range of a function is the domain of the inverse function\nThe domain of is the range of\n...\nHOW TO\nGiven a function, find the domain and range of its inverse.\n1. If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the"
	},
	{
		"chunk_id" : "00000896",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "domain of the original function as the range of the inverse.\n2. If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain\nbecomes the range of the inverse function.\nEXAMPLE4\nFinding the Inverses of Toolkit Functions\nIdentify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on"
	},
	{
		"chunk_id" : "00000897",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "which each function is one-to-one, if any. The toolkit functions are reviewed inTable 2. We restrict the domain in such a\nfashion that the function assumes ally-values exactly once.\n300 3  Functions\nConstant Identity Quadratic Cubic Reciprocal\nReciprocal squared Cube root Square root Absolute value\nTable2\nSolution\nThe constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one,\nso the constant function has no inverse."
	},
	{
		"chunk_id" : "00000898",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "so the constant function has no inverse.\nThe absolute value function can be restricted to the domain  where it is equal to the identity function.\nThe reciprocal-squared function can be restricted to the domain \nAnalysis\nWe can see that these functions (if unrestricted) are not one-to-one by looking at their graphs, shown inFigure 4. They\nboth would fail the horizontal line test. However, if a function is restricted to a certain domain so that it passes the"
	},
	{
		"chunk_id" : "00000899",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal line test, then in that restricted domain, it can have an inverse.\nFigure4 (a) Absolute value (b) Reciprocal square\nTRY IT #4 The domain of function is  and the range of function is Find the domain and\nrange of the inverse function.\nFinding and Evaluating Inverse Functions\nOnce we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a\ncomplete representation of the inverse function in many cases.\nInverting Tabular Functions"
	},
	{
		"chunk_id" : "00000900",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Inverting Tabular Functions\nSuppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is\nthe range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the\ndomain and range.\nEach row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or\ncolumn) of outputs becomes the row (or column) of inputs for the inverse function."
	},
	{
		"chunk_id" : "00000901",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n3.7  Inverse Functions 301\nEXAMPLE5\nInterpreting the Inverse of a Tabular Function\nA function is given inTable 3, showing distance in miles that a car has traveled in minutes. Find and interpret\n30 50 70 90\n20 40 60 70\nTable3\nSolution\nThe inverse function takes an output of and returns an input for So in the expression 70 is an output value\nof the original function, representing 70 miles. The inverse will return the corresponding input of the original function"
	},
	{
		"chunk_id" : "00000902",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "90 minutes, so The interpretation of this is that, to drive 70 miles, it took 90 minutes.\nAlternatively, recall that the definition of the inverse was that if then By this definition, if we are\ngiven then we are looking for a value so that In this case, we are looking for a so that\nwhich is when\nTRY IT #5 UsingTable 4, find and interpret and\n30 50 60 70 90\n20 40 50 60 70\nTable4\nEvaluating the Inverse of a Function, Given a Graph of the Original Function"
	},
	{
		"chunk_id" : "00000903",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We saw inFunctions and Function Notationthat the domain of a function can be read by observing the horizontal extent\nof its graph. We find the domain of the inverse function by observing theverticalextent of the graph of the original\nfunction, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the\ninverse function by observing thehorizontalextent of the graph of the original function, as this is the vertical extent of"
	},
	{
		"chunk_id" : "00000904",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of\nthe vertical axis of the original functions graph.\n...\nHOW TO\nGiven the graph of a function, evaluate its inverse at specific points.\n1. Find the desired input on they-axis of the given graph.\n2. Read the inverse functions output from thex-axis of the given graph.\nEXAMPLE6\nEvaluating a Function and Its Inverse from a Graph at Specific Points"
	},
	{
		"chunk_id" : "00000905",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A function is given inFigure 5. Find and\n302 3  Functions\nFigure5\nSolution\nTo evaluate we find 3 on thex-axis and find the corresponding output value on they-axis. The point tells us\nthat\nTo evaluate recall that by definition means the value ofxfor which By looking for the output\nvalue 3 on the vertical axis, we find the point on the graph, which means so by definition, See\nFigure 6.\nFigure6\nTRY IT #6 Using the graph inFigure 6, find and estimate\nFinding Inverses of Functions Represented by Formulas"
	},
	{
		"chunk_id" : "00000906",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function\nis given as a formulafor example, as a function of we can often find the inverse function by solving to obtain as\na function of\n...\nHOW TO\nGiven a function represented by a formula, find the inverse.\n1. Make sure is a one-to-one function.\n2. Solve for\n3. Interchange and\nEXAMPLE7\nInverting the Fahrenheit-to-Celsius Function"
	},
	{
		"chunk_id" : "00000907",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Inverting the Fahrenheit-to-Celsius Function\nFind a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.\nAccess for free at openstax.org\n3.7  Inverse Functions 303\nSolution\nBy solving in general, we have uncovered the inverse function. If\nthen\nIn this case, we introduced a function to represent the conversion because the input and output variables are\ndescriptive, and writing could get confusing.\nTRY IT #7 Solve for in terms of given\nEXAMPLE8"
	},
	{
		"chunk_id" : "00000908",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Solve for in terms of given\nEXAMPLE8\nSolving to Find an Inverse Function\nFind the inverse of the function\nSolution\nSo or\nAnalysis\nThe domain and range of exclude the values 3 and 4, respectively. and are equal at two points but are not the\nsame function, as we can see by creatingTable 5.\n1 2 5\n3 2 5\nTable5\nEXAMPLE9\nSolving to Find an Inverse with Radicals\nFind the inverse of the function\n304 3  Functions\nSolution\nSo"
	},
	{
		"chunk_id" : "00000909",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "304 3  Functions\nSolution\nSo\nThe domain of is  Notice that the range of is  so this means that the domain of the inverse function\nis also \nAnalysis\nThe formula we found for looks like it would be valid for all real However, itself must have an inverse\n(namely, ) so we have to restrict the domain of to  in order to make a one-to-one function. This domain\nof is exactly the range of\nTRY IT #8 What is the inverse of the function State the domains of both the function and\nthe inverse function."
	},
	{
		"chunk_id" : "00000910",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inverse function.\nFinding Inverse Functions and Their Graphs\nNow that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to\nthe quadratic function restricted to the domain  on which this function is one-to-one, and graph it as in\nFigure 7.\nFigure7 Quadratic function with domain restricted to [0, ).\nRestricting the domainto  makes the function one-to-one (it will obviously pass the horizontal line test), so it has"
	},
	{
		"chunk_id" : "00000911",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "an inverse on this restricted domain.\nWe already know that the inverse of the toolkit quadratic function is the square root function, that is,\nWhat happens if we graph both and on the same set of axes, using the axis for the input to both\nWe notice a distinct relationship: The graph of is the graph of reflected about the diagonal line which\nwe will call the identity line, shown inFigure 8.\nAccess for free at openstax.org\n3.7  Inverse Functions 305"
	},
	{
		"chunk_id" : "00000912",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.7  Inverse Functions 305\nFigure8 Square and square-root functions on the non-negative domain\nThis relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse\nswapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.\nEXAMPLE10\nFinding the Inverse of a Function Using Reflection about the Identity Line\nGiven the graph of inFigure 9, sketch a graph of\nFigure9\nSolution"
	},
	{
		"chunk_id" : "00000913",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nSolution\nThis is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain\nof  and range of   so the inverse will have a domain of   and range of \nIf we reflect this graph over the line the point reflects to and the point reflects to\nSketching the inverse on the same axes as the original graph givesFigure 10.\n306 3  Functions\nFigure10 The function and its inverse, showing reflection about the identity line"
	},
	{
		"chunk_id" : "00000914",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #9 Draw graphs of the functions and fromExample 8.\nQ&A Is there any function that is equal to its own inverse?\nYes. If then and we can think of several functions that have this property. The\nidentity function does, and so does the reciprocal function, because\nAny function where is a constant, is also equal to its own inverse.\nMEDIA\nAccess these online resources for additional instruction and practice with inverse functions.\nInverse Functions(http://openstax.org/l/inversefunction)"
	},
	{
		"chunk_id" : "00000915",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "One-to-one Functions(http://openstax.org/l/onetoone)\nInverse Function Values Using Graph(http://openstax.org/l/inversfuncgraph)\nRestricting the Domain and Finding the Inverse(http://openstax.org/l/restrictdomain)\n3.7 SECTION EXERCISES\nVerbal\n1. Describe why the horizontal 2. Why do we restrict the 3. Can a function be its own\nline test is an effective way domain of the function inverse? Explain.\nto determine whether a to find the\nfunction is one-to-one? functions inverse?"
	},
	{
		"chunk_id" : "00000916",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function is one-to-one? functions inverse?\n4. Are one-to-one functions 5. How do you find the inverse\neither always increasing or of a function algebraically?\nalways decreasing? Why or\nwhy not?\nAccess for free at openstax.org\n3.7  Inverse Functions 307\nAlgebraic\n6. Show that the function is its own\ninverse for all real numbers\nFor the following exercises, find for each function.\n7. 8. 9.\n10. 11. 12."
	},
	{
		"chunk_id" : "00000917",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7. 8. 9.\n10. 11. 12.\nFor the following exercises, find a domain on which each function is one-to-one and non-decreasing. Write the domain\nin interval notation. Then find the inverse of restricted to that domain.\n13. 14. 15.\n16. Given and\n Find and\n What does the answer tell us about the\nrelationship between and\nFor the following exercises, use function composition to verify that and are inverse functions.\n17. and 18. and\nGraphical"
	},
	{
		"chunk_id" : "00000918",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "17. and 18. and\nGraphical\nFor the following exercises, use a graphing utility to determine whether each function is one-to-one.\n19. 20. 21.\n22.\nFor the following exercises, determine whether the graph represents a one-to-one function.\n23. 24.\n308 3  Functions\nFor the following exercises, use the graph of shown inFigure 11.\nFigure11\n25. Find 26. Solve 27. Find\n28. Solve\nFor the following exercises, use the graph of the one-to-one function shown inFigure 12.\nFigure12"
	},
	{
		"chunk_id" : "00000919",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12\n29. Sketch the graph of 30. Find 31. If the complete graph of\nis shown, find the domain\nof\n32. If the complete graph of\nis shown, find the range of\nNumeric\nFor the following exercises, evaluate or solve, assuming that the function is one-to-one.\n33. If find 34. If find 35. If find\nAccess for free at openstax.org\n3.7  Inverse Functions 309\n36. If find\nFor the following exercises, use the values listed inTable 6to evaluate or solve.\n0 1 2 3 4 5 6 7 8 9\n8 0 7 4 2 6 5 3 9 1\nTable6"
	},
	{
		"chunk_id" : "00000920",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0 1 2 3 4 5 6 7 8 9\n8 0 7 4 2 6 5 3 9 1\nTable6\n37. Find 38. Solve 39. Find\n40. Solve 41. Use the tabular representation\nof inTable 7to create a table\nfor\n3 6 9 13 14\n1 4 7 12 16\nTable7\nTechnology\nFor the following exercises, find the inverse function. Then, graph the function and its inverse.\n42. 43. 44. Find the inverse function of\nUse a\ngraphing utility to find its\ndomain and range. Write\nthe domain and range in\ninterval notation.\nReal-World Applications"
	},
	{
		"chunk_id" : "00000921",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interval notation.\nReal-World Applications\n45. To convert from degrees 46. The circumference of a 47. A car travels at a constant\nCelsius to degrees circle is a function of its speed of 50 miles per hour.\nFahrenheit, we use the radius given by The distance the car travels\nformula Express the radius of a in miles is a function of\ncircle as a function of its time, in hours given by\nFind the inverse function, if\ncircumference. Call this Find the inverse\nit exists, and explain its"
	},
	{
		"chunk_id" : "00000922",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "it exists, and explain its\nfunction Find function by expressing the\nmeaning.\nand interpret its meaning. time of travel in terms of\nthe distance traveled. Call\nthis function Find\nand interpret its\nmeaning.\n310 3  Chapter Review\nChapter Review\nKey Terms\nabsolute maximum the greatest value of a function over an interval\nabsolute minimum the lowest value of a function over an interval\naverage rate of change the difference in the output values of a function found for two values of the input divided by"
	},
	{
		"chunk_id" : "00000923",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the difference between the inputs\ncomposite function the new function formed by function composition, when the output of one function is used as the\ninput of another\ndecreasing function a function is decreasing in some open interval if for any two input values and in\nthe given interval where\ndependent variable an output variable\ndomain the set of all possible input values for a relation\neven function a function whose graph is unchanged by horizontal reflection, and is symmetric about\nthe axis"
	},
	{
		"chunk_id" : "00000924",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the axis\nfunction a relation in which each input value yields a unique output value\nhorizontal compression a transformation that compresses a functions graph horizontally, by multiplying the input by\na constant\nhorizontal line test a method of testing whether a function is one-to-one by determining whether any horizontal line\nintersects the graph more than once\nhorizontal reflection a transformation that reflects a functions graph across they-axis by multiplying the input by"
	},
	{
		"chunk_id" : "00000925",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal shift a transformation that shifts a functions graph left or right by adding a positive or negative constant\nto the input\nhorizontal stretch a transformation that stretches a functions graph horizontally by multiplying the input by a\nconstant\nincreasing function a function is increasing in some open interval if for any two input values and in\nthe given interval where\nindependent variable an input variable"
	},
	{
		"chunk_id" : "00000926",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "independent variable an input variable\ninput each object or value in a domain that relates to another object or value by a relationship known as a function\ninterval notation a method of describing a set that includes all numbers between a lower limit and an upper limit; the\nlower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set,\nand a parenthesis indicating exclusion"
	},
	{
		"chunk_id" : "00000927",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and a parenthesis indicating exclusion\ninverse function for any one-to-one function the inverse is a function such that for all\nin the domain of this also implies that for all in the domain of\nlocal extrema collectively, all of a function's local maxima and minima\nlocal maximum a value of the input where a function changes from increasing to decreasing as the input value\nincreases.\nlocal minimum a value of the input where a function changes from decreasing to increasing as the input value\nincreases."
	},
	{
		"chunk_id" : "00000928",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increases.\nodd function a function whose graph is unchanged by combined horizontal and vertical reflection,\nand is symmetric about the origin\none-to-one function a function for which each value of the output is associated with a unique input value\noutput each object or value in the range that is produced when an input value is entered into a function\npiecewise function a function in which more than one formula is used to define the output"
	},
	{
		"chunk_id" : "00000929",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "range the set of output values that result from the input values in a relation\nrate of change the change of an output quantity relative to the change of the input quantity\nrelation a set of ordered pairs\nset-builder notation a method of describing a set by a rule that all of its members obey; it takes the form\nvertical compression a function transformation that compresses the functions graph vertically by multiplying the\noutput by a constant"
	},
	{
		"chunk_id" : "00000930",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output by a constant\nvertical line test a method of testing whether a graph represents a function by determining whether a vertical line\nintersects the graph no more than once\nvertical reflection a transformation that reflects a functions graph across thex-axis by multiplying the output by\nvertical shift a transformation that shifts a functions graph up or down by adding a positive or negative constant to\nthe output"
	},
	{
		"chunk_id" : "00000931",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the output\nvertical stretch a transformation that stretches a functions graph vertically by multiplying the output by a constant\nAccess for free at openstax.org\n3  Chapter Review 311\nKey Equations\nConstant function where is a constant\nIdentity function\nAbsolute value function\nQuadratic function\nCubic function\nReciprocal function\nReciprocal squared function\nSquare root function\nCube root function\nAverage rate of change\nComposite function\nVertical shift (up for )\nHorizontal shift (right for )"
	},
	{
		"chunk_id" : "00000932",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Horizontal shift (right for )\nVertical reflection\nHorizontal reflection\nVertical stretch ( )\nVertical compression\nHorizontal stretch\nHorizontal compression. ( )\nKey Concepts\n3.1Functions and Function Notation\n A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input,\nleads to exactly one range value, or output. SeeExample 1andExample 2.\n Function notation is a shorthand method for relating the input to the output in the form SeeExample 3"
	},
	{
		"chunk_id" : "00000933",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "andExample 4.\n In tabular form, a function can be represented by rows or columns that relate to input and output values. See\n312 3  Chapter Review\nExample 5.\n To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function\ncan be evaluated by replacing the input variable with a given value. SeeExample 6andExample 7.\n To solve for a specific function value, we determine the input values that yield the specific output value. See\nExample 8."
	},
	{
		"chunk_id" : "00000934",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 8.\n An algebraic form of a function can be written from an equation. SeeExample 9andExample 10.\n Input and output values of a function can be identified from a table. SeeExample 11.\n Relating input values to output values on a graph is another way to evaluate a function. SeeExample 12.\n A function is one-to-one if each output value corresponds to only one input value. SeeExample 13.\n A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one"
	},
	{
		"chunk_id" : "00000935",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "point. SeeExample 14.\n The graph of a one-to-one function passes the horizontal line test. SeeExample 15.\n3.2Domain and Range\n The domain of a function includes all real input values that would not cause us to attempt an undefined\nmathematical operation, such as dividing by zero or taking the square root of a negative number.\n The domain of a function can be determined by listing the input values of a set of ordered pairs. SeeExample 1."
	},
	{
		"chunk_id" : "00000936",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The domain of a function can also be determined by identifying the input values of a function written as an\nequation. SeeExample 2,Example 3, andExample 4.\n Interval values represented on a number line can be described using inequality notation, set-builder notation, and\ninterval notation. SeeExample 5.\n For many functions, the domain and range can be determined from a graph. SeeExample 6andExample 7."
	},
	{
		"chunk_id" : "00000937",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " An understanding of toolkit functions can be used to find the domain and range of related functions. SeeExample 8,\nExample 9, andExample 10.\n A piecewise function is described by more than one formula. SeeExample 11andExample 12.\n A piecewise function can be graphed using each algebraic formula on its assigned subdomain. SeeExample 13.\n3.3Rates of Change and Behavior of Graphs\n A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change"
	},
	{
		"chunk_id" : "00000938",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is determined using only the beginning and ending data. SeeExample 1.\n Identifying points that mark the interval on a graph can be used to find the average rate of change. SeeExample 2.\n Comparing pairs of input and output values in a table can also be used to find the average rate of change. See\nExample 3.\n An average rate of change can also be computed by determining the function values at the endpoints of an interval\ndescribed by a formula. SeeExample 4andExample 5."
	},
	{
		"chunk_id" : "00000939",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "described by a formula. SeeExample 4andExample 5.\n The average rate of change can sometimes be determined as an expression. SeeExample 6.\n A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See\nExample 7.\n A local maximum is where a function changes from increasing to decreasing and has an output value larger (more\npositive or less negative) than output values at neighboring input values."
	},
	{
		"chunk_id" : "00000940",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an\noutput value smaller (more negative or less positive) than output values at neighboring input values.\n Minima and maxima are also called extrema.\n We can find local extrema from a graph. SeeExample 8andExample 9.\n The highest and lowest points on a graph indicate the maxima and minima. SeeExample 10.\n3.4Composition of Functions"
	},
	{
		"chunk_id" : "00000941",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.4Composition of Functions\n We can perform algebraic operations on functions. SeeExample 1.\n When functions are combined, the output of the first (inner) function becomes the input of the second (outer)\nfunction.\n The function produced by combining two functions is a composite function. SeeExample 2andExample 3.\n The order of function composition must be considered when interpreting the meaning of composite functions. See\nExample 4."
	},
	{
		"chunk_id" : "00000942",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 4.\n A composite function can be evaluated by evaluating the inner function using the given input value and then\nevaluating the outer function taking as its input the output of the inner function.\n A composite function can be evaluated from a table. SeeExample 5.\n A composite function can be evaluated from a graph. SeeExample 6.\n A composite function can be evaluated from a formula. SeeExample 7.\nAccess for free at openstax.org\n3  Chapter Review 313"
	},
	{
		"chunk_id" : "00000943",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3  Chapter Review 313\n The domain of a composite function consists of those inputs in the domain of the inner function that correspond to\noutputs of the inner function that are in the domain of the outer function. SeeExample 8andExample 9.\n Just as functions can be combined to form a composite function, composite functions can be decomposed into\nsimpler functions.\n Functions can often be decomposed in more than one way. SeeExample 10.\n3.5Transformation of Functions"
	},
	{
		"chunk_id" : "00000944",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3.5Transformation of Functions\n A function can be shifted vertically by adding a constant to the output. SeeExample 1andExample 2.\n A function can be shifted horizontally by adding a constant to the input. SeeExample 3,Example 4, andExample 5.\n Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal\nshifts. SeeExample 6.\n Vertical and horizontal shifts are often combined. SeeExample 7andExample 8."
	},
	{
		"chunk_id" : "00000945",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A vertical reflection reflects a graph about the axis. A graph can be reflected vertically by multiplying the output\nby 1.\n A horizontal reflection reflects a graph about the axis. A graph can be reflected horizontally by multiplying the\ninput by 1.\n A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not\naffect the final graph. SeeExample 9."
	},
	{
		"chunk_id" : "00000946",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "affect the final graph. SeeExample 9.\n A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or\ncolumns accordingly. SeeExample 10.\n A function presented as an equation can be reflected by applying transformations one at a time. SeeExample 11.\n Even functions are symmetric about the axis, whereas odd functions are symmetric about the origin.\n Even functions satisfy the condition\n Odd functions satisfy the condition"
	},
	{
		"chunk_id" : "00000947",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Odd functions satisfy the condition\n A function can be odd, even, or neither. SeeExample 12.\n A function can be compressed or stretched vertically by multiplying the output by a constant. SeeExample 13,\nExample 14, andExample 15.\n A function can be compressed or stretched horizontally by multiplying the input by a constant. SeeExample 16,\nExample 17, andExample 18.\n The order in which different transformations are applied does affect the final function. Both vertical and horizontal"
	},
	{
		"chunk_id" : "00000948",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "transformations must be applied in the order given. However, a vertical transformation may be combined with a\nhorizontal transformation in any order. SeeExample 19andExample 20.\n3.6Absolute Value Functions\n Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See\nExample 1.\n The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes\ndirection. SeeExample 2."
	},
	{
		"chunk_id" : "00000949",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "direction. SeeExample 2.\n In an absolute value equation, an unknown variable is the input of an absolute value function.\n If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown\nvariable. SeeExample 3.\n3.7Inverse Functions\n If is the inverse of then SeeExample 1,Example 2, andExample 3.\n Only some of the toolkit functions have an inverse. SeeExample 4.\n For a function to have an inverse, it must be one-to-one (pass the horizontal line test)."
	},
	{
		"chunk_id" : "00000950",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.\n For a tabular function, exchange the input and output rows to obtain the inverse. SeeExample 5.\n The inverse of a function can be determined at specific points on its graph. SeeExample 6.\n To find the inverse of a formula, solve the equation for as a function of Then exchange the labels\nand SeeExample 7,Example 8, andExample 9."
	},
	{
		"chunk_id" : "00000951",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and SeeExample 7,Example 8, andExample 9.\n The graph of an inverse function is the reflection of the graph of the original function across the line See\nExample 10.\n314 3  Exercises\nExercises\nReview Exercises\nFunctions and Function Notation\nFor the following exercises, determine whether the relation is a function.\n1. 2. 3. for the\nindependent variable and\nthe dependent variable\n4. Is the graph inFigure 1a\nfunction?\nFigure1\nFor the following exercises, evaluate\n5. 6."
	},
	{
		"chunk_id" : "00000952",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, evaluate\n5. 6.\nFor the following exercises, determine whether the functions are one-to-one.\n7. 8.\nFor the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.\n9. 10.\nAccess for free at openstax.org\n3  Exercises 315\n11.\nFor the following exercises, graph the functions.\n12. 13.\nFor the following exercises, useFigure 2to approximate the values.\nFigure2\n14. 15. 16. If then solve for\n17. If then solve for"
	},
	{
		"chunk_id" : "00000953",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "17. If then solve for\nFor the following exercises, use the function to find the values in simplest form.\n18. 19.\nDomain and Range\nFor the following exercises, find the domain of each function, expressing answers using interval notation.\n20. 21. 22.\n316 3  Exercises\n23. Graph this piecewise\nfunction:\nRates of Change and Behavior of Graphs\nFor the following exercises, find the average rate of change of the functions from\n24. 25. 26."
	},
	{
		"chunk_id" : "00000954",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "24. 25. 26.\nFor the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing,\nor constant.\n27. 28.\n29. 30. Find the local minimum of\nthe function graphed in\nExercise 3.27.\nAccess for free at openstax.org\n3  Exercises 317\n31. Find the local extrema for 32. For the graph inFigure 3, 33. Find the absolute maximum of the\nthe function graphed in the domain of the function function graphed inFigure 3.\nExercise 3.28. is The range is\nFind the"
	},
	{
		"chunk_id" : "00000955",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Exercise 3.28. is The range is\nFind the\nabsolute minimum of the\nfunction on this interval.\nFigure3\nComposition of Functions\nFor the following exercises, find and for each pair of functions.\n34. 35. 36.\n37. 38.\nFor the following exercises, find and the domain for for each pair of functions.\n39. 40. 41.\n42.\nFor the following exercises, express each function as a composition of two functions and where\n43. 44.\nTransformation of Functions\nFor the following exercises, sketch a graph of the given function."
	},
	{
		"chunk_id" : "00000956",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "45. 46. 47.\n48. 49. 50.\n51. 52.\n318 3  Exercises\nFor the following exercises, sketch the graph of the function if the graph of the function is shown inFigure 4.\nFigure4\n53. 54.\nFor the following exercises, write the equation for the standard function represented by each of the graphs below.\n55. 56.\nFor the following exercises, determine whether each function below is even, odd, or neither.\n57. 58. 59."
	},
	{
		"chunk_id" : "00000957",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "57. 58. 59.\nFor the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.\n60. 61. 62.\nAccess for free at openstax.org\n3  Exercises 319\nAbsolute Value Functions\nFor the following exercises, write an equation for the transformation of\n63. 64. 65.\nFor the following exercises, graph the absolute value function.\n66. 67. 68.\nInverse Functions\nFor the following exercises, find for each function.\n69. 70."
	},
	{
		"chunk_id" : "00000958",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "69. 70.\nFor the following exercise, find a domain on which the function is one-to-one and non-decreasing. Write the domain in\ninterval notation. Then find the inverse of restricted to that domain.\n71. 72. Given and\n Find and\n What does the answer\ntell us about the\nrelationship between\nand\nFor the following exercises, use a graphing utility to determine whether each function is one-to-one.\n73. 74. 75. If find\n76. If find\nPractice Test"
	},
	{
		"chunk_id" : "00000959",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "73. 74. 75. If find\n76. If find\nPractice Test\nFor the following exercises, determine whether each of the following relations is a function.\n1. 2.\n320 3  Exercises\nFor the following exercises, evaluate the function at the given input.\n3. 4. 5. Show that the function\nis\nnot one-to-one.\n6. Write the domain of the 7. Given find 8. Graph the function\nfunction in in simplest\ninterval notation. form.\n9. Find the average rate of\nchange of the function\nby\nfinding in simplest\nform."
	},
	{
		"chunk_id" : "00000960",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by\nfinding in simplest\nform.\nFor the following exercises, use the functions to find the composite functions.\n10. 11. 12. Express\nas a\ncomposition of two\nfunctions, and where\nFor the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.\n13. 14.\nFor the following exercises, determine whether the functions are even, odd, or neither.\n15. 16. 17.\n18. Graph the absolute value\nfunction\nFor the following exercises, find the inverse of the function.\n19. 20."
	},
	{
		"chunk_id" : "00000961",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "19. 20.\nAccess for free at openstax.org\n3  Exercises 321\nFor the following exercises, use the graph of shown inFigure 1.\nFigure1\n21. On what intervals is the 22. On what intervals is the 23. Approximate the local\nfunction increasing? function decreasing? minimum of the function.\nExpress the answer as an\nordered pair.\n24. Approximate the local\nmaximum of the function.\nExpress the answer as an\nordered pair.\nFor the following exercises, use the graph of the piecewise function shown inFigure 2.\nFigure2"
	},
	{
		"chunk_id" : "00000962",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\n25. Find 26. Find 27. Write an equation for the\npiecewise function.\nFor the following exercises, use the values listed inTable 1.\n0 1 2 3 4 5 6 7 8\n1 3 5 7 9 11 13 15 17\nTable1\n322 3  Exercises\n28. Find 29. Solve the equation 30. Is the graph increasing or\ndecreasing on its domain?\n31. Is the function represented 32. Find 33. Given\nby the graph one-to-one? find\nAccess for free at openstax.org\n4  Introduction 323\n4 LINEAR FUNCTIONS\nA bamboo forest in China (credit: \"JFXie\"\"/Flickr)"
	},
	{
		"chunk_id" : "00000963",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Chapter Outline\n4.1Linear Functions\n4.2Modeling with Linear Functions\n4.3Fitting Linear Models to Data\nIntroduction to Linear Functions\nImagine placing a plant in the ground one day and finding that it has doubled its height just a few days later. Although it\nmay seem incredible, this can happen with certain types of bamboo species. These members of the grass family are the\nfastest-growing plants in the world. One species of bamboo has been observed to grow nearly 1.5 inches every hour.1"
	},
	{
		"chunk_id" : "00000964",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In a twenty-four hour period, this bamboo plant grows about 36 inches, or an incredible 3 feet! A constant rate of\nchange, such as the growth cycle of this bamboo plant, is a linear function.\nRecall fromFunctions and Function Notationthat a function is a relation that assigns to every element in the domain\nexactly one element in the range. Linear functions are a specific type of function that can be used to model many real-"
	},
	{
		"chunk_id" : "00000965",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "world applications, such as plant growth over time. In this chapter, we will explore linear functions, their graphs, and\nhow to relate them to data.\n4.1 Linear Functions\nLearning Objectives\nIn this section, you will:\nRepresent a linear function.\nDetermine whether a linear function is increasing, decreasing, or constant.\nInterpret slope as a rate of change.\nWrite and interpret an equation for a linear function.\nGraph linear functions.\nDetermine whether lines are parallel or perpendicular."
	},
	{
		"chunk_id" : "00000966",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write the equation of a line parallel or perpendicular to a given line.\n1 http://www.guinnessworldrecords.com/records-3000/fastest-growing-plant/\n324 4  Linear Functions\nFigure1 Shanghai MagLev Train (credit: \"kanegen\"\"/Flickr)"
	},
	{
		"chunk_id" : "00000967",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes2.\nSuppose a maglev train travels a long distance, and maintains a constant speed of 83 meters per second for a period of\ntime once it is 250 meters from the station. How can we analyze the trains distance from the station as a function of\ntime? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-"
	},
	{
		"chunk_id" : "00000968",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "world situations such as the trains distance from the station at a given point in time.\nRepresenting Linear Functions\nThe function describing the trains motion is alinear function, which is defined as a function with a constant rate of\nchange. This is a polynomial of degree 1. There are several ways to represent a linear function, including word form,\nfunction notation, tabular form, and graphical form. We will describe the trains motion as a function using each method."
	},
	{
		"chunk_id" : "00000969",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Representing a Linear Function in Word Form\nLets begin by describing the linear function in words. For the train problem we just considered, the following word\nsentence may be used to describe the function relationship.\n The trains distance from the station is a function of the time during which the train moves at a constant speed plus\nits original distance from the station when it began moving at constant speed."
	},
	{
		"chunk_id" : "00000970",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes\nwith respect to the independent variable. The rate of change for this example is constant, which means that it is the\nsame for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by\n83 meters. The train began moving at this constant speed at a distance of 250 meters from the station."
	},
	{
		"chunk_id" : "00000971",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Representing a Linear Function in Function Notation\nAnother approach to representing linear functions is by using function notation. One example of function notation is an\nequation written in theslope-intercept formof a line, where is the input value, is the rate of change, and is the\ninitial value of the dependent variable.\nIn the example of the train, we might use the notation where the total distance is a function of the time The"
	},
	{
		"chunk_id" : "00000972",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rate, is 83 meters per second. The initial value of the dependent variable is the original distance from the station,\n250 meters. We can write a generalized equation to represent the motion of the train.\nRepresenting a Linear Function in Tabular Form\nA third method of representing a linear function is through the use of a table. The relationship between the distance\nfrom the station and the time is represented inFigure 2. From the table, we can see that the distance changes by 83"
	},
	{
		"chunk_id" : "00000973",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "meters for every 1 second increase in time.\n2 http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm\nAccess for free at openstax.org\n4.1  Linear Functions 325\nFigure2 Tabular representation of the function showing selected input and output values\nQ&A Can the input in the previous example be any real number?\nNo. The input represents time so while nonnegative rational and irrational numbers are possible, negative"
	},
	{
		"chunk_id" : "00000974",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "real numbers are not possible for this example. The input consists of non-negative real numbers.\nRepresenting a Linear Function in Graphical Form\nAnother way to represent linear functions is visually, using a graph. We can use the function relationship from above,\nto draw a graph as represented inFigure 3. Notice the graph is a line.When we plot a linear function,\nthe graph is always a line."
	},
	{
		"chunk_id" : "00000975",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the graph is always a line.\nThe rate of change, which is constant, determines the slant, orslopeof the line. The point at which the input value is zero\nis the vertical intercept, ory-intercept, of the line. We can see from the graph that they-intercept in the train example we\njust saw is and represents the distance of the train from the station when it began moving at a constant speed.\nFigure3 The graph of . Graphs of linear functions are lines because the rate of change is constant."
	},
	{
		"chunk_id" : "00000976",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line\nAsk yourself what numbers can be input to the function. In other words, what is the domain of the\nfunction? The domain is comprised of all real numbers because any number may be doubled, and then have one added\nto the product.\nLinear Function\nAlinear functionis a function whose graph is a line. Linear functions can be written in theslope-intercept formof a\nline"
	},
	{
		"chunk_id" : "00000977",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "line\nwhere is the initial or starting value of the function (when input, ), and is the constant rate of change, or\nslope of the function. They-intercept is at\nEXAMPLE1\nUsing a Linear Function to Find the Pressure on a Diver\nThe pressure, in pounds per square inch (PSI) on the diver inFigure 4depends upon her depth below the water\nsurface, in feet. This relationship may be modeled by the equation, Restate this function in\nwords.\n326 4  Linear Functions\nFigure4 (credit: Ilse Reijs and Jan-Noud Hutten)"
	},
	{
		"chunk_id" : "00000978",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4 (credit: Ilse Reijs and Jan-Noud Hutten)\nSolution\nTo restate the function in words, we need to describe each part of the equation. The pressure as a function of depth\nequals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.\nAnalysis\nThe initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. The rate"
	},
	{
		"chunk_id" : "00000979",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of change, or slope, is 0.434 PSI per foot. This tells us that the pressure on the diver increases 0.434 PSI for each foot her\ndepth increases.\nDetermining Whether a Linear Function Is Increasing, Decreasing, or Constant\nThe linear functions we used in the two previous examples increased over time, but not every linear function does. A\nlinear function may be increasing, decreasing, or constant. For anincreasing function, as with the train example, the"
	},
	{
		"chunk_id" : "00000980",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with\na positive slope slants upward from left to right as inFigure 5(a). For adecreasing function, the slope is negative. The\noutput values decrease as the input values increase. A line with a negative slope slants downward from left to right as in\nFigure 5(b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with"
	},
	{
		"chunk_id" : "00000981",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a slope of zero is horizontal as inFigure 5(c).\nFigure5\nIncreasing and Decreasing Functions\nThe slope determines if the function is anincreasing linear function, adecreasing linear function, or a constant\nAccess for free at openstax.org\n4.1  Linear Functions 327\nfunction.\nis an increasing function if\nis a decreasing function if\nis a constant function if\nEXAMPLE2\nDeciding Whether a Function Is Increasing, Decreasing, or Constant"
	},
	{
		"chunk_id" : "00000982",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Studies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much\nhigher messaging rates among all users, particularly considering the various apps with which people can\ncommunicate.3. For each of the following scenarios, find the linear function that describes the relationship between the\ninput value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or\nconstant."
	},
	{
		"chunk_id" : "00000983",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "constant.\n The total number of texts a teen sends is considered a function of time in days. The input is the number of days,\nand output is the total number of texts sent.\n A person has a limit of 500 texts per month in their data plan. The input is the number of days, and output is the\ntotal number of texts remaining for the month.\n A person has an unlimited number of texts in their data plan for a cost of $50 per month. The input is the number"
	},
	{
		"chunk_id" : "00000984",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of days, and output is the total cost of texting each month.\nSolution\nAnalyze each function.\n The function can be represented as where is the number of days. The slope, 60, is positive so the\nfunction is increasing. This makes sense because the total number of texts increases with each day.\n The function can be represented as where is the number of days. In this case, the slope is\nnegative so the function is decreasing. This makes sense because the number of texts remaining decreases each day"
	},
	{
		"chunk_id" : "00000985",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and this function represents the number of texts remaining in the data plan after days.\n The cost function can be represented as because the number of days does not affect the total cost. The\nslope is 0 so the function is constant.\nInterpreting Slope as a Rate of Change\nIn the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given\ninput and output values. Recall that given two values for the input, and and two corresponding values for the"
	},
	{
		"chunk_id" : "00000986",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output, and which can be represented by a set of points, and we can calculate the slope\nNote that in function notation we can obtain two corresponding values for the output and for the function\nand so we could equivalently write\nFigure 6indicates how the slope of the line between the points, and is calculated. Recall that the slope\nmeasures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is."
	},
	{
		"chunk_id" : "00000987",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3 http://www.cbsnews.com/8301-501465_162-57400228-501465/teens-are-sending-60-texts-a-day-study-says/\n328 4  Linear Functions\nFigure6 The slope of a function is calculated by the change in divided by the change in It does not matter which\ncoordinate is used as the and which is the as long as each calculation is started with the elements from\nthe same coordinate pair.\nQ&A Are the units for slope always"
	},
	{
		"chunk_id" : "00000988",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Are the units for slope always\nYes. Think of the units as the change of output value for each unit of change in input value. An example of\nslope could be miles per hour or dollars per day. Notice the units appear as a ratio of units for the output\nper units for the input.\nCalculate Slope\nThe slope, or rate of change, of a function can be calculated according to the following:\nwhere and are input values, and are output values.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00000989",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven two points from a linear function, calculate and interpret the slope.\n1. Determine the units for output and input values.\n2. Calculate the change of output values and change of input values.\n3. Interpret the slope as the change in output values per unit of the input value.\nEXAMPLE3\nFinding the Slope of a Linear Function\nIf is a linear function, and and are points on the line, find the slope. Is this function increasing or\ndecreasing?\nSolution"
	},
	{
		"chunk_id" : "00000990",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decreasing?\nSolution\nThe coordinate pairs are and To find the rate of change, we divide the change in output by the change in\ninput.\nWe could also write the slope as The function is increasing because\nAccess for free at openstax.org\n4.1  Linear Functions 329\nAnalysis\nAs noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long\nas the first output value, ory-coordinate, used corresponds with the first input value, orx-coordinate, used. Note that if"
	},
	{
		"chunk_id" : "00000991",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we had reversed them, we would have obtained the same slope.\nTRY IT #1 If is a linear function, and and are points on the line, find the slope. Is this\nfunction increasing or decreasing?\nEXAMPLE4\nFinding the Population Change from a Linear Function\nThe population of a city increased from 23,400 to 27,800 between 2008 and 2012. Find the change of population per year\nif we assume the change was constant from 2008 to 2012.\nSolution"
	},
	{
		"chunk_id" : "00000992",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe rate of change relates the change in population to the change in time. The population increased by\npeople over the four-year time interval. To find the rate of change, divide the change in the\nnumber of people by the number of years.\nSo the population increased by 1,100 people per year.\nAnalysis\nBecause we are told that the population increased, we would expect the slope to be positive. This positive slope we\ncalculated is therefore reasonable."
	},
	{
		"chunk_id" : "00000993",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "calculated is therefore reasonable.\nTRY IT #2 The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Find the\nchange of population per year if we assume the change was constant from 2009 to 2012.\nWriting and Interpreting an Equation for a Linear Function\nRecall fromEquations and Inequalitiesthat we wrote equations in both theslope-intercept formand thepoint-slope\nform. Now we can choose which method to use to write equations for linear functions based on the information we are"
	},
	{
		"chunk_id" : "00000994",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Look at the\ngraph of the function inFigure 7.\nFigure7\n330 4  Linear Functions\nWe are not given the slope of the line, but we can choose any two points on the line to find the slope. Lets choose\nand\nNow we can substitute the slope and the coordinates of one of the points into the point-slope form.\nIf we want to rewrite the equation in the slope-intercept form, we would find"
	},
	{
		"chunk_id" : "00000995",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the\nline crosses they-axis when the output value is 7. Therefore, We now have the initial value and the slope so\nwe can substitute and into the slope-intercept form of a line.\nSo the function is and the linear equation would be\n...\nHOW TO\nGiven the graph of a linear function, write an equation to represent the function.\n1. Identify two points on the line."
	},
	{
		"chunk_id" : "00000996",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify two points on the line.\n2. Use the two points to calculate the slope.\n3. Determine where the line crosses they-axis to identify they-intercept by visual inspection.\n4. Substitute the slope andy-intercept into the slope-intercept form of a line equation.\nEXAMPLE5\nWriting an Equation for a Linear Function\nWrite an equation for a linear function given a graph of shown inFigure 8.\nAccess for free at openstax.org\n4.1  Linear Functions 331\nFigure8\nSolution"
	},
	{
		"chunk_id" : "00000997",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4.1  Linear Functions 331\nFigure8\nSolution\nIdentify two points on the line, such as and Use the points to calculate the slope.\nSubstitute the slope and the coordinates of one of the points into the point-slope form.\nWe can use algebra to rewrite the equation in the slope-intercept form.\nAnalysis\nThis makes sense because we can see fromFigure 9that the line crosses they-axis at the point which is the\ny-intercept, so\n332 4  Linear Functions\nFigure9\nEXAMPLE6\nWriting an Equation for a Linear Cost Function"
	},
	{
		"chunk_id" : "00000998",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Writing an Equation for a Linear Cost Function\nSuppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his\noffice rent. His production costs are $37.50 per item. Write a linear function where is the cost for items\nproduced in a given month.\nSolution\nThe fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is"
	},
	{
		"chunk_id" : "00000999",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$37.50. The variable cost, called the marginal cost, is represented by The cost Ben incurs is the sum of these two\ncosts, represented by\nAnalysis\nIf Ben produces 100 items in a month, his monthly cost is found by substituting 100 for\nSo his monthly cost would be $5,000.\nEXAMPLE7\nWriting an Equation for a Linear Function Given Two Points\nIf is a linear function, with and find an equation for the function in slope-intercept form.\nSolution\nWe can write the given points using coordinates."
	},
	{
		"chunk_id" : "00001000",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can write the given points using coordinates.\nWe can then use the points to calculate the slope.\nSubstitute the slope and the coordinates of one of the points into the point-slope form.\nAccess for free at openstax.org\n4.1  Linear Functions 333\nWe can use algebra to rewrite the equation in the slope-intercept form.\nTRY IT #3 If is a linear function, with and write an equation for the function in\nslope-intercept form.\nModeling Real-World Problems with Linear Functions"
	},
	{
		"chunk_id" : "00001001",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the real world, problems are not always explicitly stated in terms of a function or represented with a graph.\nFortunately, we can analyze the problem by first representing it as a linear function and then interpreting the\ncomponents of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear\nfunction, we can solve many different kinds of real-world problems.\n...\nHOW TO\nGiven a linear function and the initial value and rate of change, evaluate"
	},
	{
		"chunk_id" : "00001002",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Determine the initial value and the rate of change (slope).\n2. Substitute the values into\n3. Evaluate the function at\nEXAMPLE8\nUsing a Linear Function to Determine the Number of Songs in a Music Collection\nMarcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the\nnumber of songs, in his collection as a function of time, the number of months. How many songs will he own at the\nend of one year?\nSolution"
	},
	{
		"chunk_id" : "00001003",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "end of one year?\nSolution\nThe initial value for this function is 200 because he currently owns 200 songs, so which means that\nThe number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know\nthat We can substitute the initial value and the rate of change into the slope-intercept form of a line.\nFigure10\nWe can write the formula\nWith this formula, we can then predict how many songs Marcus will have at the end of one year (12 months). In other"
	},
	{
		"chunk_id" : "00001004",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "words, we can evaluate the function at\n334 4  Linear Functions\nMarcus will have 380 songs in 12 months.\nAnalysis\nNotice thatNis an increasing linear function. As the input (the number of months) increases, the output (number of\nsongs) increases as well.\nEXAMPLE9\nUsing a Linear Function to Calculate Salary Based on Commission\nWorking as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilyas"
	},
	{
		"chunk_id" : "00001005",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "weekly income depends on the number of new policies, he sells during the week. Last week he sold 3 new policies,\nand earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for and\ninterpret the meaning of the components of the equation.\nSolution\nThe given information gives us two input-output pairs: and We start by finding the rate of change.\nKeeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies"
	},
	{
		"chunk_id" : "00001006",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increased by 2, so the rate of change is $80 per policy. Therefore, Ilya earns a commission of $80 for each policy sold\nduring the week.\nWe can then solve for the initial value.\nThe value of is the starting value for the function and represents Ilyas income when or when no new policies\nare sold. We can interpret this as Ilyas base salary for the week, which does not depend upon the number of policies\nsold.\nWe can now write the final equation."
	},
	{
		"chunk_id" : "00001007",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sold.\nWe can now write the final equation.\nOur final interpretation is that Ilyas base salary is $520 per week and he earns an additional $80 commission for each\npolicy sold.\nEXAMPLE10\nUsing Tabular Form to Write an Equation for a Linear Function\nTable 1relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.\nnumber of weeks,w 0 2 4 6\nnumber of rats,P(w) 1000 1080 1160 1240\nTable1\nSolution"
	},
	{
		"chunk_id" : "00001008",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nSolution\nWe can see from the table that the initial value for the number of rats is 1000, so\nAccess for free at openstax.org\n4.1  Linear Functions 335\nRather than solving for we can tell from looking at the table that the population increases by 80 for every 2 weeks that\npass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.\nIf we did not notice the rate of change from the table we could still solve for the slope using any two points from the"
	},
	{
		"chunk_id" : "00001009",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "table. For example, using and\nQ&A Is the initial value always provided in a table of values likeTable 1?\nNo. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input\nof 0, then the initial value would be the corresponding output. If the initial value is not provided because\nthere is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the\nslope into and solve for"
	},
	{
		"chunk_id" : "00001010",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "slope into and solve for\nTRY IT #4 A new plant food was introduced to a young tree to test its effect on the height of the tree.Table 2\nshows the height of the tree, in feet, months since the measurements began. Write a linear\nfunction, where is the number of months since the start of the experiment.\nx 0 2 4 8 12\nH(x) 12.5 13.5 14.5 16.5 18.5\nTable2\nGraphing Linear Functions\nNow that weve seen and interpreted graphs of linear functions, lets take a look at how to create the graphs. There are"
	},
	{
		"chunk_id" : "00001011",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the\npoints. The second is by using they-intercept and slope. And the third method is by using transformations of the identity\nfunction\nGraphing a Function by Plotting Points\nTo find points of a function, we can choose input values, evaluate the function at these input values, and calculate output"
	},
	{
		"chunk_id" : "00001012",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values. The input values and corresponding output values form coordinate pairs. We then plot the coordinate pairs on a\ngrid. In general, we should evaluate the function at a minimum of two inputs in order to find at least two points on the\ngraph. For example, given the function, we might use the input values 1 and 2. Evaluating the function for an\ninput value of 1 yields an output value of 2, which is represented by the point Evaluating the function for an input"
	},
	{
		"chunk_id" : "00001013",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value of 2 yields an output value of 4, which is represented by the point Choosing three points is often advisable\nbecause if all three points do not fall on the same line, we know we made an error.\n...\nHOW TO\nGiven a linear function, graph by plotting points.\n1. Choose a minimum of two input values.\n2. Evaluate the function at each input value.\n3. Use the resulting output values to identify coordinate pairs.\n4. Plot the coordinate pairs on a grid.\n5. Draw a line through the points."
	},
	{
		"chunk_id" : "00001014",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. Draw a line through the points.\n336 4  Linear Functions\nEXAMPLE11\nGraphing by Plotting Points\nGraph by plotting points.\nSolution\nBegin by choosing input values. This function includes a fraction with a denominator of 3, so lets choose multiples of 3\nas input values. We will choose 0, 3, and 6.\nEvaluate the function at each input value, and use the output value to identify coordinate pairs.\nPlot the coordinate pairs and draw a line through the points.Figure 11represents the graph of the function"
	},
	{
		"chunk_id" : "00001015",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure11 The graph of the linear function\nAnalysis\nThe graph of the function is a line as expected for a linear function. In addition, the graph has a downward slant, which\nindicates a negative slope. This is also expected from the negative, constant rate of change in the equation for the\nfunction.\nTRY IT #5 Graph by plotting points.\nGraphing a Function Usingy-intercept and Slope\nAnother way to graph linear functions is by using specific characteristics of the function rather than plotting points. The"
	},
	{
		"chunk_id" : "00001016",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "first characteristic is itsy-intercept, which is the point at which the input value is zero. To find they-intercept, we can set\nin the equation.\nThe other characteristic of the linear function is its slope.\nLets consider the following function.\nThe slope is Because the slope is positive, we know the graph will slant upward from left to right. They-intercept is\nthe point on the graph when The graph crosses they-axis at Now we know the slope and they-intercept."
	},
	{
		"chunk_id" : "00001017",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can begin graphing by plotting the point We know that the slope is the change in they-coordinate over the\nchange in thex-coordinate. This is commonly referred to as rise over run, From our example, we have\nwhich means that the rise is 1 and the run is 2. So starting from oury-intercept we can rise 1 and then run 2, or\nAccess for free at openstax.org\n4.1  Linear Functions 337\nrun 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in\nFigure 12."
	},
	{
		"chunk_id" : "00001018",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 12.\nFigure12\nGraphical Interpretation of a Linear Function\nIn the equation\n is they-intercept of the graph and indicates the point at which the graph crosses they-axis.\n is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run)\nbetween each successive pair of points. Recall the formula for the slope:\nQ&A Do all linear functions havey-intercepts?"
	},
	{
		"chunk_id" : "00001019",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Do all linear functions havey-intercepts?\nYes. All linear functions cross the y-axis and therefore have y-intercepts.(Note:A vertical line is parallel to\nthe y-axis does not have a y-intercept, but it is not a function.)\n...\nHOW TO\nGiven the equation for a linear function, graph the function using they-intercept and slope.\n1. Evaluate the function at an input value of zero to find they-intercept.\n2. Identify the slope as the rate of change of the input value."
	},
	{
		"chunk_id" : "00001020",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Plot the point represented by they-intercept.\n4. Use to determine at least two more points on the line.\n5. Sketch the line that passes through the points.\nEXAMPLE12\nGraphing by Using they-intercept and Slope\nGraph using they-intercept and slope.\nSolution\nEvaluate the function at to find they-intercept. The output value when is 5, so the graph will cross they-axis\nat\nAccording to the equation for the function, the slope of the line is This tells us that for each vertical decrease in the"
	},
	{
		"chunk_id" : "00001021",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rise of units, the run increases by 3 units in the horizontal direction. We can now graph the function by first\nplotting they-intercept on the graph inFigure 13. From the initial value we move down 2 units and to the right 3\nunits. We can extend the line to the left and right by repeating, and then drawing a line through the points.\n338 4  Linear Functions\nFigure13 Graph of and shows how to calculate the rise over run for the slope.\nAnalysis"
	},
	{
		"chunk_id" : "00001022",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nThe graph slants downward from left to right, which means it has a negative slope as expected.\nTRY IT #6 Find a point on the graph we drew inExample 12that has a negativex-value.\nGraphing a Function Using Transformations\nAnother option for graphing is to use atransformationof the identity function A function may be transformed\nby a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.\nVertical Stretch or Compression"
	},
	{
		"chunk_id" : "00001023",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Vertical Stretch or Compression\nIn the equation the is acting as thevertical stretchorcompressionof the identity function. When is\nnegative, there is also a vertical reflection of the graph. Notice inFigure 14that multiplying the equation of by\nstretches the graph of by a factor of units if and compresses the graph of by a factor of units if\nThis means the larger the absolute value of the steeper the slope.\nFigure14 Vertical stretches and compressions and reflections on the function\nVertical Shift"
	},
	{
		"chunk_id" : "00001024",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Vertical Shift\nIn the acts as thevertical shift, moving the graph up and down without affecting the slope of the line.\nNotice inFigure 15that adding a value of to the equation of shifts the graph of a total of units up if is\npositive and units down if is negative.\nAccess for free at openstax.org\n4.1  Linear Functions 339\nFigure15 This graph illustrates vertical shifts of the function"
	},
	{
		"chunk_id" : "00001025",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of\nlinear functions. Although this may not be the easiest way to graph this type of function, it is still important to practice\neach method.\n...\nHOW TO\nGiven the equation of a linear function, use transformations to graph the linear function in the form\n1. Graph\n2. Vertically stretch or compress the graph by a factor\n3. Shift the graph up or down units.\nEXAMPLE13"
	},
	{
		"chunk_id" : "00001026",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Shift the graph up or down units.\nEXAMPLE13\nGraphing by Using Transformations\nGraph using transformations.\nSolution\nThe equation for the function shows that so the identity function is vertically compressed by The equation for\nthe function also shows that so the identity function is vertically shifted down 3 units. First, graph the identity\nfunction, and show the vertical compression as inFigure 16.\nFigure16 The function, compressed by a factor of .\nThen show the vertical shift as inFigure 17."
	},
	{
		"chunk_id" : "00001027",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then show the vertical shift as inFigure 17.\n340 4  Linear Functions\nFigure17 The function shifted down 3 units.\nTRY IT #7 Graph using transformations.\nQ&A InExample 15, could we have sketched the graph by reversing the order of the transformations?\nNo. The order of the transformations follows the order of operations. When the function is evaluated at a\ngiven input, the corresponding output is calculated by following the order of operations. This is why we"
	},
	{
		"chunk_id" : "00001028",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "performed the compression first. For example, following the order: Let the input be 2.\nWriting the Equation for a Function from the Graph of a Line\nEarlier, we wrote the equation for a linear function from a graph. Now we can extend what we know about graphing\nlinear functions to analyze graphs a little more closely. Begin by taking a look atFigure 18. We can see right away that\nthe graph crosses they-axis at the point so this is they-intercept.\nFigure18"
	},
	{
		"chunk_id" : "00001029",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure18\nThen we can calculate the slope by finding the rise and run. We can choose any two points, but lets look at the point\nTo get from this point to they-intercept, we must move up 4 units (rise) and to the right 2 units (run). So the\nslope must be\nSubstituting the slope andy-intercept into the slope-intercept form of a line gives\nAccess for free at openstax.org\n4.1  Linear Functions 341\n...\nHOW TO\nGiven a graph of linear function, find the equation to describe the function."
	},
	{
		"chunk_id" : "00001030",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify they-intercept of an equation.\n2. Choose two points to determine the slope.\n3. Substitute they-intercept and slope into the slope-intercept form of a line.\nEXAMPLE14\nMatching Linear Functions to Their Graphs\nMatch each equation of the linear functions with one of the lines inFigure 19.\n   \nFigure19\nSolution\nAnalyze the information for each function.\n This function has a slope of 2 and ay-intercept of 3. It must pass through the point (0, 3) and slant upward from"
	},
	{
		"chunk_id" : "00001031",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "left to right. We can use two points to find the slope, or we can compare it with the other functions listed. Function\nhas the same slope, but a differenty-intercept. Lines I and III have the same slant because they have the same slope.\nLine III does not pass through so must be represented by line I.\n This function also has a slope of 2, but ay-intercept of It must pass through the point and slant\nupward from left to right. It must be represented by line III."
	},
	{
		"chunk_id" : "00001032",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " This function has a slope of 2 and ay-intercept of 3. This is the only function listed with a negative slope, so it\nmust be represented by line IV because it slants downward from left to right.\n This function has a slope of and ay-intercept of 3. It must pass through the point (0, 3) and slant upward from\nleft to right. Lines I and II pass through but the slope of is less than the slope of so the line for must be\nflatter. This function is represented by Line II."
	},
	{
		"chunk_id" : "00001033",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "flatter. This function is represented by Line II.\nNow we can re-label the lines as inFigure 20.\n342 4  Linear Functions\nFigure20\nFinding thex-intercept of a Line\nSo far we have been finding they-intercepts of a function: the point at which the graph of the function crosses they-axis.\nRecall that a function may also have anx-intercept, which is thex-coordinate of the point where the graph of the\nfunction crosses thex-axis. In other words, it is the input value when the output value is zero."
	},
	{
		"chunk_id" : "00001034",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To find thex-intercept, set a function equal to zero and solve for the value of For example, consider the function\nshown.\nSet the function equal to 0 and solve for\nThe graph of the function crosses thex-axis at the point\nQ&A Do all linear functions havex-intercepts?\nNo. However, linear functions of the form where is a nonzero real number are the only examples\nof linear functions with no x-intercept. For example, is a horizontal line 5 units above the x-axis. This"
	},
	{
		"chunk_id" : "00001035",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function has no x-intercepts, as shown inFigure 21.\nAccess for free at openstax.org\n4.1  Linear Functions 343\nFigure21\nx-intercept\nThex-intercept of the function is value of when It can be solved by the equation\nEXAMPLE15\nFinding anx-intercept\nFind thex-intercept of\nSolution\nSet the function equal to zero to solve for\nThe graph crosses thex-axis at the point\nAnalysis\nA graph of the function is shown inFigure 22. We can see that thex-intercept is as we expected.\n344 4  Linear Functions\nFigure22"
	},
	{
		"chunk_id" : "00001036",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "344 4  Linear Functions\nFigure22\nTRY IT #8 Find thex-intercept of\nDescribing Horizontal and Vertical Lines\nThere are two special cases of lines on a graphhorizontal and vertical lines. Ahorizontal lineindicates a constant\noutput, ory-value. InFigure 23, we see that the output has a value of 2 for every input value. The change in outputs\nbetween any two points, therefore, is 0. In the slope formula, the numerator is 0, so the slope is 0. If we use in the"
	},
	{
		"chunk_id" : "00001037",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation the equation simplifies to In other words, the value of the function is a constant. This\ngraph represents the function\nFigure23 A horizontal line representing the function\nAvertical lineindicates a constant input, orx-value. We can see that the input value for every point on the line is 2, but\nthe output value varies. Because this input value is mapped to more than one output value, a vertical line does not"
	},
	{
		"chunk_id" : "00001038",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represent a function. Notice that between any two points, the change in the input values is zero. In the slope formula,\nthe denominator will be zero, so the slope of a vertical line is undefined.\nAccess for free at openstax.org\n4.1  Linear Functions 345\nFigure24 Example of how a line has a vertical slope. 0 in the denominator of the slope.\nA vertical line, such as the one inFigure 25,has anx-intercept, but noy-intercept unless its the line This graph\nrepresents the line"
	},
	{
		"chunk_id" : "00001039",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represents the line\nFigure25 The vertical line, which does not represent a function\nHorizontal and Vertical Lines\nLines can be horizontal or vertical.\nAhorizontal lineis a line defined by an equation in the form\nAvertical lineis a line defined by an equation in the form\nEXAMPLE16\nWriting the Equation of a Horizontal Line\nWrite the equation of the line graphed inFigure 26.\n346 4  Linear Functions\nFigure26\nSolution\nFor anyx-value, they-value is so the equation is\nEXAMPLE17"
	},
	{
		"chunk_id" : "00001040",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE17\nWriting the Equation of a Vertical Line\nWrite the equation of the line graphed inFigure 27.\nFigure27\nSolution\nThe constantx-value is so the equation is\nDetermining Whether Lines are Parallel or Perpendicular\nThe two lines inFigure 28areparallel lines: they will never intersect. They have exactly the same steepness, which means\ntheir slopes are identical. The only difference between the two lines is they-intercept. If we shifted one line vertically\ntoward the other, they would become coincident."
	},
	{
		"chunk_id" : "00001041",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "toward the other, they would become coincident.\nAccess for free at openstax.org\n4.1  Linear Functions 347\nFigure28 Parallel lines\nWe can determine from their equations whether two lines are parallel by comparing their slopes. If the slopes are the\nsame and they-intercepts are different, the lines are parallel. If the slopes are different, the lines are not parallel.\nUnlike parallel lines,perpendicular linesdo intersect. Their intersection forms a right, or 90-degree, angle. The two lines"
	},
	{
		"chunk_id" : "00001042",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inFigure 29are perpendicular.\nFigure29 Perpendicular lines\nPerpendicular lines do not have the same slope. The slopes of perpendicular lines are different from one another in a\nspecific way. The slope of one line is the negative reciprocal of the slope of the other line. The product of a number and\nits reciprocal is So, if are negative reciprocals of one another, they can be multiplied together to yield"
	},
	{
		"chunk_id" : "00001043",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To find the reciprocal of a number, divide 1 by the number. So the reciprocal of 8 is and the reciprocal of is 8. To\nfind the negative reciprocal, first find the reciprocal and then change the sign.\nAs with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that\nthe lines are neither horizontal nor vertical. The slope of each line below is the negative reciprocal of the other so the\nlines are perpendicular.\n348 4  Linear Functions"
	},
	{
		"chunk_id" : "00001044",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "lines are perpendicular.\n348 4  Linear Functions\nThe product of the slopes is 1.\nParallel and Perpendicular Lines\nTwo lines areparallel linesif they do not intersect. The slopes of the lines are the same.\nIf and only if and we say the lines coincide. Coincident lines are the same line.\nTwo lines areperpendicular linesif they intersect to form a right angle.\nEXAMPLE18\nIdentifying Parallel and Perpendicular Lines"
	},
	{
		"chunk_id" : "00001045",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Parallel and Perpendicular Lines\nGiven the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular\nlines.\nSolution\nParallel lines have the same slope. Because the functions and each have a slope of 2, they\nrepresent parallel lines. Perpendicular lines have negative reciprocal slopes. Because 2 and are negative reciprocals,\nthe functions and represent perpendicular lines.\nAnalysis\nA graph of the lines is shown inFigure 30.\nFigure30"
	},
	{
		"chunk_id" : "00001046",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure30\nThe graph shows that the lines and are parallel, and the lines and\nare perpendicular.\nWriting the Equation of a Line Parallel or Perpendicular to a Given Line\nIf we know the equation of a line, we can use what we know about slope to write the equation of a line that is either\nAccess for free at openstax.org\n4.1  Linear Functions 349\nparallel or perpendicular to the given line.\nWriting Equations of Parallel Lines\nSuppose for example, we are given the equation shown."
	},
	{
		"chunk_id" : "00001047",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We know that the slope of the line formed by the function is 3. We also know that they-intercept is Any other line\nwith a slope of 3 will be parallel to So the lines formed by all of the following functions will be parallel to\nSuppose then we want to write the equation of a line that is parallel to and passes through the point This type of\nproblem is often described as a point-slope problem because we have a point and a slope. In our example, we know that"
	},
	{
		"chunk_id" : "00001048",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the slope is 3. We need to determine which value of will give the correct line. We can begin with the point-slope form of\nan equation for a line, and then rewrite it in the slope-intercept form.\nSo is parallel to and passes through the point\n...\nHOW TO\nGiven the equation of a function and a point through which its graph passes, write the equation of a line\nparallel to the given line that passes through the given point.\n1. Find the slope of the function."
	},
	{
		"chunk_id" : "00001049",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Find the slope of the function.\n2. Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.\n3. Simplify.\nEXAMPLE19\nFinding a Line Parallel to a Given Line\nFind a line parallel to the graph of that passes through the point\nSolution\nThe slope of the given line is 3. If we choose the slope-intercept form, we can substitute and into\nthe slope-intercept form to find they-intercept.\nThe line parallel to that passes through is\nAnalysis"
	},
	{
		"chunk_id" : "00001050",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe can confirm that the two lines are parallel by graphing them.Figure 31shows that the two lines will never intersect.\n350 4  Linear Functions\nFigure31\nWriting Equations of Perpendicular Lines\nWe can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same\nslope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown."
	},
	{
		"chunk_id" : "00001051",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The slope of the line is 2, and its negative reciprocal is Any function with a slope of will be perpendicular to\nSo the lines formed by all of the following functions will be perpendicular to\nAs before, we can narrow down our choices for a particular perpendicular line if we know that it passes through a given\npoint. Suppose then we want to write the equation of a line that is perpendicular to and passes through the point"
	},
	{
		"chunk_id" : "00001052",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We already know that the slope is Now we can use the point to find they-intercept by substituting the given\nvalues into the slope-intercept form of a line and solving for\nThe equation for the function with a slope of and ay-intercept of 2 is\nSo is perpendicular to and passes through the point Be aware that perpendicular\nlines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature."
	},
	{
		"chunk_id" : "00001053",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are\nperpendicular, but the product of their slopes is not 1. Doesnt this fact contradict the definition of\nperpendicular lines?\nNo. For two perpendicular linear functions, the product of their slopes is 1. However, a vertical line is not\na function so the definition is not contradicted.\n...\nHOW TO\nGiven the equation of a function and a point through which its graph passes, write the equation of a line"
	},
	{
		"chunk_id" : "00001054",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n4.1  Linear Functions 351\nperpendicular to the given line.\n1. Find the slope of the function.\n2. Determine the negative reciprocal of the slope.\n3. Substitute the new slope and the values for and from the coordinate pair provided into\n4. Solve for\n5. Write the equation of the line.\nEXAMPLE20\nFinding the Equation of a Perpendicular Line\nFind the equation of a line perpendicular to that passes through the point\nSolution"
	},
	{
		"chunk_id" : "00001055",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe original line has slope so the slope of the perpendicular line will be its negative reciprocal, or Using this\nslope and the given point, we can find the equation of the line.\nThe line perpendicular to that passes through is\nAnalysis\nA graph of the two lines is shown inFigure 32.\nFigure32\nNote that that if we graph perpendicular lines on a graphing calculator using standard zoom, the lines may not appear"
	},
	{
		"chunk_id" : "00001056",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to be perpendicular. Adjusting the window will make it possible to zoom in further to see the intersection more closely.\nTRY IT #9 Given the function write an equation for the line passing through that is\n parallel to  perpendicular to\n...\nHOW TO\nGiven two points on a line and a third point, write the equation of the perpendicular line that passes through\nthe point.\n1. Determine the slope of the line passing through the points.\n2. Find the negative reciprocal of the slope.\n352 4  Linear Functions"
	},
	{
		"chunk_id" : "00001057",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "352 4  Linear Functions\n3. Use the slope-intercept form or point-slope form to write the equation by substituting the known values.\n4. Simplify.\nEXAMPLE21\nFinding the Equation of a Line Perpendicular to a Given Line Passing through a Point\nA line passes through the points and Find the equation of a perpendicular line that passes through the\npoint\nSolution\nFrom the two points of the given line, we can calculate the slope of that line.\nFind the negative reciprocal of the slope."
	},
	{
		"chunk_id" : "00001058",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the negative reciprocal of the slope.\nWe can then solve for they-intercept of the line passing through the point\nThe equation for the line that is perpendicular to the line passing through the two given points and also passes through\npoint is\nTRY IT #10 A line passes through the points, and Find the equation of a perpendicular line\nthat passes through the point,\nMEDIA\nAccess this online resource for additional instruction and practice with linear functions."
	},
	{
		"chunk_id" : "00001059",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Linear Functions(http://openstax.org/l/linearfunctions)\nFinding Input of Function from the Output and Graph(http://openstax.org/l/findinginput)\nGraphing Functions using Tables(http://openstax.org/l/graphwithtable)\nAccess for free at openstax.org\n4.1  Linear Functions 353\n4.1 SECTION EXERCISES\nVerbal\n1. Terry is skiing down a steep 2. Jessica is walking home 3. A boat is 100 miles away\nhill. Terry's elevation, from a friends house. After from the marina, sailing"
	},
	{
		"chunk_id" : "00001060",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "in feet after seconds is 2 minutes she is 1.4 miles directly toward it at 10 miles\ngiven by from home. Twelve minutes per hour. Write an equation\nWrite a complete sentence after leaving, she is 0.9 miles for the distance of the boat\ndescribing Terrys starting from home. What is her rate from the marina aftert\nelevation and how it is in miles per hour? hours.\nchanging over time.\n4. If the graphs of two linear 5. If a horizontal line has the\nfunctions are perpendicular, equation and a"
	},
	{
		"chunk_id" : "00001061",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions are perpendicular, equation and a\ndescribe the relationship vertical line has the\nbetween the slopes and the equation what is the\ny-intercepts. point of intersection?\nExplain why what you found\nis the point of intersection.\nAlgebraic\nFor the following exercises, determine whether the equation of the curve can be written as a linear function.\n6. 7. 8.\n9. 10. 11.\n12. 13.\nFor the following exercises, determine whether each function is increasing or decreasing.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23."
	},
	{
		"chunk_id" : "00001062",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23.\nFor the following exercises, find the slope of the line that passes through the two given points.\n24. and 25. and 26. and\n27. and 28. and\n354 4  Linear Functions\nFor the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.\n29. and 30. and 31. Passes through and\n32. Passes through and 33. Passes through and 34. Passes through and\n35. xintercept at andy 36. xintercept at andy\nintercept at intercept at"
	},
	{
		"chunk_id" : "00001063",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intercept at intercept at\nFor the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or\nneither.\n37. 38. 39.\n40.\nFor the following exercises, find thex- andy-intercepts of each equation.\n41. 42. 43.\n44. 45. 46.\nFor the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2.\nIs each pair of lines parallel, perpendicular, or neither?"
	},
	{
		"chunk_id" : "00001064",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. Line 1: Passes through 48. Line 1: Passes through 49. Line 1: Passes through\nand and and\nLine 2: Passes through Line 2: Passes through Line 2: Passes through\nand and and\n50. Line 1: Passes through 51. Line 1: Passes through\nand and\nLine 2: Passes through Line 2: Passes through\nand and\nFor the following exercises, write an equation for the line described.\n52. Write an equation for a line 53. Write an equation for a line 54. Write an equation for a line\nparallel to parallel to perpendicular to"
	},
	{
		"chunk_id" : "00001065",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parallel to parallel to perpendicular to\nand passing through the and passing through the and passing\npoint point through the point\n55. Write an equation for a line\nperpendicular to\nand passing\nthrough the point\nAccess for free at openstax.org\n4.1  Linear Functions 355\nGraphical\nFor the following exercises, find the slope of the linegraphed.\n56. 57.\nFor the following exercises, write an equation for the line graphed.\n58. 59. 60.\n61. 62. 63.\n356 4  Linear Functions"
	},
	{
		"chunk_id" : "00001066",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "58. 59. 60.\n61. 62. 63.\n356 4  Linear Functions\nFor the following exercises, match the given linear equation with its graph inFigure 33.\nFigure33\n64. 65. 66.\n67. 68. 69.\nFor the following exercises, sketch a line with the given features.\n70. Anx-intercept of 71. Anx-intercept and 72. Ay-intercept of and\nandy-intercept of y-intercept of slope\n73. Ay-intercept of and 74. Passing through the points 75. Passing through the points\nslope and and\nFor the following exercises, sketch the graph of each equation."
	},
	{
		"chunk_id" : "00001067",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "76. 77. 78.\n79. 80. 81.\n82. 83. 84.\nAccess for free at openstax.org\n4.1  Linear Functions 357\nFor the following exercises, write the equation of the line shown in the graph.\n85. 86. 87.\n88.\nNumeric\nFor the following exercises, which of the tables could represent a linear function? For each that could be linear, find a\nlinear equation that models the data.\n89. 90. 91.\n0 5 10 15 0 5 10 15 0 5 10 15\n5 10 25 40 5 30 105 230 5 20 45 70\n92. 93. 94.\n5 10 20 25 0 2 4 6 2 4 8 10"
	},
	{
		"chunk_id" : "00001068",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "92. 93. 94.\n5 10 20 25 0 2 4 6 2 4 8 10\n13 28 58 73 6 19 44 69 13 23 43 53\n95. 96.\n2 4 6 8 0 2 6 8\n4 16 36 56 6 31 106 231\n358 4  Linear Functions\nTechnology\nFor the following exercises, use a calculator or graphing technology to complete the task.\n97. If is a linear function, 98. Graph the function on a 99. Graph the function on a\n, and domain of domain of\n, find an\nequation for the function. Enter the function in a graphing\nutility. For the viewing window,\nset the minimum value of to"
	},
	{
		"chunk_id" : "00001069",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "set the minimum value of to\nbe and the maximum\nvalue of to be .\n100. Table 3shows the input, 101. Table 4shows the input, and 102. Graph the linear function\nand output, for a linear output, for a linear function on a domain of\nfunction for the function\n Fill in the missing values of the\nwhose slope is and\n Fill in the missing values of table.\ny-intercept is Label\nthe table.  Write the linear function\n Write the linear function the points for the input\nvalues of and"
	},
	{
		"chunk_id" : "00001070",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values of and\nround to 3 decimal places. p 0.5 0.8 12 b\nw 10 5.5 67.5 b q 400 700 a 1,000,000\nk 30 26 a 44 Table4\nTable3\n103. Graph the linear function 104. Graph the linear function\non a domain of where\nfor the on the same set of axes\nfunction whose slope is 75 on a domain of for\nandy-intercept is the following values of\nLabel the points for the and\ninput values of and\n\n\n\n\nExtensions\n105. Find the value of if a 106. Find the value ofyif a 107. Find the equation of the"
	},
	{
		"chunk_id" : "00001071",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "linear function goes linear function goes line that passes through\nthrough the following through the following the following points:\npoints and has the points and has the\nand\nfollowing slope: following slope:\n108. Find the equation of the 109. Find the equation of the 110. Find the equation of the\nline that passes through line that passes through line parallel to the line\nthe following points: the following points:\nthrough the point\nand and\nAccess for free at openstax.org\n4.1  Linear Functions 359"
	},
	{
		"chunk_id" : "00001072",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4.1  Linear Functions 359\n111. Find the equation of the\nline perpendicular to the\nline\nthrough the point\nFor the following exercises, use the functions\n112. Find the point of 113. Where is greater\nintersection of the lines than Where is\nand greater than\nReal-World Applications\n114. At noon, a barista notices 115. A gym membership with 116. A clothing business finds\nthat they have $20 in her two personal training there is a linear\ntip jar. If they maks an sessions costs $125, while relationship between the"
	},
	{
		"chunk_id" : "00001073",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "average of $0.50 from gym membership with number of shirts, it can\neach customer, how much five personal training sell and the price, it can\nwill the barista have in the sessions costs $260. What charge per shirt. In\ntip jar if they serve more is cost per session? particular, historical data\ncustomers during the shows that 1,000 shirts\nshift? can be sold at a price of\nwhile 3,000 shirts can\nbe sold at a price of $22.\nFind a linear equation in\nthe form\nthat gives the price they\ncan charge for shirts."
	},
	{
		"chunk_id" : "00001074",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that gives the price they\ncan charge for shirts.\n117. A phone company 118. A farmer finds there is a 119. A citys population in the\ncharges for service linear relationship year 1960 was 287,500. In\naccording to the formula: between the number of 1989 the population was\nwhere bean stalks, she plants 275,900. Compute the rate\nis the number of and the yield, each of growth of the\nminutes talked, and plant produces. When she population and make a"
	},
	{
		"chunk_id" : "00001075",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the monthly charge, in plants 30 stalks, each statement about the\ndollars. Find and interpret plant yields 30 oz of population rate of change\nthe rate of change and beans. When she plants in people per year.\ninitial value. 34 stalks, each plant\nproduces 28 oz of beans.\nFind a linear relationships\nin the form\nthat gives the yield when\nstalks are planted.\n360 4  Linear Functions\n120. A towns population has 121. Suppose that average 122. When temperature is 0"
	},
	{
		"chunk_id" : "00001076",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "been growing linearly. In annual income (in dollars) degrees Celsius, the\n2003, the population was for the years 1990 Fahrenheit temperature is\n45,000, and the through 1999 is given by 32. When the Celsius\npopulation has been the linear function: temperature is 100, the\ngrowing by 1,700 people , corresponding Fahrenheit\neach year. Write an where is the number of temperature is 212.\nequation, for the years after 1990. Which of Express the Fahrenheit"
	},
	{
		"chunk_id" : "00001077",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "population years after the following interprets temperature as a linear\n2003. the slope in the context of function of the Celsius\nthe problem? temperature,\n As of 1990, average  Find the rate of\nannual income was change of Fahrenheit\n$23,286. temperature for each unit\n In the ten-year period change temperature of\nfrom 19901999, average Celsius.\nannual income increased  Find and interpret\nby a total of $1,054.\n Each year in the  Find and interpret\ndecade of the 1990s,\naverage annual income"
	},
	{
		"chunk_id" : "00001078",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decade of the 1990s,\naverage annual income\nincreased by $1,054.\n Average annual\nincome rose to a level of\n$23,286 by the end of\n1999.\n4.2 Modeling with Linear Functions\nLearning Objectives\nIn this section, you will:\nBuild linear models from verbal descriptions.\nModel a set of data with a linear function.\nFigure1 (credit: EEK Photography/Flickr)\nElan is a college student who plans to spend a summer in Seattle. Elan has saved $3,500 for their trip and anticipates"
	},
	{
		"chunk_id" : "00001079",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "spending $400 each week on rent, food, and activities. How can we write a linear model to represent this situation? What\nwould be thex-intercept, and what can Elan learn from it? To answer these and related questions, we can create a model\nusing a linear function. Models such as this one can be extremely useful for analyzing relationships and making\nAccess for free at openstax.org\n4.2  Modeling with Linear Functions 361"
	},
	{
		"chunk_id" : "00001080",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4.2  Modeling with Linear Functions 361\npredictions based on those relationships. In this section, we will explore examples of linear function models.\nBuilding Linear Models from Verbal Descriptions\nWhen building linear models to solve problems involving quantities with a constant rate of change, we typically follow\nthe same problem strategies that we would use for any type of function. Lets briefly review them:"
	},
	{
		"chunk_id" : "00001081",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate,\nsketch a picture or define a coordinate system.\n2. Carefully read the problem to identify important information. Look for information that provides values for the\nvariables or values for parts of the functional model, such as slope and initial value.\n3. Carefully read the problem to determine what we are trying to find, identify, solve, or interpret."
	},
	{
		"chunk_id" : "00001082",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve\nchecking and tracking units, building a table, or even finding a formula for the function being used to model the\nproblem.\n5. When needed, write a formula for the function.\n6. Solve or evaluate the function using the formula.\n7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically."
	},
	{
		"chunk_id" : "00001083",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.\nNow lets take a look at the student in Seattle. In Elans situation, there are two changing quantities: time and money.\nThe amount of money they have remaining while on vacation depends on how long they stay. We can use this\ninformation to define our variables, including units.\nSo, the amount of money remaining depends on the number of weeks: ."
	},
	{
		"chunk_id" : "00001084",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also,\nbecause the slope is negative, the linear function is decreasing. This should make sense because she is spending money\neach week.\nTherate of changeis constant, so we can start with thelinear model Then we can substitute the\nintercept and slope provided.\nTo find thet-intercept (horizontal axis intercept), we set the output to zero, and solve for the input."
	},
	{
		"chunk_id" : "00001085",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thet-intercept (horizontal axis intercept) is 8.75 weeks. Because this represents the input value when the output will be\nzero, we could say that Elan will have no money left after 8.75 weeks.\nWhen modeling any real-life scenario with functions, there is typically a limited domain over which that model will be\nvalidalmost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesnt"
	},
	{
		"chunk_id" : "00001086",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before\nElan saved $3,500, but the scenario discussed poses the question once they saved $3,500 because this is when the trip\nand subsequent spending starts. It is also likely that this model is not valid after thet-intercept (horizontal axis\nintercept), unless Elan uses a credit card and goes into debt. The domain represents the set of input values, so the\nreasonable domain for this function is"
	},
	{
		"chunk_id" : "00001087",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reasonable domain for this function is\nIn this example, we were given a written description of the situation. We followed the steps of modeling a problem to\n362 4  Linear Functions\nanalyze the information. However, the information provided may not always be the same. Sometimes we might be\nprovided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the\ninformation we are given, and use it appropriately to build a linear model."
	},
	{
		"chunk_id" : "00001088",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using a Given Intercept to Build a Model\nSome real-world problems provide the vertical axis intercept, which is the constant or initial value. Once the vertical axis\nintercept is known, thet-intercept (horizontal axis intercept) can be calculated. Suppose, for example, that Hannah plans\nto pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her"
	},
	{
		"chunk_id" : "00001089",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "balance is $0. They-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per\nmonth. We can then use the slope-intercept form and the given information to develop a linear model.\nNow we can set the function equal to 0, and solve for to find thex-intercept.\nThex-intercept is the number of months it takes her to reach a balance of $0. Thex-intercept is 4 months, so it will take\nHannah four months to pay off her loan.\nUsing a Given Input and Output to Build a Model"
	},
	{
		"chunk_id" : "00001090",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using a Given Input and Output to Build a Model\nMany real-world applications are not as direct as the ones we just considered. Instead they require us to identify some\naspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the\nequation of the linear model equal to a specified output.\n...\nHOW TO\nGiven a word problem that includes two pairs of input and output values, use the linear function to solve a\nproblem."
	},
	{
		"chunk_id" : "00001091",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "problem.\n1. Identify the input and output values.\n2. Convert the data to two coordinate pairs.\n3. Find the slope.\n4. Write the linear model.\n5. Use the model to make a prediction by evaluating the function at a givenx-value.\n6. Use the model to identify anx-value that results in a giveny-value.\n7. Answer the question posed.\nEXAMPLE1\nUsing a Linear Model to Investigate a Towns Population\nA towns population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to"
	},
	{
		"chunk_id" : "00001092",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8,100. Assume this trend continues.\n Predict the population in 2013.  Identify the year in which the population will reach 15,000.\nSolution\nThe two changing quantities are the population size and time. While we could use the actual year value as the input\nquantity, doing so tends to lead to very cumbersome equations because they-intercept would correspond to the year 0,\nmore than 2000 years ago!\nTo make computation a little nicer, we will define our input as the number of years since 2004."
	},
	{
		"chunk_id" : "00001093",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n4.2  Modeling with Linear Functions 363\nTo predict the population in 2013 ( ), we would first need an equation for the population. Likewise, to find when the\npopulation would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an\nequation, we need the initial value and the rate of change, or slope.\nTo determine the rate of change, we will use the change in output per change in input."
	},
	{
		"chunk_id" : "00001094",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would\ncorrespond to giving the point Notice that through our clever choice of variable definition, we have\ngiven ourselves they-intercept of the function. The year 2009 would correspond to giving the point\nThe two coordinate pairs are and Recall that we encountered examples in which we were provided\ntwo points earlier in the chapter. We can use these values to calculate the slope."
	},
	{
		"chunk_id" : "00001095",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We already know they-intercept of the line, so we can immediately write the equation:\nTo predict the population in 2013, we evaluate our function at\nIf the trend continues, our model predicts a population of 9,620 in 2013.\nTo find when the population will reach 15,000, we can set and solve for\nOur model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the\nyear 2027."
	},
	{
		"chunk_id" : "00001096",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "year 2027.\nTRY IT #1 A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other\nexpenses. It costs $0.25 to produce each doughnut.\n Write a linear model to represent the cost of the company as a function of the number of\ndoughnuts produced.\n Find and interpret they-intercept.\nTRY IT #2 A citys population has been growing linearly. In 2008, the population was 28,200. By 2012, the\npopulation was 36,800. Assume this trend continues.\n Predict the population in 2014."
	},
	{
		"chunk_id" : "00001097",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Predict the population in 2014.\n Identify the year in which the population will reach 54,000.\nUsing a Diagram to Build a Model\nIt is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input\nand output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then,\ndetermine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are"
	},
	{
		"chunk_id" : "00001098",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "364 4  Linear Functions\ndrawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a\nrectangle is sketched, labeling width and height is helpful.\nEXAMPLE2\nUsing a Diagram to Model Distance Walked\nAnna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3\nmiles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start"
	},
	{
		"chunk_id" : "00001099",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "walking will they fall out of radio contact?\nSolution\nIn essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart,\nwhich leads us to ask a new question:\n\"Howlongwillittakethemtobe2milesapart\"\"?"
	},
	{
		"chunk_id" : "00001100",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because it is not obvious how to define our output variable, well start by drawing a picture such asFigure 2.\nFigure2\nInitial Value: They both start at the same intersection so when the distance traveled by each person should also be\n0. Thus the initial value for each is 0.\nRate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of\nchange. The slope for is 4 and the slope for is 3."
	},
	{
		"chunk_id" : "00001101",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using those values, we can write formulas for the distance each person has walked.\nFor this problem, the distances from the starting point are important. To notate these, we can define a coordinate\nsystem, identifying the starting point at the intersection where they both started. Then we can use the variable,\nwhich we introduced above, to represent Annas position, and define it to be a measurement from the starting point in"
	},
	{
		"chunk_id" : "00001102",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the eastward direction. Likewise, can use the variable, to represent Emanuels position, measured from the starting\npoint in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the\nmeasurement and the direction of measure.\nWe can then define a third variable, to be the measurement of the distance between Anna and Emanuel. Showing the\nvariables on the diagram is often helpful, as we can see fromFigure 3."
	},
	{
		"chunk_id" : "00001103",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Recall that we need to know how long it takes for the distance between them, to equal 2 miles. Notice that for any\ngiven input the outputs and represent distances.\nAccess for free at openstax.org\n4.2  Modeling with Linear Functions 365\nFigure3\nFigure 2shows us that we can use the Pythagorean Theorem because we have drawn a right angle.\nUsing the Pythagorean Theorem, we get:\nIn this scenario we are considering only positive values of so our distance will always be positive. We can simplify"
	},
	{
		"chunk_id" : "00001104",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "this answer to This means that the distance between Anna and Emanuel is also a linear function. Because is\na linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the\noutput and solve for\nThey will fall out of radio contact in 0.4 hour, or 24 minutes.\nQ&A Should I draw diagrams when given information based on a geometric shape?\nYes. Sketch the figure and label the quantities and unknowns on the sketch.\nEXAMPLE3"
	},
	{
		"chunk_id" : "00001105",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nUsing a Diagram to Model Distance Between Cities\nThere is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway\ndown this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the\ntown of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from\nWestborough?\nSolution"
	},
	{
		"chunk_id" : "00001106",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Westborough?\nSolution\nIt might help here to draw a picture of the situation. SeeFigure 4. It would then be helpful to introduce a coordinate\nsystem. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at\ncoordinates and Eastborough at\n366 4  Linear Functions\nFigure4\nUsing this point along with the origin, we can find the slope of the line from Westborough to Agritown.\nNow we can write an equation to describe the road from Westborough to Agritown."
	},
	{
		"chunk_id" : "00001107",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "From this, we can determine the perpendicular road to Eastborough will have slope Because the town of\nEastborough is at the point (20, 0), we can find the equation.\nWe can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them\nequal,\nThe roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to\nthe junction.\nAnalysis"
	},
	{
		"chunk_id" : "00001108",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the junction.\nAnalysis\nOne nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means\nreal-world applications discussing maps need linear functions to model the distances between reference points.\nTRY IT #3 There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles\nnorth. Partway down the road, it junctions with a second road, perpendicular to the first, leading"
	},
	{
		"chunk_id" : "00001109",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of\nTimpson, how far is the road junction from Timpson?\nModeling a Set of Data with Linear Functions\nReal-world situations including two or more linear functions may be modeled with asystem of linear equations.\nRemember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically,\nthere are three types of answers possible, as shown inFigure 5."
	},
	{
		"chunk_id" : "00001110",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n4.2  Modeling with Linear Functions 367\nFigure5\n...\nHOW TO\nGiven a situation that represents a system of linear equations, write the system of equations and identify the\nsolution.\n1. Identify the input and output of each linear model.\n2. Identify the slope andy-intercept of each linear model.\n3. Find the solution by setting the two linear functions equal to another and solving for or find the point of\nintersection on a graph.\nEXAMPLE4"
	},
	{
		"chunk_id" : "00001111",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intersection on a graph.\nEXAMPLE4\nBuilding a System of Linear Models to Choose a Truck Rental Company\nJamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20,\nthen 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile4. When will\nKeep on Trucking, Inc. be the better choice for Jamal?\nSolution"
	},
	{
		"chunk_id" : "00001112",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe two important quantities in this problem are the cost and the number of miles driven. Because we have two\ncompanies to consider, we will define two functions inTable 1.\nInput distance driven in miles\nOutputs cost, in dollars, for renting from Keep on Trucking\ncost, in dollars, for renting from Move It Your Way\nInitial Value Up-front fee: and\nRate of Change /mile and /mile\nTable1\nA linear function is of the form Using the rates of change and initial charges, we can write the equations"
	},
	{
		"chunk_id" : "00001113",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to\nmake that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when\nThe solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where\n4 Rates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/\n368 4  Linear Functions\nthe function is smaller."
	},
	{
		"chunk_id" : "00001114",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "368 4  Linear Functions\nthe function is smaller.\nThese graphs are sketched inFigure 6, with in blue.\nFigure6\nTo find the intersection, we set the equations equal and solve:\nThis tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph,\nor noting that is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price\nwhen more than 100 miles are driven, that is .\nMEDIA"
	},
	{
		"chunk_id" : "00001115",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess this online resource for additional instruction and practice with linear function models.\nInterpreting a Linear Function(http://openstax.org/l/interpretlinear)\n4.2 SECTION EXERCISES\nVerbal\n1. Explain how to find the 2. Explain how to find the 3. Explain how to interpret the\ninput variable in a word output variable in a word initial value in a word\nproblem that uses a linear problem that uses a linear problem that uses a linear\nfunction. function. function.\n4. Explain how to determine"
	},
	{
		"chunk_id" : "00001116",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Explain how to determine\nthe slope in a word problem\nthat uses a linear function.\nAccess for free at openstax.org\n4.2  Modeling with Linear Functions 369\nAlgebraic\n5. Find the area of a 6. Find the area of a triangle 7. Find the area of a triangle\nparallelogram bounded by bounded by thex-axis, the bounded by they-axis, the\nthey-axis, the line the line and the line and the\nline and the line perpendicular to line perpendicular to\nline parallel to passing that passes through the that passes through the"
	},
	{
		"chunk_id" : "00001117",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "through origin. origin.\n8. Find the area of a parallelogram bounded by the\nx-axis, the line the line and the\nline parallel to passing through\nFor the following exercises, consider this scenario: A towns population has been decreasing at a constant rate. In 2010\nthe population was 5,900. By 2012 the population had dropped to 4,700. Assume this trend continues.\n9. Predict the population in 10. Identify the year in which\n2016. the population will reach 0."
	},
	{
		"chunk_id" : "00001118",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2016. the population will reach 0.\nFor the following exercises, consider this scenario: A towns population has been increased at a constant rate. In 2010\nthe population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.\n11. Predict the population in 12. Identify the year in which\n2016. the population will reach\n75,000.\nFor the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate"
	},
	{
		"chunk_id" : "00001119",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 2,500 per year for 5 years.\n13. Find the linear function 14. Find a reasonable domain 15. If the function is\nthat models the towns and range for the function graphed, find and interpret\npopulation as a function thex- andy-intercepts.\nof the year, where is the\nnumber of years since the\nmodel began.\n16. If the function is 17. When will the population 18. What is the population in\ngraphed, find and interpret reach 100,000? the year 12 years from the\nthe slope of the function. onset of the model?"
	},
	{
		"chunk_id" : "00001120",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the slope of the function. onset of the model?\nFor the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half\npound a month for its first year.\n19. Find the linear function 20. Find a reasonable domain 21. If the function is\nthat models the babys and range for the function graphed, find and interpret\nweight as a function of thex- andy-intercepts.\nthe age of the baby, in\nmonths,"
	},
	{
		"chunk_id" : "00001121",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the age of the baby, in\nmonths,\n22. If the functionWis 23. When did the baby weight 24. What is the output when\ngraphed, find and interpret 10.4 pounds? the input is 6.2?\nthe slope of the function.\n370 4  Linear Functions\nFor the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter\nmonths steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were inflicted."
	},
	{
		"chunk_id" : "00001122",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "25. Find the linear function 26. Find a reasonable domain 27. If the function is\nthat models the number of and range for the function graphed, find and interpret\npeople inflicted with the thex- andy-intercepts.\ncommon cold as a\nfunction of the year,\n28. If the function is 29. When will the output reach 30. In what year will the\ngraphed, find and interpret 0? number of people be\nthe slope of the function. 9,700?\nGraphical"
	},
	{
		"chunk_id" : "00001123",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the slope of the function. 9,700?\nGraphical\nFor the following exercises, use the graph inFigure 7, which shows the profit, in thousands of dollars, of a company in\na given year, where represents the number of years since 1980.\nFigure7\n31. Find the linear function 32. Find and interpret the 33. Find and interpret the\nwhere depends on the y-intercept. x-intercept.\nnumber of years since\n1980.\n34. Find and interpret the\nslope."
	},
	{
		"chunk_id" : "00001124",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1980.\n34. Find and interpret the\nslope.\nFor the following exercises, use the graph inFigure 8, which shows the profit, in thousands of dollars, of a company in\na given year, where represents the number of years since 1980.\nFigure8\nAccess for free at openstax.org\n4.2  Modeling with Linear Functions 371\n35. Find the linear function 36. Find and interpret the 37. Find and interpret the\nwhere depends on the y-intercept. x-intercept.\nnumber of years since\n1980.\n38. Find and interpret the\nslope.\nNumeric"
	},
	{
		"chunk_id" : "00001125",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1980.\n38. Find and interpret the\nslope.\nNumeric\nFor the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown inTable\n2. Assume that the house values are changing linearly.\nYear Mississippi Hawaii\n1950 $25,200 $74,400\n2000 $71,400 $272,700\nTable2\n39. In which state have home 40. If these trends were to 41. If we assume the linear\nvalues increased at a continue, what would be trend existed before 1950"
	},
	{
		"chunk_id" : "00001126",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "higher rate? the median home value in and continues after 2000,\nMississippi in 2010? the two states median\nhouse values will be (or\nwere) equal in what year?\n(The answer might be\nabsurd.)\nFor the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown inTable\n3. Assume that the house values are changing linearly.\nYear Indiana Alabama\n1950 $37,700 $27,100\n2000 $94,300 $85,100\nTable3"
	},
	{
		"chunk_id" : "00001127",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1950 $37,700 $27,100\n2000 $94,300 $85,100\nTable3\n42. In which state have home 43. If these trends were to 44. If we assume the linear\nvalues increased at a continue, what would be trend existed before 1950\nhigher rate? the median home value in and continues after 2000,\nIndiana in 2010? the two states median\nhouse values will be (or\nwere) equal in what year?\n(The answer might be\nabsurd.)\n372 4  Linear Functions\nReal-World Applications\n45. In 2004, a school 46. In 2003, a towns 47. A phone company has a"
	},
	{
		"chunk_id" : "00001128",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "population was 1001. By population was 1431. By monthly cellular plan\n2008 the population had 2007 the population had where a customer pays a\ngrown to 1697. Assume the grown to 2134. Assume the flat monthly fee and then a\npopulation is changing population is changing certain amount of money\nlinearly. linearly. per minute used for voice\nor video calling. If a\n How much did the  How much did the\ncustomer uses 410\npopulation grow between population grow between\nminutes, the monthly cost"
	},
	{
		"chunk_id" : "00001129",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minutes, the monthly cost\nthe year 2004 and 2008? the year 2003 and 2007?\nwill be $71.50. If the\n How long did it take the  How long did it take the customer uses 720\npopulation to grow from population to grow from minutes, the monthly cost\n1001 students to 1697 1431 people to 2134 will be $118.\nstudents? people?\n Find a linear equation\n What is the average  What is the average\nfor the monthly cost of the\npopulation growth per population growth per\ncell plan as a function ofx,\nyear? year?"
	},
	{
		"chunk_id" : "00001130",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cell plan as a function ofx,\nyear? year?\nthe number of monthly\n What was the  What was the\nminutes used.\npopulation in the year population in the year\n Interpret the slope and\n2000? 2000?\ny-intercept of the equation.\n Find an equation for the  Find an equation for the\n Use your equation to\npopulation, of the population, of the town\nfind the total monthly cost\nschooltyears after 2000. years after 2000.\nif 687 minutes are used.\n Using your equation,  Using your equation,"
	},
	{
		"chunk_id" : "00001131",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Using your equation,  Using your equation,\npredict the population of predict the population of\nthe school in 2011. the town in 2014.\n48. A phone company has a 49. In 1991, the moose 50. In 2003, the owl population\nmonthly cellular data plan population in a park was in a park was measured to\nwhere a customer pays a measured to be 4,360. By be 340. By 2007, the\nflat monthly fee of $10 and 1999, the population was population was measured\nthen a certain amount of measured again to be again to be 285. The"
	},
	{
		"chunk_id" : "00001132",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "money per megabyte (MB) 5,880. Assume the population changes\nof data used on the phone. population continues to linearly. Let the input be\nIf a customer uses 20 MB, change linearly. years since 2003.\nthe monthly cost will be\n Find a formula for the  Find a formula for the\n$11.20. If the customer\nmoose population,Psince owl population, Let the\nuses 130 MB, the monthly\n1990. input be years since 2003.\ncost will be $17.80.\n What does your model  What does your model\n Find a linear equation"
	},
	{
		"chunk_id" : "00001133",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Find a linear equation\npredict the moose predict the owl population\nfor the monthly cost of the population to be in 2003? to be in 2012?\ndata plan as a function of\nthe number of MB used.\n Interpret the slope and\ny-intercept of the equation.\n Use your equation to\nfind the total monthly cost\nif 250 MB are used.\nAccess for free at openstax.org\n4.2  Modeling with Linear Functions 373\n51. The Federal Helium 52. Suppose the worlds oil 53. You are choosing between"
	},
	{
		"chunk_id" : "00001134",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Reserve held about 16 reserves in 2014 are 1,820 two different prepaid cell\nbillion cubic feet of helium billion barrels. If, on phone plans. The first plan\nin 2010 and is being average, the total reserves charges a rate of 26 cents\ndepleted by about 2.1 are decreasing by 25 billion per minute. The second\nbillion cubic feet each year. barrels of oil each year: plan charges a monthly fee\nof $19.95plus11 cents per\n Give a linear equation  Give a linear equation\nminute. How many"
	},
	{
		"chunk_id" : "00001135",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minute. How many\nfor the remaining federal for the remaining oil\nminutes would you have to\nhelium reserves, in reserves, in terms of\nuse in a month in order for\nterms of the number of the number of years since\nthe second plan to be\nyears since 2010. now.\npreferable?\n In 2015, what will the  Seven years from now,\nhelium reserves be? what will the oil reserves\n If the rate of depletion be?\ndoesnt change, in what  If the rate at which the\nyear will the Federal reserves are decreasing is"
	},
	{
		"chunk_id" : "00001136",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "year will the Federal reserves are decreasing is\nHelium Reserve be constant, when will the\ndepleted? worlds oil reserves be\ndepleted?\n54. You are choosing between 55. When hired at a new job 56. When hired at a new job\ntwo different window selling jewelry, you are selling electronics, you are\nwashing companies. The given two pay options: given two pay options:\nfirst charges $5 per\nOption A: Base salary of Option A: Base salary of\nwindow. The second\n$17,000 a year with a $14,000 a year with a"
	},
	{
		"chunk_id" : "00001137",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$17,000 a year with a $14,000 a year with a\ncharges a base fee of $40\ncommission of 12% of your commission of 10% of your\nplus $3 per window. How\nsales sales\nmany windows would you\nneed to have for the Option B: Base salary of Option B: Base salary of\nsecond company to be $20,000 a year with a $19,000 a year with a\npreferable? commission of 5% of your commission of 4% of your\nsales sales\nHow much jewelry would How much electronics\nyou need to sell for option would you need to sell for"
	},
	{
		"chunk_id" : "00001138",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A to produce a larger option A to produce a\nincome? larger income?\n57. When hired at a new job 58. When hired at a new job\nselling electronics, you are selling electronics, you are\ngiven two pay options: given two pay options:\nOption A: Base salary of Option A: Base salary of\n$20,000 a year with a $10,000 a year with a\ncommission of 12% of your commission of 9% of your\nsales sales\nOption B: Base salary of Option B: Base salary of\n$26,000 a year with a $20,000 a year with a"
	},
	{
		"chunk_id" : "00001139",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$26,000 a year with a $20,000 a year with a\ncommission of 3% of your commission of 4% of your\nsales sales\nHow much electronics How much electronics\nwould you need to sell for would you need to sell for\noption A to produce a option A to produce a\nlarger income? larger income?\n374 4  Linear Functions\n4.3 Fitting Linear Models to Data\nLearning Objectives\nIn this section, you will:\nDraw and interpret scatter diagrams.\nUse a graphing utility to find the line of best fit."
	},
	{
		"chunk_id" : "00001140",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Distinguish between linear and nonlinear relations.\nFit a regression line to a set of data and use the linear model to make predictions.\nA professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders\nif there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a\ndiagram that relates the age of each student to the exam score received. In this section, we will examine one such"
	},
	{
		"chunk_id" : "00001141",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "diagram known as a scatter plot.\nDrawing and Interpreting Scatter Plots\nAscatter plotis a graph of plotted points that may show a relationship between two sets of data. If the relationship is\nfrom alinear model, or a model that is nearly linear, the professor can draw conclusions using his knowledge of linear\nfunctions.Figure 1shows a sample scatter plot.\nFigure1 A scatter plot of age and final exam score variables"
	},
	{
		"chunk_id" : "00001142",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice this scatter plot doesnotindicate alinear relationship. The points do not appear to follow a trend. In other words,\nthere does not appear to be a relationship between the age of the student and the score on the final exam.\nEXAMPLE1\nUsing a Scatter Plot to Investigate Cricket Chirps\nTable 1shows the number of cricket chirps in 15 seconds, for several different air temperatures, in degrees Fahrenheit5.\nPlot this data, and determine whether the data appears to be linearly related."
	},
	{
		"chunk_id" : "00001143",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Chirps 44 35 20.4 33 31 35 18.5 37 26\nTemperature 80.5 70.5 57 66 68 72 52 73.5 53\nTable1 Cricket Chirps vs Air Temperature\nSolution\nPlotting this data, as depicted inFigure 2suggests that there may be a trend. We can see from the trend in the data that\nthe number of chirps increases as the temperature increases. The trend appears to be roughly linear, though certainly\nnot perfectly so.\n5 Selected data from http://classic.globe.gov/fsl/scientistsblog/2007/10/. Retrieved Aug 3, 2010"
	},
	{
		"chunk_id" : "00001144",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n4.3  Fitting Linear Models to Data 375\nFigure2\nFinding the Line of Best Fit\nOnce we recognize a need for a linear function to model that data, the natural follow-up question is what is that linear\nfunction? One way to approximate our linear function is to sketch the line that seems to best fit the data. Then we can\nextend the line until we can verify they-intercept. We can approximate the slope of the line by extending it until we can\nestimate the\nEXAMPLE2"
	},
	{
		"chunk_id" : "00001145",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "estimate the\nEXAMPLE2\nFinding a Line of Best Fit\nFind a linear function that fits the data inTable 1by eyeballing a line that seems to fit.\nSolution\nOn a graph, we could try sketching a line. Using the starting and ending points of our hand drawn line, points (0, 30) and\n(50, 90), this graph has a slope of\nand ay-intercept at 30. This gives an equation of\nwhere is the number of chirps in 15 seconds, and is the temperature in degrees Fahrenheit. The resulting\nequation is represented inFigure 3."
	},
	{
		"chunk_id" : "00001146",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation is represented inFigure 3.\n376 4  Linear Functions\nFigure3\nAnalysis\nThis linear equation can then be used to approximate answers to various questions we might ask about the trend.\nRecognizing Interpolation or Extrapolation\nWhile the data for most examples does not fall perfectly on the line, the equation is our best guess as to how the\nrelationship will behave outside of the values for which we have data. We use a process known asinterpolationwhen"
	},
	{
		"chunk_id" : "00001147",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we predict a value inside the domain and range of the data. The process ofextrapolationis used when we predict a\nvalue outside the domain and range of the data.\nFigure 4compares the two processes for the cricket-chirp data addressed inExample 2. We can see that interpolation\nwould occur if we used our model to predict temperature when the values for chirps are between 18.5 and 44.\nExtrapolation would occur if we used our model to predict temperature when the values for chirps are less than 18.5 or"
	},
	{
		"chunk_id" : "00001148",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "greater than 44.\nThere is a difference between making predictions inside the domain and range of values for which we have data and\noutside that domain and range. Predicting a value outside of the domain and range has its limitations. When our model\nno longer applies after a certain point, it is sometimes calledmodel breakdown. For example, predicting a cost function\nfor a period of two years may involve examining the data where the input is the time in years and the output is the cost."
	},
	{
		"chunk_id" : "00001149",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "But if we try to extrapolate a cost when that is in 50 years, the model would not apply because we could not\naccount for factors fifty years in the future.\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 377\nFigure4 Interpolation occurs within the domain and range of the provided data whereas extrapolation occurs outside.\nInterpolation and Extrapolation\nDifferent methods of making predictions are used to analyze data."
	},
	{
		"chunk_id" : "00001150",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The method ofinterpolationinvolves predicting a value inside the domain and/or range of the data.\nThe method ofextrapolationinvolves predicting a value outside the domain and/or range of the data.\nModel breakdown occurs at the point when the model no longer applies.\nEXAMPLE3\nUnderstanding Interpolation and Extrapolation\nUse the cricket data fromTable 1to answer the following questions:\n Would predicting the temperature when crickets are chirping 30 times in 15 seconds be interpolation or"
	},
	{
		"chunk_id" : "00001151",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "extrapolation? Make the prediction, and discuss whether it is reasonable.\n Would predicting the number of chirps crickets will make at 40 degrees be interpolation or extrapolation? Make\nthe prediction, and discuss whether it is reasonable.\nSolution\n The number of chirps in the data provided varied from 18.5 to 44. A prediction at 30 chirps per 15 seconds is\ninside the domain of our data, so would be interpolation. Using our model:\nBased on the data we have, this value seems reasonable."
	},
	{
		"chunk_id" : "00001152",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The temperature values varied from 52 to 80.5. Predicting the number of chirps at 40 degrees is extrapolation\nbecause 40 is outside the range of our data. Using our model:\nWe can compare the regions of interpolation and extrapolation usingFigure 5.\n378 4  Linear Functions\nFigure5\nAnalysis\nOur model predicts the crickets would chirp 8.33 times in 15 seconds. While this might be possible, we have no reason to"
	},
	{
		"chunk_id" : "00001153",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "believe our model is valid outside the domain and range. In fact, generally crickets stop chirping altogether below\naround 50 degrees.\nTRY IT #1 According to the data fromTable 1, what temperature can we predict it is if we counted 20 chirps\nin 15 seconds?\nFinding the Line of Best Fit Using a Graphing Utility\nWhile eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the"
	},
	{
		"chunk_id" : "00001154",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "differences between the line and data values6. One such technique is calledleast squares regressionand can be\ncomputed by many graphing calculators, spreadsheet software, statistical software, and many web-based calculators7.\nLeast squares regression is one means to determine the line that best fits the data, and here we will refer to this method\nas linear regression.\n...\nHOW TO\nGiven data of input and corresponding outputs from a linear function, find the best fit line using linear\nregression."
	},
	{
		"chunk_id" : "00001155",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "regression.\n1. Enter the input in List 1 (L1).\n2. Enter the output in List 2 (L2).\n3. On a graphing utility, select Linear Regression (LinReg).\nEXAMPLE4\nFinding a Least Squares Regression Line\nFind the least squaresregression lineusing the cricket-chirp data inTable 2.\nSolution\n1. Enter the input (chirps) in List 1 (L1).\n6 Technically, the method minimizes the sum of the squared differences in the vertical direction between the line and the data values."
	},
	{
		"chunk_id" : "00001156",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7 For example, http://www.shodor.org/unchem/math/lls/leastsq.html\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 379\n2. Enter the output (temperature) in List 2 (L2). SeeTable 2.\nL1 44 35 20.4 33 31 35 18.5 37 26\nL2 80.5 70.5 57 66 68 72 52 73.5 53\nTable2\n3. On a graphing utility, select Linear Regression (LinReg). Using the cricket chirp data from earlier, with technology\nwe obtain the equation:\nAnalysis"
	},
	{
		"chunk_id" : "00001157",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we obtain the equation:\nAnalysis\nNotice that this line is quite similar to the equation we eyeballed but should fit the data better. Notice also that using\nthis equation would change our prediction for the temperature when hearing 30 chirps in 15 seconds from 66 degrees\nto:\nThe graph of the scatter plot with the least squares regression line is shown inFigure 6.\nFigure6\nQ&A Will there ever be a case where two different lines will serve as the best fit for the data?\nNo. There is only one best fit line."
	},
	{
		"chunk_id" : "00001158",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "No. There is only one best fit line.\nDistinguishing Between Linear and Nonlinear Models\nAs we saw above with the cricket-chirp model, some data exhibit strong linear trends, but other data, like the final exam\nscores plotted by age, are clearly nonlinear. Most calculators and computer software can also provide us with the\ncorrelation coefficient, which is a measure of how closely the line fits the data. Many graphing calculators require the"
	},
	{
		"chunk_id" : "00001159",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "user to turn a \"diagnostic on\"\" selection to find the correlation coefficient"
	},
	{
		"chunk_id" : "00001160",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "meaningless. To get a sense for the relationship between the value of and the graph of the data,Figure 7shows some\nlarge data sets with their correlation coefficients. Remember, for all plots, the horizontal axis shows the input and the\nvertical axis shows the output.\n380 4  Linear Functions\nFigure7 Plotted data and related correlation coefficients. (credit: DenisBoigelot, Wikimedia Commons)\nCorrelation Coefficient\nThecorrelation coefficientis a value, between 1 and 1."
	},
	{
		"chunk_id" : "00001161",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " suggests a positive (increasing) relationship\n suggests a negative (decreasing) relationship\n The closer the value is to 0, the more scattered the data.\n The closer the value is to 1 or 1, the less scattered the data is.\nEXAMPLE5\nFinding a Correlation Coefficient\nCalculate the correlation coefficient for cricket-chirp data inTable 1.\nSolution\nBecause the data appear to follow a linear pattern, we can use technology to calculate Enter the inputs and"
	},
	{
		"chunk_id" : "00001162",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "corresponding outputs and select the Linear Regression. The calculator will also provide you with the correlation\ncoefficient, This value is very close to 1, which suggests a strong increasing linear relationship.\nNote: For some calculators, the Diagnostics must be turned \"on\"\" in order to get the correlation coefficient when linear"
	},
	{
		"chunk_id" : "00001163",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Fitting a Regression Line to a Set of Data\nOnce we determine that a set of data is linear using the correlation coefficient, we can use the regression line to make\npredictions. As we learned above, a regression line is a line that is closest to the data in the scatter plot, which means\nthat only one such line is a best fit for the data.\nEXAMPLE6\nUsing a Regression Line to Make Predictions\nGasoline consumption in the United States has been steadily increasing. Consumption data from 1994 to 2004 is shown"
	},
	{
		"chunk_id" : "00001164",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inTable 3.8 Determine whether the trend is linear, and if so, find a model for the data. Use the model to predict the\nconsumption in 2008.\n8 http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 381\nYear '94 '95 '96 '97 '98 '99 '00 '01 '02 '03 '04\nConsumption (billions of gallons) 113 116 118 119 123 125 126 128 131 133 136\nTable3"
	},
	{
		"chunk_id" : "00001165",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table3\nThe scatter plot of the data, including the least squares regression line, is shown inFigure 8.\nFigure8\nSolution\nWe can introduce a new input variable, representing years since 1994.\nThe least squares regression equation is:\nUsing technology, the correlation coefficient was calculated to be 0.9965, suggesting a very strong increasing linear\ntrend.\nUsing this to predict consumption in 2008\nThe model predicts 144.244 billion gallons of gasoline consumption in 2008."
	},
	{
		"chunk_id" : "00001166",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Use the model we created using technology inExample 6to predict the gas consumption in 2011.\nIs this an interpolation or an extrapolation?\nMEDIA\nAccess these online resources for additional instruction and practice with fitting linear models to data.\nIntroduction to Regression Analysis(http://openstax.org/l/introregress)\nLinear Regression(http://openstax.org/l/linearregress)\n382 4  Linear Functions\n4.3 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00001167",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4.3 SECTION EXERCISES\nVerbal\n1. Describe what it means if 2. What is interpolation when 3. What is extrapolation when\nthere is a model breakdown using a linear model? using a linear model?\nwhen using a linear model.\n4. Explain the difference 5. Explain how to interpret the\nbetween a positive and a absolute value of a\nnegative correlation correlation coefficient.\ncoefficient.\nAlgebraic\n6. A regression was run to determine whether there is 7. A regression was run to determine whether there is"
	},
	{
		"chunk_id" : "00001168",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a relationship between hours of TV watched per a relationship between the diameter of a tree ( ,\nday and number of sit-ups a person can do in inches) and the trees age ( , in years). The\nThe results of the regression are given below. Use results of the regression are given below. Use this\nthis to predict the number of sit-ups a person who to predict the age of a tree with diameter 10 inches.\nwatches 11 hours of TV can do."
	},
	{
		"chunk_id" : "00001169",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "watches 11 hours of TV can do.\nFor the following exercises, draw a scatter plot for the data provided. Does the data appear to be linearly related?\n8. 9.\n0 2 4 6 8 10 1 2 3 4 5 6\n22 19 15 11 6 2 46 50 59 75 100 136\n10. 11.\n100 250 300 450 600 750 1 3 5 7 9 11\n12 12.6 13.1 14 14.5 15.2 1 9 28 65 125 216\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 383\n12. For the following data, draw a scatter plot. If we 13. For the following data, draw a scatter plot. If"
	},
	{
		"chunk_id" : "00001170",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "wanted to know when the population would reach we wanted to know when the temperature\n15,000, would the answer involve interpolation or would reach 28F, would the answer involve\nextrapolation? Eyeball the line, and estimate the interpolation or extrapolation? Eyeball the line\nanswer. and estimate the answer.\nYear Population Temperature,F 16 18 20 25 30\n1990 11,500 Time, seconds 46 50 54 55 62\n1995 12,100\n2000 12,700\n2005 13,000\n2010 13,750\nGraphical"
	},
	{
		"chunk_id" : "00001171",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2000 12,700\n2005 13,000\n2010 13,750\nGraphical\nFor the following exercises, match each scatterplot with one of the four specified correlations inFigure 9andFigure 10.\nFigure9\nFigure10\n14. 15. 16.\n17.\n384 4  Linear Functions\nFor the following exercises, draw a best-fit line for the plotted data.\n18.\n19.\n20.\n21.\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 385\nNumeric\n22. The U.S. Census tracks the 23. The U.S. import of wine (in 24. Table 6shows the year and"
	},
	{
		"chunk_id" : "00001172",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "percentage of persons 25 hectoliters) for several the number of people\nyears or older who are years is given inTable 5. unemployed in a particular\ncollege graduates. That Determine whether the city for several years.\ndata for several years is trend appears linear. If so, Determine whether the\ngiven inTable 4.9 and assuming the trend trend appears linear. If so,\nDetermine whether the continues, in what year will and assuming the trend"
	},
	{
		"chunk_id" : "00001173",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "trend appears linear. If so, imports exceed 12,000 continues, in what year will\nand assuming the trend hectoliters? the number of unemployed\ncontinues, in what year will reach 5?\nthe percentage exceed Year Imports\n35%? Number\nYear\n1992 2665 Unemployed\nPercent 1994 2688 1990 750\nYear\nGraduates\n1996 3565 1992 670\n1990 21.3\n1998 4129 1994 650\n1992 21.4\n2000 4584 1996 605\n1994 22.2\n2002 5655 1998 550\n1996 23.6\n2004 6549 2000 510\n1998 24.4\n2006 7950 2002 460\n2000 25.6\n2008 8487 2004 420\n2002 26.7"
	},
	{
		"chunk_id" : "00001174",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2000 25.6\n2008 8487 2004 420\n2002 26.7\n2009 9462 2006 380\n2004 27.7\nTable5 2008 320\n2006 28\nTable6\n2008 29.4\nTable4\nTechnology\nFor the following exercises, use each set of data to calculate the regression line using a calculator or other technology\ntool, and determine the correlation coefficient to 3 decimal places of accuracy.\n25. 26.\n8 15 26 31 56 5 7 10 12 15\n23 41 53 72 103 4 12 17 22 24\n9 Based on data from http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/2014."
	},
	{
		"chunk_id" : "00001175",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "386 4  Linear Functions\n27. 28.\n3 21.9 10 18.54 4 44.8\n4 22.22 11 15.76 5 43.1\n5 22.74 12 13.68 6 38.8\n6 22.26 13 14.1 7 39\n7 20.78 14 14.02 8 38\n8 17.6 15 11.94 9 32.7\n9 16.52 16 12.76 10 30.1\n11 29.3\n12 27\n13 25.8\n29. 30.\n21 25 30 31 40 50\n17 11 2 1 18 40 100 2000\n80 1798\n60 1589\n55 1580\n40 1390\n20 1202\n31.\n900 988 1000 1010 1200 1205\n70 80 82 84 105 108\nAccess for free at openstax.org\n4.3  Fitting Linear Models to Data 387\nExtensions"
	},
	{
		"chunk_id" : "00001176",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Extensions\n32. Graph Pick a set of five ordered 33. Graph Pick a set of five ordered\npairs using inputs and use linear pairs using inputs and use linear\nregression to verify that the function is a good fit regression to verify the function.\nfor the data.\nFor the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The\nfollowing ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands of over the"
	},
	{
		"chunk_id" : "00001177",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ten-year span, (number of units sold, profit) for specific recorded years:\n34. Use linear regression to 35. Find to the nearest tenth 36. Find to the nearest tenth\ndetermine a function and interpret the and interpret the\nwhere the profit in x-intercept. y-intercept.\nthousands of dollars\ndepends on the number of\nunits sold in hundreds.\nReal-World Applications\nFor the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The"
	},
	{
		"chunk_id" : "00001178",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following ordered pairs shows the population and the year over the ten-year span, (population, year) for specific\nrecorded years:\n37. Use linear regression to determine a function 38. Predict when the population will hit 8,000.\nwhere the year depends on the population. Round\nto three decimal places of accuracy.\nFor the following exercises, consider this scenario: The profit of a company increased steadily over a ten-year span. The"
	},
	{
		"chunk_id" : "00001179",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following ordered pairs show the number of units sold in hundreds and the profit in thousands of over the ten year\nspan, (number of units sold, profit) for specific recorded years:\n39. Use linear regression to determine a functiony, 40. Predict when the profit will exceed one million\nwhere the profit in thousands of dollars depends dollars.\non the number of units sold in hundreds.\nFor the following exercises, consider this scenario: The profit of a company decreased steadily over a ten-year span. The"
	},
	{
		"chunk_id" : "00001180",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following ordered pairs show dollars and the number of units sold in hundreds and the profit in thousands of over the\nten-year span (number of units sold, profit) for specific recorded years:\n41. Use linear regression to determine a functiony, 42. Predict when the profit will dip below the $25,000\nwhere the profit in thousands of dollars depends threshold.\non the number of units sold in hundreds.\n388 4  Chapter Review\nChapter Review\nKey Terms"
	},
	{
		"chunk_id" : "00001181",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "388 4  Chapter Review\nChapter Review\nKey Terms\ncorrelation coefficient a value, between 1 and 1 that indicates the degree of linear correlation of variables, or how\nclosely a regression line fits a data set.\ndecreasing linear function a function with a negative slope: If\nextrapolation predicting a value outside the domain and range of the data\nhorizontal line a line defined by where is a real number. The slope of a horizontal line is 0.\nincreasing linear function a function with a positive slope: If"
	},
	{
		"chunk_id" : "00001182",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interpolation predicting a value inside the domain and range of the data\nleast squares regression a statistical technique for fitting a line to data in a way that minimizes the differences\nbetween the line and data values\nlinear function a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a\nstraight line\nmodel breakdown when a model no longer applies after a certain point\nparallel lines two or more lines with the same slope"
	},
	{
		"chunk_id" : "00001183",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "perpendicular lines two lines that intersect at right angles and have slopes that are negative reciprocals of each other\npoint-slope form the equation for a line that represents a linear function of the form\nslope the ratio of the change in output values to the change in input values; a measure of the steepness of a line\nslope-intercept form the equation for a line that represents a linear function in the form"
	},
	{
		"chunk_id" : "00001184",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertical line a line defined by where is a real number. The slope of a vertical line is undefined.\nKey Concepts\n4.1Linear Functions\n Linear functions can be represented in words, function notation, tabular form, and graphical form. SeeExample 1.\n An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A\ndecreasing linear function results in a graph that slants downward from left to right and has a negative slope. A"
	},
	{
		"chunk_id" : "00001185",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "constant linear function results in a graph that is a horizontal line. SeeExample 2.\n Slope is a rate of change. The slope of a linear function can be calculated by dividing the difference between\ny-values by the difference in correspondingx-values of any two points on the line. SeeExample 3andExample 4.\n An equation for a linear function can be written from a graph. SeeExample 5.\n The equation for a linear function can be written if the slope and initial value are known. SeeExample 6and\nExample 7."
	},
	{
		"chunk_id" : "00001186",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 7.\n A linear function can be used to solve real-world problems given information in different forms. SeeExample 8,\nExample 9,andExample 10.\n Linear functions can be graphed by plotting points or by using they-intercept and slope. SeeExample 11and\nExample 12.\n Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches,\ncompressions, and reflections. SeeExample 13."
	},
	{
		"chunk_id" : "00001187",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compressions, and reflections. SeeExample 13.\n The equation for a linear function can be written by interpreting the graph. SeeExample 14.\n Thex-intercept is the point at which the graph of a linear function crosses thex-axis. SeeExample 15.\n Horizontal lines are written in the form, SeeExample 16.\n Vertical lines are written in the form, SeeExample 17.\n Parallel lines have the same slope. Perpendicular lines have negative reciprocal slopes, assuming neither is vertical.\nSeeExample 18."
	},
	{
		"chunk_id" : "00001188",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 18.\n A line parallel to another line, passing through a given point, may be found by substituting the slope value of the\nline and thex- andy-values of the given point into the equation, and using the that results.\nSimilarly, the point-slope form of an equation can also be used. SeeExample 19.\n A line perpendicular to another line, passing through a given point, may be found in the same manner, with the\nexception of using the negative reciprocal slope. SeeExample 20andExample 21."
	},
	{
		"chunk_id" : "00001189",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4.2Modeling with Linear Functions\n We can use the same problem strategies that we would use for any type of function.\n When modeling and solving a problem, identify the variables and look for key values, including the slope and\ny-intercept. SeeExample 1.\n Draw a diagram, where appropriate. SeeExample 2andExample 3.\n Check for reasonableness of the answer.\nAccess for free at openstax.org\n4  Exercises 389\n Linear models may be built by identifying or calculating the slope and using they-intercept."
	},
	{
		"chunk_id" : "00001190",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Thex-intercept may be found by setting which is setting the expression equal to 0.\n The point of intersection of a system of linear equations is the point where thex- andy-values are the same. See\nExample 4.\n A graph of the system may be used to identify the points where one line falls below (or above) the other line.\n4.3Fitting Linear Models to Data\n Scatter plots show the relationship between two sets of data. SeeExample 1.\n Scatter plots may represent linear or non-linear models."
	},
	{
		"chunk_id" : "00001191",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The line of best fit may be estimated or calculated, using a calculator or statistical software. SeeExample 2.\n Interpolation can be used to predict values inside the domain and range of the data, whereas extrapolation can be\nused to predict values outside the domain and range of the data. SeeExample 3.\n The correlation coefficient, indicates the degree of linear relationship between data. SeeExample 4.\n A regression line best fits the data. SeeExample 5."
	},
	{
		"chunk_id" : "00001192",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The least squares regression line is found by minimizing the squares of the distances of points from a line passing\nthrough the data and may be used to make predictions regarding either of the variables. SeeExample 6.\nExercises\nReview Exercises\nLinear Functions\n1. Determine whether the 2. Determine whether the 3. Determine whether the\nalgebraic equation is linear. algebraic equation is linear. function is increasing or\ndecreasing.\n4. Determine whether the 5. Given each set of 6. Given each set of"
	},
	{
		"chunk_id" : "00001193",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function is increasing or information, find a linear information, find a linear\ndecreasing. equation that satisfies the equation that satisfies the\ngiven conditions, if possible. given conditions, if possible.\nPasses through and x-intercept at and\ny-intercept at\n7. Find the slope of the line shown in the graph. 8. Find the slope of the line graphed.\n390 4  Exercises\n9. Write an equation in slope-intercept form for the 10. Does the following table\nline shown. represent a linear function? If"
	},
	{
		"chunk_id" : "00001194",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "line shown. represent a linear function? If\nso, find the linear equation\nthat models the data.\nx 4 0 2 10\ng(x) 18 2 12 52\n11. Does the following table 12. On June 1st, a company has\nrepresent a linear function? If $4,000,000 profit. If the\nso, find the linear equation that company then loses\nmodels the data. 150,000 dollars per day\nthereafter in the month of\nx 6 8 12 26 June, what is the\ncompanys profitnthday\ng(x) 8 12 18 46 after June 1st?"
	},
	{
		"chunk_id" : "00001195",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "g(x) 8 12 18 46 after June 1st?\nFor the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or\nneither parallel nor perpendicular:\n13. 14.\nFor the following exercises, find thex- andy- intercepts of the given equation\n15. 16.\nFor the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of\nlines parallel, perpendicular, or neither?"
	},
	{
		"chunk_id" : "00001196",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "lines parallel, perpendicular, or neither?\n17. Line 1: Passes through 18. Line 1: Passes through 19. Write an equation for a line\nand and perpendicular to\nand passing\nLine 2: Passes through Line 2: Passes through\nthrough the point (5, 20).\nand and\nAccess for free at openstax.org\n4  Exercises 391\n20. Find the equation of a line 21. Sketch a graph of the linear 22. Find the point of\nwith ay- intercept of function intersection for the 2 linear\nand slope\nfunctions:\n23. A car rental company"
	},
	{
		"chunk_id" : "00001197",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and slope\nfunctions:\n23. A car rental company\noffers two plans for renting\na car.\nPlan A: 25 dollars per day\nand 10 cents per mile\nPlan B: 50 dollars per day\nwith free unlimited mileage\nHow many miles would you\nneed to drive for plan B to\nsave you money?\nModeling with Linear Functions\n24. Find the area of a triangle 25. A towns population 26. The number of people\nbounded by theyaxis, the increases at a constant afflicted with the common\nline and rate. In 2010 the cold in the winter months"
	},
	{
		"chunk_id" : "00001198",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the line perpendicular to population was 55,000. By dropped steadily by 50\nthat passes through the 2012 the population had each year since 2004 until\norigin. increased to 76,000. If this 2010. In 2004, 875 people\ntrend continues, predict were inflicted.\nthe population in 2016.\nFind the linear function\nthat models the number of\npeople afflicted with the\ncommon coldCas a\nfunction of the year,\nWhen will no one be\nafflicted?"
	},
	{
		"chunk_id" : "00001199",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "When will no one be\nafflicted?\nFor the following exercises, use the graph inFigure 1showing the profit, in thousands of dollars, of a company in a\ngiven year, where represents years since 1980.\nFigure1\n27. Find the linear functiony, whereydepends on 28. Find and interpret they-intercept.\nthe number of years since 1980.\n392 4  Exercises\nFor the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had\ngrown to 2,500."
	},
	{
		"chunk_id" : "00001200",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "grown to 2,500.\n29. Assume the population is changing linearly.\n How much did the population grow between\nthe year 2004 and 2012?\n What is the average population growth per\nyear?\n Find an equation for the population,P, of the\nschooltyears after 2004.\nFor the following exercises, consider this scenario: In 2000, the moose population in a park was measured to be 6,500. By\n2010, the population was measured to be 12,500. Assume the population continues to change linearly."
	},
	{
		"chunk_id" : "00001201",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "30. Find a formula for the 31. What does your model\nmoose population, . predict the moose\npopulation to be in 2020?\nFor the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley\n(adjusted for inflation) are shown inTable 1. Assume that the house values are changing linearly.\nYear Pima Central East Valley\n1970 32,000 120,250\n2010 85,000 150,000\nTable1\n32. In which subdivision have 33. If these trends were to"
	},
	{
		"chunk_id" : "00001202",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "home values increased at a continue, what would be\nhigher rate? the median home value in\nPima Central in 2015?\nAccess for free at openstax.org\n4  Exercises 393\nFitting Linear Models to Data\n34. Draw a scatter plot for the 35. Draw a scatter plot for the 36. Eight students were asked\ndata inTable 2. Then data inTable 3. If we to estimate their score on a\ndetermine whether the wanted to know when the 10-point quiz. Their\ndata appears to be linearly population would reach estimated and actual"
	},
	{
		"chunk_id" : "00001203",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "related. 15,000, would the answer scores are given inTable 4.\ninvolve interpolation or Plot the points, then sketch\n0 -105 extrapolation? a line that fits the data.\n2 -50 Year Population Predicted Actual\n4 1 1990 5,600 6 6\n6 55 1995 5,950 7 7\n8 105 2000 6,300 7 8\n10 160 2005 6,600 8 8\nTable2 2010 6,900 7 9\nTable3 9 10\n10 10\n10 9\nTable4\n37. Draw a best-fit line for the plotted\ndata.\nFor the following exercises, consider the data inTable 5, which shows the percent of unemployed in a city of people 25"
	},
	{
		"chunk_id" : "00001204",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "years or older who are college graduates is given below, by year.\nYear 2000 2002 2005 2007 2010\nPercent Graduates 6.5 7.0 7.4 8.2 9.0\nTable5\n394 4  Exercises\n38. Determine whether the 39. In what year will the 40. Based on the set of data given\ntrend appears to be linear. percentage exceed 12%? inTable 6, calculate the\nIf so, and assuming the regression line using a\ntrend continues, find a calculator or other technology\nlinear regression model to tool, and determine the"
	},
	{
		"chunk_id" : "00001205",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "predict the percent of correlation coefficient to three\nunemployed in a given decimal places.\nyear to three decimal\nplaces. 17 20 23 26 29\n15 25 31 37 40\nTable6\n41. Based on the set of data given\ninTable 7, calculate the\nregression line using a\ncalculator or other technology\ntool, and determine the\ncorrelation coefficient to three\ndecimal places.\n10 12 15 18 20\n36 34 30 28 22\nTable7\nFor the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The"
	},
	{
		"chunk_id" : "00001206",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following ordered pairs show the population and the year over the ten-year span (population, year) for specific recorded\nyears:\n42. Use linear regression to 43. Predict when the 44. What is the correlation\ndetermine a function population will hit 12,000. coefficient for this model to\nwhere the year depends on three decimal places of\nthe population, to three accuracy?\ndecimal places of accuracy.\n45. According to the model,\nwhat is the population in\n2014?\nPractice Test"
	},
	{
		"chunk_id" : "00001207",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "what is the population in\n2014?\nPractice Test\n1. Determine whether the 2. Determine whether the 3. Determine whether the\nfollowing algebraic equation following function is following function is\ncan be written as a linear increasing or decreasing. increasing or decreasing.\nfunction.\nAccess for free at openstax.org\n4  Exercises 395\n4. Find a linear equation that 5. Find a linear equation, that 6. Find the slope of the line inFigure\npasses through (5, 1) and (3, has anxintercept at (4, 0) 1."
	},
	{
		"chunk_id" : "00001208",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9), if possible. and ay-intercept at (0, 6), if\npossible.\nFigure1\n7. Write an equation for line inFigure 2. 8. DoesTable 1represent a linear function? If so, find\na linear equation that models the data.\n6 0 2 4\n14 32 38 44\nTable1\nFigure2\n9. DoesTable 2represent a linear function? If so, find 10. At 6 am, an online\na linear equation that models the data. company has sold 120\nitems that day. If the\nx 1 3 7 11 company sells an average\nof 30 items per hour for\ng(x) 4 9 19 12 the remainder of the day,"
	},
	{
		"chunk_id" : "00001209",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "g(x) 4 9 19 12 the remainder of the day,\nwrite an expression to\nTable2 represent the number of\nitems that were sold after\n6 am.\n396 4  Exercises\nFor the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or\nneither parallel nor perpendicular.\n11. 12. 13. Find thex- andy-intercepts\nof the equation\n14. Given below are 15. Write an equation for a line 16. Sketch a line with a\ndescriptions of two lines. perpendicular to y-intercept of and"
	},
	{
		"chunk_id" : "00001210",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the slopes of Line 1 and passing slope\nand Line 2. Is the pair of through the point\nlines parallel,\nperpendicular, or neither?\nLine 1: Passes through\nand\nLine 2: Passes through\nand\n17. Graph of the linear 18. For the two linear 19. A car rental company\nfunction functions, find the point of offers two plans for renting\nintersection: a car.\nPlan A: $25 per day and\n$0.10 per mile\nPlan B: $40 per day with\nfree unlimited mileage\nHow many miles would you\nneed to drive for plan B to\nsave you money?"
	},
	{
		"chunk_id" : "00001211",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "need to drive for plan B to\nsave you money?\n20. Find the area of a triangle 21. A towns population 22. The number of people\nbounded by theyaxis, the increases at a constant afflicted with the common\nline and rate. In 2010 the cold in the winter months\nthe line perpendicular to population was 65,000. By dropped steadily by 25\nthat passes through the 2012 the population had each year since 2002 until\norigin. increased to 90,000. 2012. In 2002, 8,040 people\nAssuming this trend were inflicted. Find the"
	},
	{
		"chunk_id" : "00001212",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Assuming this trend were inflicted. Find the\ncontinues, predict the linear function that models\npopulation in 2018. the number of people\nafflicted with the common\ncold as a function of the\nyear, When will less than\n6,000 people be afflicted?\nAccess for free at openstax.org\n4  Exercises 397\nFor the following exercises, use the graph inFigure 3, showing the profit, in thousands of dollars, of a company in a\ngiven year, where represents years since 1980.\nFigure3"
	},
	{
		"chunk_id" : "00001213",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure3\n23. Find the linear function 24. Find and interpret the 25. In 2004, a school\nwhere depends on the y-intercept. population was 1250. By\nnumber of years since 2012 the population had\n1980. dropped to 875. Assume\nthe population is changing\nlinearly.\n How much did the\npopulation drop between\nthe year 2004 and 2012?\n What is the average\npopulation decline per\nyear?\n Find an equation for the\npopulation,P, of the school\ntyears after 2004."
	},
	{
		"chunk_id" : "00001214",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "population,P, of the school\ntyears after 2004.\n26. Draw a scatter plot for the data provided inTable 27. Draw a best-fit line for the plotted data.\n3. Then determine whether the data appears to be\nlinearly related.\n0 2 4 6 8 10\n450 200 10 265 500 755\nTable3\nFor the following exercises, useTable 4, which shows the percent of unemployed persons 25 years or older who are\ncollege graduates in a particular city, by year.\nYear 2000 2002 2005 2007 2010\nPercent Graduates 8.5 8.0 7.2 6.7 6.4\nTable4"
	},
	{
		"chunk_id" : "00001215",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Percent Graduates 8.5 8.0 7.2 6.7 6.4\nTable4\n398 4  Exercises\n28. Determine whether the 29. In what year will the 30. Based on the set of data given in\ntrend appears linear. If so, percentage drop below Table 5, calculate the regression line\nand assuming the trend 4%? using a calculator or other\ncontinues, find a linear technology tool, and determine the\nregression model to correlation coefficient. Round to\npredict the percent of three decimal places of accuracy.\nunemployed in a given"
	},
	{
		"chunk_id" : "00001216",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unemployed in a given\nyear to three decimal x 16 18 20 24 26\nplaces.\ny 106 110 115 120 125\nTable5\nFor the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The\nfollowing ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for\nspecific recorded years:\n31. Use linear regression to 32. Predict when the 33. What is the correlation"
	},
	{
		"chunk_id" : "00001217",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determine a functiony, population will hit 20,000. coefficient for this model?\nwhere the year depends on\nthe population. Round to\nthree decimal places of\naccuracy.\nAccess for free at openstax.org\n5  Introduction 399\n5 POLYNOMIAL AND RATIONAL FUNCTIONS\nWhether they think about it in mathematical terms or not, scuba divers must consider the impact of functional\nrelationships in order to remain safe. The gas laws, which are a series of relations and equations that describe the"
	},
	{
		"chunk_id" : "00001218",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "behavior of most gases, play a core role in diving. This diver, near the wreck of a World War II Japanese ocean liner\nturned troop transport, must remain attentive to gas laws during their dive and as they ascend to the surface. (credit:\n\"Aikoku - Aft Gun\"\": modification of work by montereydiver/flickr)"
	},
	{
		"chunk_id" : "00001219",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.6Rational Functions\n5.7Inverses and Radical Functions\n5.8Modeling Using Variation\nIntroduction to Polynomial and Rational Functions\nYou don't need to dive very deep to feel the effects of pressure. As a person in their neighborhood pool moves eight,\nten, twelve feet down, they often feel pain in their ears as a result of water and air pressure differentials. Pressure plays\na much greater role at ocean diving depths."
	},
	{
		"chunk_id" : "00001220",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a much greater role at ocean diving depths.\nid=\"scuban\"\">Scuba and free divers are constantly negotiating the effects of pressure in order to experience enjoyable"
	},
	{
		"chunk_id" : "00001221",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "simple, such as the inverse relationship regarding pressure and volume, and others are more complex. While their\nformulas seem more straightforward than many you will encounter in this chapter, the gas laws are generally\npolynomial expressions.\n400 5  Polynomial and Rational Functions\n5.1 Quadratic Functions\nLearning Objectives\nIn this section, you will:\nRecognize characteristics of parabolas.\nUnderstand how the graph of a parabola is related to its quadratic function."
	},
	{
		"chunk_id" : "00001222",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determine a quadratic functions minimum or maximum value.\nSolve problems involving a quadratic functions minimum or maximum value.\nFigure1 An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)\nCurved antennas, such as the ones shown inFigure 1, are commonly used to focus microwaves and radio waves to\ntransmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the"
	},
	{
		"chunk_id" : "00001223",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "antenna is in the shape of a parabola, which can be described by a quadratic function.\nIn this section, we will investigate quadratic functions, which frequently model problems involving area and projectile\nmotion. Working with quadratic functions can be less complex than working with higher degree functions, so they\nprovide a good opportunity for a detailed study of function behavior.\nRecognizing Characteristics of Parabolas"
	},
	{
		"chunk_id" : "00001224",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Recognizing Characteristics of Parabolas\nThe graph of a quadratic function is a U-shaped curve called aparabola. One important feature of the graph is that it has\nan extreme point, called thevertex. If the parabola opens up, the vertex represents the lowest point on the graph, or\ntheminimum valueof the quadratic function. If the parabola opens down, the vertex represents the highest point on the"
	},
	{
		"chunk_id" : "00001225",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph, or themaximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with\na vertical line drawn through the vertex, called theaxis of symmetry. These features are illustrated inFigure 2.\nFigure2\nThey-intercept is the point at which the parabola crosses they-axis. Thex-intercepts are the points at which the\nparabola crosses thex-axis. If they exist, thex-intercepts represent thezeros,orroots, of the quadratic function, the\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001226",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n5.1  Quadratic Functions 401\nvalues of at which\nEXAMPLE1\nIdentifying the Characteristics of a Parabola\nDetermine the vertex, axis of symmetry, zeros, and intercept of the parabola shown inFigure 3.\nFigure3\nSolution\nThe vertex is the turning point of the graph. We can see that the vertex is at Because this parabola opens upward,\nthe axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is This"
	},
	{
		"chunk_id" : "00001227",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parabola does not cross the axis, so it has no zeros. It crosses the axis at so this is they-intercept.\nUnderstanding How the Graphs of Parabolas are Related to Their Quadratic\nFunctions\nThegeneral formof a quadratic functionpresents the function in the form\nwhere and are real numbers and If the parabola opens upward. If the parabola opens\ndownward. We can use the general form of a parabola to find the equation for the axis of symmetry."
	},
	{
		"chunk_id" : "00001228",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The axis of symmetry is defined by If we use the quadratic formula, to solve\nfor the intercepts, or zeros, we find the value of halfway between them is always the\nequation for the axis of symmetry.\nFigure 4represents the graph of the quadratic function written in general form as In this form,\nand Because the parabola opens upward. The axis of symmetry is This also\nmakes sense because we can see from the graph that the vertical line divides the graph in half. The vertex"
	},
	{
		"chunk_id" : "00001229",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on\nthe graph, in this instance, The intercepts, those points where the parabola crosses the axis, occur at\nand\n402 5  Polynomial and Rational Functions\nFigure4\nThestandard form of a quadratic functionpresents the function in the form\nwhere is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also\nknown as thevertex form of a quadratic function."
	},
	{
		"chunk_id" : "00001230",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "known as thevertex form of a quadratic function.\nAs with the general form, if the parabola opens upward and the vertex is a minimum. If the parabola opens\ndownward, and the vertex is a maximum.Figure 5represents the graph of the quadratic function written in standard\nform as Since in this example, In this form, and\nBecause the parabola opens downward. The vertex is at\nFigure5\nThe standard form is useful for determining how the graph is transformed from the graph of Figure 6is the"
	},
	{
		"chunk_id" : "00001231",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph of this basic function.\nFigure6\nIf the graph shifts upward, whereas if the graph shifts downward. InFigure 5, so the graph is shifted\n4 units upward. If the graph shifts toward the right and if the graph shifts to the left. InFigure 5, so\nthe graph is shifted 2 units to the left. The magnitude of indicates the stretch of the graph. If the point\nassociated with a particular value shifts farther from thex-axis, so the graph appears to become narrower, and there"
	},
	{
		"chunk_id" : "00001232",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is a vertical stretch. But if the point associated with a particular value shifts closer to thex-axis, so the graph\nAccess for free at openstax.org\n5.1  Quadratic Functions 403\nappears to become wider, but in fact there is a vertical compression. InFigure 5, so the graph becomes\nnarrower.\nThe standard form and the general form are equivalent methods of describing the same function. We can see this by\nexpanding out the general form and setting it equal to the standard form."
	},
	{
		"chunk_id" : "00001233",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the linear terms to be equal, the coefficients must be equal.\nThis is theaxis of symmetrywe defined earlier. Setting the constant terms equal:\nIn practice, though, it is usually easier to remember thatkis the output value of the function when the input is so\nForms of Quadratic Functions\nA quadratic function is a polynomial function of degree two. The graph of aquadratic functionis a parabola.\nThegeneral form of a quadratic functionis where and are real numbers and"
	},
	{
		"chunk_id" : "00001234",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thestandard form of a quadratic functionis where\nThe vertex is located at\n...\nHOW TO\nGiven a graph of a quadratic function, write the equation of the function in general form.\n1. Identify the horizontal shift of the parabola; this value is Identify the vertical shift of the parabola; this value is\n2. Substitute the values of the horizontal and vertical shift for and in the function\n3. Substitute the values of any point, other than the vertex, on the graph of the parabola for and"
	},
	{
		"chunk_id" : "00001235",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Solve for the stretch factor,\n5. Expand and simplify to write in general form.\nEXAMPLE2\nWriting the Equation of a Quadratic Function from the Graph\nWrite an equation for the quadratic function inFigure 7as a transformation of and then expand the\nformula, and simplify terms to write the equation in general form.\n404 5  Polynomial and Rational Functions\nFigure7\nSolution\nWe can see the graph ofgis the graph of shifted to the left 2 and down 3, giving a formula in the form"
	},
	{
		"chunk_id" : "00001236",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substituting the coordinates of a point on the curve, such as we can solve for the stretch factor.\nIn standard form, the algebraic model for this graph is\nTo write this in general polynomial form, we can expand the formula and simplify terms.\nNotice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the\nvertex of the parabola; the vertex is unaffected by stretches and compressions.\nAnalysis"
	},
	{
		"chunk_id" : "00001237",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe can check our work using the table feature on a graphing utility. First enter Next, select\nthen use and and select SeeTable 1.\n6 4 2 0 2\n5 1 3 1 5\nTable1\nThe ordered pairs in the table correspond to points on the graph.\nTRY IT #1 A coordinate grid has been superimposed over the quadratic path of a basketball inFigure 8. Find\nAccess for free at openstax.org\n5.1  Quadratic Functions 405\nan equation for the path of the ball. Does the shooter make the basket?"
	},
	{
		"chunk_id" : "00001238",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8 (credit: modification of work by Dan Meyer)\n...\nHOW TO\nGiven a quadratic function in general form, find the vertex of the parabola.\n1. Identify\n2. Find thex-coordinate of the vertex, by substituting and into\n3. Find they-coordinate of the vertex, by evaluating\nEXAMPLE3\nFinding the Vertex of a Quadratic Function\nFind the vertex of the quadratic function Rewrite the quadratic in standard form (vertex form).\nSolution"
	},
	{
		"chunk_id" : "00001239",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nRewriting into standard form, the stretch factor will be the same as the in the original quadratic. First, find the\nhorizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard\nform, using the \" \"\" from the general form."
	},
	{
		"chunk_id" : "00001240",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nOne reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or\nminimum value of the output occurs, and where it occurs,\nTRY IT #2 Given the equation write the equation in general form and then in standard\nform.\nFinding the Domain and Range of a Quadratic Function\nAny number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real"
	},
	{
		"chunk_id" : "00001241",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a\nparabola will be either a maximum or a minimum, the range will consist of ally-values greater than or equal to the\ny-coordinate at the turning point or less than or equal to they-coordinate at the turning point, depending on whether\nthe parabola opens up or down.\nDomain and Range of a Quadratic Function"
	},
	{
		"chunk_id" : "00001242",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Domain and Range of a Quadratic Function\nThe domain of anyquadratic functionis all real numbers unless the context of the function presents some\nrestrictions.\nThe range of a quadratic function written in general form with a positive value is\nor  the range of a quadratic function written in general form with a negative value\nis or \nThe range of a quadratic function written in standard form with a positive value is\nthe range of a quadratic function written in standard form with a negative value is\n..."
	},
	{
		"chunk_id" : "00001243",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a quadratic function, find the domain and range.\n1. Identify the domain of any quadratic function as all real numbers.\n2. Determine whether is positive or negative. If is positive, the parabola has a minimum. If is negative, the\nparabola has a maximum.\n3. Determine the maximum or minimum value of the parabola,\n4. If the parabola has a minimum, the range is given by or  If the parabola has a maximum, the\nrange is given by or \nEXAMPLE4\nFinding the Domain and Range of a Quadratic Function"
	},
	{
		"chunk_id" : "00001244",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain and range of\nAccess for free at openstax.org\n5.1  Quadratic Functions 407\nSolution\nAs with any quadratic function, the domain is all real numbers.\nBecause is negative, the parabola opens downward and has a maximum value. We need to determine the maximum\nvalue. We can begin by finding the value of the vertex.\nThe maximum value is given by\nThe range is or \nTRY IT #3 Find the domain and range of\nDetermining the Maximum and Minimum Values of Quadratic Functions"
	},
	{
		"chunk_id" : "00001245",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the\norientation of theparabola. We can see the maximum and minimum values inFigure 9.\nFigure9\nThere are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such\nas applications involving area and revenue.\nEXAMPLE5\nFinding the Maximum Value of a Quadratic Function"
	},
	{
		"chunk_id" : "00001246",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Maximum Value of a Quadratic Function\nA backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has\npurchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth\nside.\n Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have\nlength\n What dimensions should she make her garden to maximize the enclosed area?\nSolution"
	},
	{
		"chunk_id" : "00001247",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nLets use a diagram such asFigure 10to record the given information. It is also helpful to introduce a temporary\nvariable, to represent the width of the garden and the length of the fence section parallel to the backyard fence.\n408 5  Polynomial and Rational Functions\nFigure10\n We know we have only 80 feet of fence available, and or more simply, This\nallows us to represent the width, in terms of"
	},
	{
		"chunk_id" : "00001248",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "allows us to represent the width, in terms of\nNow we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length\nmultiplied by width, so\nThis formula represents the area of the fence in terms of the variable length The function, written in general form,\nis\n The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the"
	},
	{
		"chunk_id" : "00001249",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard\npolynomial form with decreasing powers. This is why we rewrote the function in general form above. Since is the\ncoefficient of the squared term, and\nTo find the vertex:\nThe maximum value of the function is an area of 800 square feet, which occurs when feet. When the shorter"
	},
	{
		"chunk_id" : "00001250",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden\nso the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.\nAnalysis\nThis problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a\ngraph of the quadratic function inFigure 11.\nAccess for free at openstax.org\n5.1  Quadratic Functions 409\nFigure11\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001251",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.1  Quadratic Functions 409\nFigure11\n...\nHOW TO\nGiven an application involving revenue, use a quadratic equation to find the maximum.\n1. Write a quadratic equation for a revenue function.\n2. Find the vertex of the quadratic equation.\n3. Determine they-value of the vertex.\nEXAMPLE6\nFinding Maximum Revenue\nThe unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will"
	},
	{
		"chunk_id" : "00001252",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market\nresearch has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that\nsubscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to\nmaximize their revenue?\nSolution\nRevenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price"
	},
	{
		"chunk_id" : "00001253",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "per subscription times the number of subscribers, or quantity. We can introduce variables, for price per subscription\nand for quantity, giving us the equation\nBecause the number of subscribers changes with the price, we need to find a relationship between the variables. We\nknow that currently and We also know that if the price rises to $32, the newspaper would lose 5,000\nsubscribers, giving a second pair of values, and From this we can find a linear equation relating the\ntwo quantities. The slope will be"
	},
	{
		"chunk_id" : "00001254",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "two quantities. The slope will be\nThis tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the\ny-intercept.\nThis gives us the linear equation relating cost and subscribers. We now return to our revenue\nequation.\n410 5  Polynomial and Rational Functions\nWe now have a quadratic function for revenue as a function of the subscription charge. To find the price that will\nmaximize revenue for the newspaper, we can find the vertex."
	},
	{
		"chunk_id" : "00001255",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what\nthe maximum revenue is, we evaluate the revenue function.\nAnalysis\nThis could also be solved by graphing the quadratic as inFigure 12. We can see the maximum revenue on a graph of the\nquadratic function.\nFigure12\nFinding thex- andy-Intercepts of a Quadratic Function\nMuch as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing"
	},
	{
		"chunk_id" : "00001256",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parabolas. Recall that we find the intercept of a quadratic by evaluating the function at an input of zero, and we find\nthe intercepts at locations where the output is zero. Notice inFigure 13that the number of intercepts can vary\ndepending upon the location of the graph.\nFigure13 Number ofx-intercepts of a parabola\n...\nHOW TO\nGiven a quadratic function find the andx-intercepts.\n1. Evaluate to find they-intercept.\nAccess for free at openstax.org\n5.1  Quadratic Functions 411"
	},
	{
		"chunk_id" : "00001257",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.1  Quadratic Functions 411\n2. Solve the quadratic equation to find thex-intercepts.\nEXAMPLE7\nFinding they- andx-Intercepts of a Parabola\nFind they- andx-intercepts of the quadratic\nSolution\nWe find they-intercept by evaluating\nSo they-intercept is at\nFor thex-intercepts, we find all solutions of\nIn this case, the quadratic can be factored easily, providing the simplest method for solution.\nSo thex-intercepts are at and\nAnalysis"
	},
	{
		"chunk_id" : "00001258",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So thex-intercepts are at and\nAnalysis\nBy graphing the function, we can confirm that the graph crosses they-axis at We can also confirm that the graph\ncrosses thex-axis at and SeeFigure 14\nFigure14\nRewriting Quadratics in Standard Form\nInExample 7, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored.\nWe can solve these quadratics by first rewriting them in standard form.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001259",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a quadratic function, find the intercepts by rewriting in standard form.\n1. Substitute and into\n2. Substitute into the general form of the quadratic function to find\n3. Rewrite the quadratic in standard form using and\n4. Solve for when the output of the function will be zero to find the intercepts.\n412 5  Polynomial and Rational Functions\nEXAMPLE8\nFinding thex-Intercepts of a Parabola\nFind the intercepts of the quadratic function\nSolution"
	},
	{
		"chunk_id" : "00001260",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe begin by solving for when the output will be zero.\nBecause the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in\nstandard form.\nWe know that Then we solve for and\nSo now we can rewrite in standard form.\nWe can now solve for when the output will be zero.\nThe graph hasx-intercepts at and\nWe can check our work by graphing the given function on a graphing utility and observing the intercepts. SeeFigure\n15.\nFigure15\nAnalysis"
	},
	{
		"chunk_id" : "00001261",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "15.\nFigure15\nAnalysis\nWe could have achieved the same results using the quadratic formula. Identify and\nAccess for free at openstax.org\n5.1  Quadratic Functions 413\nSo thex-intercepts occur at and\nTRY IT #4 In aTry It, we found the standard and general form for the function Now\nfind they- andx-intercepts (if any).\nEXAMPLE9\nApplying the Vertex andx-Intercepts of a Parabola\nA ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The balls height above"
	},
	{
		"chunk_id" : "00001262",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ground can be modeled by the equation\n When does the ball reach the maximum height?  What is the maximum height of the ball?\n When does the ball hit the ground?\nSolution\n The ball reaches the maximum height at the vertex of the parabola.\nThe ball reaches a maximum height after 2.5 seconds.\n To find the maximum height, find the coordinate of the vertex of the parabola.\nThe ball reaches a maximum height of 140 feet.\n414 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001263",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "414 5  Polynomial and Rational Functions\n To find when the ball hits the ground, we need to determine when the height is zero,\nWe use the quadratic formula.\nBecause the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.\nThe second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after\nabout 5.458 seconds. SeeFigure 16.\nFigure16"
	},
	{
		"chunk_id" : "00001264",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "about 5.458 seconds. SeeFigure 16.\nFigure16\nNote that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on\neach axis in mind while interpreting the graph.\nTRY IT #5 A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of\n96 feet per second. The rocks height above ocean can be modeled by the equation\n When does the rock reach the maximum height?"
	},
	{
		"chunk_id" : "00001265",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " When does the rock reach the maximum height?\n What is the maximum height of the rock?  When does the rock hit the ocean?\nMEDIA\nAccess these online resources for additional instruction and practice with quadratic equations.\nGraphing Quadratic Functions in General Form(http://openstax.org/l/graphquadgen)\nGraphing Quadratic Functions in Standard Form(http://openstax.org/l/graphquadstan)\nQuadratic Function Review(http://openstax.org/l/quadfuncrev)"
	},
	{
		"chunk_id" : "00001266",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Characteristics of a Quadratic Function(http://openstax.org/l/characterquad)\nAccess for free at openstax.org\n5.1  Quadratic Functions 415\n5.1 SECTION EXERCISES\nVerbal\n1. Explain the advantage of 2. How can the vertex of a 3. Explain why the condition of\nwriting a quadratic function parabola be used in solving is imposed in the\nin standard form. real-world problems? definition of the quadratic\nfunction.\n4. What is another name for 5. What two algebraic methods\nthe standard form of a can be used to find the"
	},
	{
		"chunk_id" : "00001267",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the standard form of a can be used to find the\nquadratic function? horizontal intercepts of a\nquadratic function?\nAlgebraic\nFor the following exercises, rewrite the quadratic functions in standard form and give the vertex.\n6. 7. 8.\n9. 10. 11.\n12. 13.\nFor the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find\nthe value and the axis of symmetry.\n14. 15. 16.\n17. 18. 19.\n20."
	},
	{
		"chunk_id" : "00001268",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "14. 15. 16.\n17. 18. 19.\n20.\nFor the following exercises, determine the domain and range of the quadratic function.\n21. 22. 23.\n24. 25.\nFor the following exercises, use the vertex and a point on the graph to find the general form of the equation\nof the quadratic function.\n26. 27. 28.\n29. 30. 31.\n32. 33.\n416 5  Polynomial and Rational Functions\nGraphical\nFor the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and\nintercepts.\n34. 35. 36.\n37. 38. 39."
	},
	{
		"chunk_id" : "00001269",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intercepts.\n34. 35. 36.\n37. 38. 39.\nFor the following exercises, write the equation for the graphed quadratic function.\n40. 41. 42.\n43. 44. 45.\nNumeric\nFor the following exercises, use the table of values that represent points on the graph of a quadratic function. By\ndetermining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.\n46. 47. 48.\n2 1 0 1 2 2 1 0 1 2 2 1 0 1 2\n5 2 1 2 5 1 0 1 4 9 2 1 2 1 2\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001270",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n5.1  Quadratic Functions 417\n49. 50.\n2 1 0 1 2 2 1 0 1 2\n8 3 0 1 0 8 2 0 2 8\nTechnology\nFor the following exercises, use a calculator to find the answer.\n51. Graph on the same set of axes the functions 52. Graph on the same set of axes\n, , and . and\nand\nWhat appears to be the effect of changing the\nWhat appears to be the effect of adding a\ncoefficient?\nconstant?\n53. Graph on the same set of axes 54. The path of an object projected at a 45 degree"
	},
	{
		"chunk_id" : "00001271",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : ", and angle with initial velocity of 80 feet per second is\ngiven by the function where\nWhat appears to be the effect of adding or is the horizontal distance traveled and is the\nsubtracting those numbers? height in feet. Use the TRACE feature of your\ncalculator to determine the height of the object\nwhen it has traveled 100 feet away horizontally.\n55. A suspension bridge can be modeled by the\nquadratic function with\nwhere is the number of\nfeet from the center and is height in feet."
	},
	{
		"chunk_id" : "00001272",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "feet from the center and is height in feet.\nUse the TRACE feature of your calculator to\nestimate how far from the center does the bridge\nhave a height of 100 feet.\nExtensions\nFor the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to\nfind the domain and range of the function.\n56. Vertex opens up. 57. Vertex opens down. 58. Vertex opens\ndown.\n59. Vertex opens\nup."
	},
	{
		"chunk_id" : "00001273",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "down.\n59. Vertex opens\nup.\nFor the following exercises, write the equation of the quadratic function that contains the given point and has the same\nshape as the given function.\n60. Contains and has 61. Contains and has 62. Contains and has the\nshape of the shape of shape of\nVertex is on the axis. Vertex is on the axis. Vertex is on the axis.\n418 5  Polynomial and Rational Functions\n63. Contains and has 64. Contains and has the 65. Contains has the\nthe shape of shape of shape of"
	},
	{
		"chunk_id" : "00001274",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the shape of shape of shape of\nVertex is on the axis. Vertex is on the axis. Vertex has x-coordinate of\nReal-World Applications\n66. Find the dimensions of the 67. Find the dimensions of the 68. Find the dimensions of the\nrectangular dog park rectangular dog park split rectangular dog park\nproducing the greatest into 2 pens of the same producing the greatest\nenclosed area given 200 size producing the greatest enclosed area split into 3\nfeet of fencing. possible enclosed area sections of the same size"
	},
	{
		"chunk_id" : "00001275",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given 300 feet of fencing. given 500 feet of fencing.\n69. Among all of the pairs of 70. Among all of the pairs of 71. Suppose that the price per\nnumbers whose sum is 6, numbers whose difference unit in dollars of a cell\nfind the pair with the is 12, find the pair with the phone production is\nlargest product. What is smallest product. What is modeled by\nthe product? the product? where\nis in thousands of phones\nproduced, and the revenue\nrepresented by thousands\nof dollars is Find\nthe production level that"
	},
	{
		"chunk_id" : "00001276",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of dollars is Find\nthe production level that\nwill maximize revenue.\n72. A rocket is launched in the 73. A ball is thrown in the air 74. A soccer stadium holds\nair. Its height, in meters from the top of a building. 62,000 spectators. With a\nabove sea level, as a Its height, in meters above ticket price of $11, the\nfunction of time, in seconds, ground, as a function of average attendance has\nis given by time, in seconds, is given been 26,000. When the\nby price dropped to $9, the"
	},
	{
		"chunk_id" : "00001277",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by price dropped to $9, the\nFind the maximum height How long does it take to average attendance rose to\nthe rocket attains. reach maximum height? 31,000. Assuming that\nattendance is linearly\nrelated to ticket price, what\nticket price would\nmaximize revenue?\n75. A farmer finds that if she\nplants 75 trees per acre,\neach tree will yield 20\nbushels of fruit. She\nestimates that for each\nadditional tree planted per\nacre, the yield of each tree\nwill decrease by 3 bushels.\nHow many trees should"
	},
	{
		"chunk_id" : "00001278",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will decrease by 3 bushels.\nHow many trees should\nshe plant per acre to\nmaximize her harvest?\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 419\n5.2 Power Functions and Polynomial Functions\nLearning Objectives\nIn this section, you will:\nIdentify power functions.\nIdentify end behavior of power functions.\nIdentify polynomial functions.\nIdentify the degree and leading coefficient of polynomial functions.\nFigure1 (credit: Jason Bay, Flickr)"
	},
	{
		"chunk_id" : "00001279",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1 (credit: Jason Bay, Flickr)\nSuppose a certain species of bird thrives on a small island. Its population over the last few years is shown inTable 1.\nYear\nBird Population\nTable1\nThe population can be estimated using the function where represents the bird\npopulation on the island years after 2009. We can use this model to estimate the maximum bird population and when it\nwill occur. We can also use this model to predict when the bird population will disappear from the island. In this section,"
	},
	{
		"chunk_id" : "00001280",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we will examine functions that we can use to estimate and predict these types of changes.\nIdentifying Power Functions\nBefore we can understand the bird problem, it will be helpful to understand a different type of function. Apower\nfunctionis a function with a single term that is the product of a real number, acoefficient,and a variable raised to a\nfixed real number.\nAs an example, consider functions for area or volume. The function for thearea of a circlewith radius is"
	},
	{
		"chunk_id" : "00001281",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and the function for thevolume of a spherewith radius is\nBoth of these are examples of power functions because they consist of a coefficient, or multiplied by a variable\nraised to a power.\nPower Function\nApower functionis a function that can be represented in the form\n420 5  Polynomial and Rational Functions\nwhere and are real numbers, and is known as thecoefficient.\nQ&A Is a power function?\nNo. A power function contains a variable base raised to a fixed power. This function has a constant base"
	},
	{
		"chunk_id" : "00001282",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "raised to a variable power. This is called an exponential function, not a power function.\nEXAMPLE1\nIdentifying Power Functions\nWhich of the following functions are power functions?\nSolution\nAll of the listed functions are power functions.\nThe constant and identity functions are power functions because they can be written as and\nrespectively.\nThe quadratic and cubic functions are power functions with whole number powers and"
	},
	{
		"chunk_id" : "00001283",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thereciprocaland reciprocal squared functions are power functions with negative whole number powers because they\ncan be written as and\nThe square andcube rootfunctions are power functions with fractional powers because they can be written as\nor\nTRY IT #1 Which functions are power functions?\nIdentifying End Behavior of Power Functions\nFigure 2shows the graphs of and which are all power functions with even, whole-"
	},
	{
		"chunk_id" : "00001284",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the\ntoolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from\nthe origin.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 421\nFigure2 Even-power functions\nTo describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbolfor"
	},
	{
		"chunk_id" : "00001285",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "positive infinity and for negative infinity. When we say that  approaches infinity, which can be symbolically\nwritten as  we are describing a behavior; we are saying that is increasing without bound.\nWith the positive even-power function, as the input increases or decreases without bound, the output values become\nvery large, positive numbers. Equivalently, we could describe this behavior by saying that as approaches positive or"
	},
	{
		"chunk_id" : "00001286",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "negative infinity, the values increase without bound. In symbolic form, we could write\n \nFigure 3shows the graphs of and which are all power functions with odd, whole-\nnumber powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases,\nthe graphs flatten near the origin and become steeper away from the origin.\nFigure3 Odd-power functions\nThese examples illustrate that functions of the form reveal symmetry of one kind or another. First, inFigure 2"
	},
	{
		"chunk_id" : "00001287",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we see that even functions of the form even, are symmetric about the axis. InFigure 3we see that odd\nfunctions of the form odd, are symmetric about the origin.\nFor these odd power functions, as approaches negative infinity, decreases without bound. As approaches\npositive infinity, increases without bound. In symbolic form we write\n \n \nThe behavior of the graph of a function as the input values get very small ( ) and get very large ( ) is"
	},
	{
		"chunk_id" : "00001288",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "referred to as theend behaviorof the function. We can use words or symbols to describe end behavior.\nFigure 4shows the end behavior of power functions in the form where is a non-negative integer\ndepending on the power and the constant.\n422 5  Polynomial and Rational Functions\nFigure4\n...\nHOW TO\nGiven a power function where is a non-negative integer, identify the end behavior.\n1. Determine whether the power is even or odd.\n2. Determine whether the constant is positive or negative."
	},
	{
		"chunk_id" : "00001289",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. UseFigure 4to identify the end behavior.\nEXAMPLE2\nIdentifying the End Behavior of a Power Function\nDescribe the end behavior of the graph of\nSolution\nThe coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As approaches infinity, the\noutput (value of ) increases without bound. We write as   As approaches negative infinity, the\noutput increases without bound. In symbolic form, as   We can graphically represent the function as\nshown inFigure 5."
	},
	{
		"chunk_id" : "00001290",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shown inFigure 5.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 423\nFigure5\nEXAMPLE3\nIdentifying the End Behavior of a Power Function.\nDescribe the end behavior of the graph of\nSolution\nThe exponent of the power function is 9 (an odd number). Because the coefficient is (negative), the graph is the\nreflection about the axis of the graph of Figure 6shows that as approaches infinity, the output decreases"
	},
	{
		"chunk_id" : "00001291",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "without bound. As approaches negative infinity, the output increases without bound. In symbolic form, we would write\n \n \nFigure6\nAnalysis\nWe can check our work by using the table feature on a graphing utility.\n10 1,000,000,000\nTable2\n424 5  Polynomial and Rational Functions\n5 1,953,125\n0 0\n5 1,953,125\n10 1,000,000,000\nTable2\nWe can see fromTable 2that, when we substitute very small values for the output is very large, and when we"
	},
	{
		"chunk_id" : "00001292",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitute very large values for the output is very small (meaning that it is a very large negative value).\nTRY IT #2 Describe in words and symbols the end behavior of\nIdentifying Polynomial Functions\nAn oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles\nin radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil"
	},
	{
		"chunk_id" : "00001293",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "slick by combining two functions. The radius of the spill depends on the number of weeks that have passed. This\nrelationship is linear.\nWe can combine this with the formula for the area of a circle.\nComposing these functions gives a formula for the area in terms of weeks.\nMultiplying gives the formula.\nThis formula is an example of apolynomial function. A polynomial function consists of either zero or the sum of a finite"
	},
	{
		"chunk_id" : "00001294",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of non-zeroterms, each of which is a product of a number, called thecoefficientof the term, and a variable\nraised to a non-negative integer power.\nPolynomial Functions\nLet be a non-negative integer. Apolynomial functionis a function that can be written in the form\nThis is called the general form of a polynomial function. Each is a coefficient and can be any real number, but .\nEach expression is aterm of a polynomial function.\nEXAMPLE4\nIdentifying Polynomial Functions"
	},
	{
		"chunk_id" : "00001295",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nIdentifying Polynomial Functions\nWhich of the following are polynomial functions?\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 425\nSolution\nThe first two functions are examples of polynomial functions because they can be written in the form\nwhere the powers are non-negative integers and the coefficients are real\nnumbers.\n can be written as\n can be written as\n cannot be written in this form and is therefore not a polynomial function."
	},
	{
		"chunk_id" : "00001296",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying the Degree and Leading Coefficient of a Polynomial Function\nBecause of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the\nvariable. Although the order of the terms in the polynomial function is not important for performing operations, we\ntypically arrange the terms in descending order of power, or in general form. Thedegreeof the polynomial is the"
	},
	{
		"chunk_id" : "00001297",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in\ngeneral form. Theleading termis the term containing the highest power of the variable, or the term with the highest\ndegree. Theleading coefficientis the coefficient of the leading term.\nTerminology of Polynomial Functions\nWe often rearrange polynomials so that the powers are descending.\nWhen a polynomial is written in this way, we say that it is in general form.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001298",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a polynomial function, identify the degree and leading coefficient.\n1. Find the highest power of to determine the degree of the function.\n2. Identify the term containing the highest power of to find the leading term.\n3. Identify the coefficient of the leading term.\nEXAMPLE5\nIdentifying the Degree and Leading Coefficient of a Polynomial Function\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\nSolution"
	},
	{
		"chunk_id" : "00001299",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nFor the function the highest power of is 3, so the degree is 3. The leading term is the term containing that\n426 5  Polynomial and Rational Functions\ndegree, The leading coefficient is the coefficient of that term,\nFor the function the highest power of is so the degree is The leading term is the term containing that degree,\nThe leading coefficient is the coefficient of that term,\nFor the function the highest power of is so the degree is The leading term is the term containing that"
	},
	{
		"chunk_id" : "00001300",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "degree, The leading coefficient is the coefficient of that term,\nTRY IT #3 Identify the degree, leading term, and leading coefficient of the polynomial\nIdentifying End Behavior of Polynomial Functions\nKnowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end\nbehavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that"
	},
	{
		"chunk_id" : "00001301",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "term will grow significantly faster than the other terms as gets very large or very small, so its behavior will dominate\nthe graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function\nconsisting of the leading term. SeeTable 3.\nPolynomial Function Leading Term Graph of Polynomial Function\nTable3\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 427\nPolynomial Function Leading Term Graph of Polynomial Function\nTable3\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001302",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table3\nEXAMPLE6\nIdentifying End Behavior and Degree of a Polynomial Function\nDescribe the end behavior and determine a possible degree of the polynomial function inFigure 7.\n428 5  Polynomial and Rational Functions\nFigure7\nSolution\nAs the input values get very large, the output values increase without bound. As the input values get very small,\nthe output values decrease without bound. We can describe the end behavior symbolically by writing\n \n "
	},
	{
		"chunk_id" : "00001303",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \n \nIn words, we could say that as values approach infinity, the function values approach infinity, and as values approach\nnegative infinity, the function values approach negative infinity.\nWe can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the\npolynomial creating this graph must be odd and the leading coefficient must be positive.\nTRY IT #4 Describe the end behavior, and determine a possible degree of the polynomial function inFigure\n8."
	},
	{
		"chunk_id" : "00001304",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8.\nFigure8\nEXAMPLE7\nIdentifying End Behavior and Degree of a Polynomial Function\nGiven the function express the function as a polynomial in general form, and determine the\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 429\nleading term, degree, and end behavior of the function.\nSolution\nObtain the general form by expanding the given expression for\nThe general form is The leading term is therefore, the degree of the polynomial is 4."
	},
	{
		"chunk_id" : "00001305",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The degree is even (4) and the leading coefficient is negative (3), so the end behavior is\n \n \nTRY IT #5 Given the function express the function as a polynomial in\ngeneral form and determine the leading term, degree, and end behavior of the function.\nIdentifying Local Behavior of Polynomial Functions\nIn addition to the end behavior of polynomial functions, we are also interested in what happens in the middle of the"
	},
	{
		"chunk_id" : "00001306",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function. In particular, we are interested in locations where graph behavior changes. Aturning pointis a point at which\nthe function values change from increasing to decreasing or decreasing to increasing.\nWe are also interested in the intercepts. As with all functions, they-intercept is the point at which the graph intersects\nthe vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a"
	},
	{
		"chunk_id" : "00001307",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, only one output value corresponds to each input value so there can be only oney-intercept The\nx-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one\nx-intercept. SeeFigure 9.\nFigure9\nIntercepts and Turning Points of Polynomial Functions\nAturning pointof a graph is a point at which the graph changes direction from increasing to decreasing or"
	},
	{
		"chunk_id" : "00001308",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decreasing to increasing. They-intercept is the point at which the function has an input value of zero. Thex-intercepts\nare the points at which the output value is zero.\n430 5  Polynomial and Rational Functions\n...\nHOW TO\nGiven a polynomial function, determine the intercepts.\n1. Determine they-intercept by setting and finding the corresponding output value.\n2. Determine thex-intercepts by solving for the input values that yield an output value of zero.\nEXAMPLE8"
	},
	{
		"chunk_id" : "00001309",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE8\nDetermining the Intercepts of a Polynomial Function\nGiven the polynomial function written in factored form for your convenience, determine\nthey- andx-intercepts.\nSolution\nThey-intercept occurs when the input is zero so substitute 0 for\nThey-intercept is (0, 8).\nThex-intercepts occur when the output is zero.\nThex-intercepts are and\nWe can see these intercepts on the graph of the function shown inFigure 10.\nFigure10\nEXAMPLE9\nDetermining the Intercepts of a Polynomial Function with Factoring"
	},
	{
		"chunk_id" : "00001310",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given the polynomial function determine they- andx-intercepts.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 431\nSolution\nThey-intercept occurs when the input is zero.\nThey-intercept is\nThex-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the\npolynomial.\nThex-intercepts are and\nWe can see these intercepts on the graph of the function shown inFigure 11. We can see that the function is even\nbecause\nFigure11"
	},
	{
		"chunk_id" : "00001311",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "because\nFigure11\nTRY IT #6 Given the polynomial function determine they- andx-intercepts.\nComparing Smooth and Continuous Graphs\nThe degree of a polynomial function helps us to determine the number ofx-intercepts and the number of turning points.\nA polynomial function of degree is the product of factors, so it will have at most roots or zeros, orx-intercepts.\nThe graph of the polynomial function of degree must have at most turning points. This means the graph has at"
	},
	{
		"chunk_id" : "00001312",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.\nAcontinuous functionhas no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A\nsmooth curveis a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded\ncurves. The graphs of polynomial functions are both continuous and smooth.\nIntercepts and Turning Points of Polynomials"
	},
	{
		"chunk_id" : "00001313",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Intercepts and Turning Points of Polynomials\nA polynomial of degree will have, at most, x-intercepts and turning points.\n432 5  Polynomial and Rational Functions\nEXAMPLE10\nDetermining the Number of Intercepts and Turning Points of a Polynomial\nWithout graphing the function, determine the local behavior of the function by finding the maximum number of\nx-intercepts and turning points for\nSolution\nThe polynomial has a degree of so there are at most 10x-intercepts and at most 9 turning points."
	},
	{
		"chunk_id" : "00001314",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Without graphing the function, determine the maximum number ofx-intercepts and turning\npoints for\nEXAMPLE11\nDrawing Conclusions about a Polynomial Function from the Graph\nWhat can we conclude about the polynomial represented by the graph shown inFigure 12based on its intercepts and\nturning points?\nFigure12\nSolution\nThe end behavior of the graph tells us this is the graph of an even-degree polynomial. SeeFigure 13.\nFigure13"
	},
	{
		"chunk_id" : "00001315",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure13\nThe graph has 2x-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or\ngreater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 433\nTRY IT #8 What can we conclude about the polynomial represented by the graph shown inFigure 14based\non its intercepts and turning points?\nFigure14\nEXAMPLE12"
	},
	{
		"chunk_id" : "00001316",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure14\nEXAMPLE12\nDrawing Conclusions about a Polynomial Function from the Factors\nGiven the function determine the local behavior.\nSolution\nThey-intercept is found by evaluating\nThey-intercept is\nThex-intercepts are found by determining the zeros of the function.\nThex-intercepts are and\nThe degree is 3 so the graph has at most 2 turning points.\nTRY IT #9 Given the function determine the local behavior.\nMEDIA"
	},
	{
		"chunk_id" : "00001317",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with power and polynomial functions.\nFind Key Information about a Given Polynomial Function(http://openstax.org/l/keyinfopoly)\nEnd Behavior of a Polynomial Function(http://openstax.org/l/endbehavior)\nTurning Points and intercepts of Polynomial Functions(http://openstax.org/l/turningpoints)\nLeast Possible Degree of a Polynomial Function(http://openstax.org/l/leastposdegree)\n434 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001318",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "434 5  Polynomial and Rational Functions\n5.2 SECTION EXERCISES\nVerbal\n1. Explain the difference 2. If a polynomial function is in 3. In general, explain the end\nbetween the coefficient of a factored form, what would behavior of a power\npower function and its be a good first step in order function with odd degree if\ndegree. to determine the degree of the leading coefficient is\nthe function? positive.\n4. What is the relationship 5. What can we conclude if, in\nbetween the degree of a general, the graph of a"
	},
	{
		"chunk_id" : "00001319",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "between the degree of a general, the graph of a\npolynomial function and the polynomial function exhibits\nmaximum number of the following end behavior?\nturning points in its graph? As  \nand as\n \nAlgebraic\nFor the following exercises, identify the function as a power function, a polynomial function, or neither.\n6. 7. 8.\n9. 10. 11.\nFor the following exercises, find the degree and leading coefficient for the given polynomial.\n12. 13. 14.\n15. 16."
	},
	{
		"chunk_id" : "00001320",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16.\nFor the following exercises, determine the end behavior of the functions.\n17. 18. 19.\n20. 21. 22.\n23. 24.\nFor the following exercises, find the intercepts of the functions.\n25. 26. 27.\n28. 29. 30.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 435\nGraphical\nFor the following exercises, determine the least possible degree of the polynomial function shown.\n31. 32. 33.\n34. 35. 36.\n37. 38.\n436 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001321",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "37. 38.\n436 5  Polynomial and Rational Functions\nFor the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If\nso, determine the number of turning points and the least possible degree for the function.\n39. 40. 41.\n42. 43. 44.\n45.\nNumeric\nFor the following exercises, make a table to confirm the end behavior of the function.\n46. 47. 48.\n49. 50.\nAccess for free at openstax.org\n5.2  Power Functions and Polynomial Functions 437\nTechnology"
	},
	{
		"chunk_id" : "00001322",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Technology\nFor the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the\nintercepts and the end behavior.\n51. 52. 53.\n54. 55. 56.\n57. 58. 59.\n60.\nExtensions\nFor the following exercises, use the information about the graph of a polynomial function to determine the function.\nAssume the leading coefficient is 1 or 1. There may be more than one correct answer.\n61. The intercept is The intercepts are 62. The intercept is The intercepts are"
	},
	{
		"chunk_id" : "00001323",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : ", Degree is 2. , Degree is 2.\nEnd behavior: as , ; as , End behavior: as , , as\n , \n63. The intercept is The intercepts are 64. The intercept is The intercept is\n, Degree is 3. Degree is 3.\nEnd behavior: as , , as End behavior: as , , as ,\n,  \n65. The intercept is There is no intercept.\nDegree is 4.\nEnd behavior: as , , as ,\n\nReal-World Applications\nFor the following exercises, use the written statements to construct a polynomial function that represents the required\ninformation."
	},
	{
		"chunk_id" : "00001324",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "information.\n66. An oil slick is expanding as 67. A cube has an edge of 3 68. A rectangle has a length of\na circle. The radius of the feet. The edge is increasing 10 inches and a width of 6\ncircle is increasing at the at the rate of 2 feet per inches. If the length is\nrate of 20 meters per day. minute. Express the increased by inches and\nExpress the area of the volume of the cube as a the width increased by\ncircle as a function of the function of the number twice that amount, express"
	},
	{
		"chunk_id" : "00001325",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of days elapsed. of minutes elapsed. the area of the rectangle as\na function of\n438 5  Polynomial and Rational Functions\n69. An open box is to be 70. A rectangle is twice as long\nconstructed by cutting out as it is wide. Squares of\nsquare corners of inch side 2 feet are cut out from\nsides from a piece of each corner. Then the sides\ncardboard 8 inches by 8 are folded up to make an\ninches and then folding up open box. Express the\nthe sides. Express the volume of the box as a"
	},
	{
		"chunk_id" : "00001326",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the sides. Express the volume of the box as a\nvolume of the box as a function of the width ( ).\nfunction of\n5.3 Graphs of Polynomial Functions\nLearning Objectives\nIn this section, you will:\nRecognize characteristics of graphs of polynomial functions.\nUse factoring to find zeros of polynomial functions.\nIdentify zeros and their multiplicities.\nDetermine end behavior.\nUnderstand the relationship between degree and turning points.\nGraph polynomial functions.\nUse the Intermediate Value Theorem."
	},
	{
		"chunk_id" : "00001327",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use the Intermediate Value Theorem.\nThe revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown inTable 1.\nYear 2006 2007 2008 2009 2010 2011 2012 2013\nRevenues 52.4 52.8 51.2 49.5 48.6 48.6 48.7 47.1\nTable1\nThe revenue can be modeled by the polynomial function\nwhere represents the revenue in millions of dollars and represents the year, with corresponding to 2006. Over"
	},
	{
		"chunk_id" : "00001328",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company\ndecreasing? These questions, along with many others, can be answered by examining the graph of the polynomial\nfunction. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will\nexplore the local behavior of polynomials in general.\nRecognizing Characteristics of Graphs of Polynomial Functions"
	},
	{
		"chunk_id" : "00001329",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs\nare called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called\ncontinuous.Figure 1shows a graph that represents apolynomial functionand a graph that represents a function that is\nnot a polynomial.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 439\nFigure1\nEXAMPLE1\nRecognizing Polynomial Functions"
	},
	{
		"chunk_id" : "00001330",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nEXAMPLE1\nRecognizing Polynomial Functions\nWhich of the graphs inFigure 2represents a polynomial function?\nFigure2\n440 5  Polynomial and Rational Functions\nSolution\nThe graphs of and are graphs of polynomial functions. They are smooth andcontinuous.\nThe graphs of and are graphs of functions that are not polynomials. The graph of function has a sharp corner. The\ngraph of function is not continuous.\nQ&A Do all polynomial functions have as their domain all real numbers?"
	},
	{
		"chunk_id" : "00001331",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Yes. Any real number is a valid input for a polynomial function.\nUsing Factoring to Find Zeros of Polynomial Functions\nRecall that if is a polynomial function, the values of for which are calledzerosof If the equation of the\npolynomial function can be factored, we can set each factor equal to zero and solve for the zeros.\nWe can use this method to find intercepts because at the intercepts we find the input values when the output value"
	},
	{
		"chunk_id" : "00001332",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively\nsimple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough\nto remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to\nthree cases:\n1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring."
	},
	{
		"chunk_id" : "00001333",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. The polynomial is given in factored form.\n3. Technology is used to determine the intercepts.\n...\nHOW TO\nGiven a polynomial function find thex-intercepts by factoring.\n1. Set\n2. If the polynomial function is not given in factored form:\na. Factor out any common monomial factors.\nb. Factor any factorable binomials or trinomials.\n3. Set each factor equal to zero and solve to find the intercepts.\nEXAMPLE2\nFinding thex-Intercepts of a Polynomial Function by Factoring\nFind thex-intercepts of\nSolution"
	},
	{
		"chunk_id" : "00001334",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find thex-intercepts of\nSolution\nWe can attempt to factor this polynomial to find solutions for\nThis gives us fivex-intercepts: and SeeFigure 3. We can see that this is an even\nfunction because it is symmetric about they-axis.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 441\nFigure3\nEXAMPLE3\nFinding thex-Intercepts of a Polynomial Function by Factoring\nFind thex-intercepts of\nSolution\nFind solutions for by factoring.\nThere are threex-intercepts: and SeeFigure 4.\nFigure4"
	},
	{
		"chunk_id" : "00001335",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\n442 5  Polynomial and Rational Functions\nEXAMPLE4\nFinding they- andx-Intercepts of a Polynomial in Factored Form\nFind they- andx-intercepts of\nSolution\nThey-intercept can be found by evaluating\nSo they-intercept is\nThex-intercepts can be found by solving\nSo thex-intercepts are and\nAnalysis\nWe can always check that our answers are reasonable by using a graphing calculator to graph the polynomial as shown\ninFigure 5.\nFigure5\nEXAMPLE5\nFinding thex-Intercepts of a Polynomial Function Using a Graph"
	},
	{
		"chunk_id" : "00001336",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find thex-intercepts of\nSolution\nThis polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques\npreviously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make\ngraphing difficult by hand. In these cases, we can take advantage of graphing utilities.\nLooking at the graph of this function, as shown inFigure 6, it appears that there arex-intercepts at and\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001337",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n5.3  Graphs of Polynomial Functions 443\nFigure6\nWe can check whether these are correct by substituting these values for and verifying that\nSince we have:\nEachx-intercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the\npolynomial in factored form.\nTRY IT #1 Find they- andx-intercepts of the function\nIdentifying Zeros and Their Multiplicities"
	},
	{
		"chunk_id" : "00001338",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Zeros and Their Multiplicities\nGraphs behave differently at variousx-intercepts. Sometimes, the graph will cross over the horizontal axis at an\nintercept. Other times, the graph will touch the horizontal axis and \"bounce\"\" off."
	},
	{
		"chunk_id" : "00001339",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "444 5  Polynomial and Rational Functions\nFigure7 Identifying the behavior of the graph at anx-intercept by examining the multiplicity of the zero.\nThex-intercept is the solution of equation The graph passes directly through thex-intercept at\nThe factor is linear (has a degree of 1), so the behavior near the intercept is like that of a lineit passes directly\nthrough the intercept. We call this a single zero because the zero corresponds to a single factor of the function."
	},
	{
		"chunk_id" : "00001340",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thex-intercept is the repeated solution of equation The graph touches the axis at the intercept and\nchanges direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadraticit\nbounces off of the horizontal axis at the intercept.\nThe factor is repeated, that is, the factor appears twice. The number of times a given factor appears in the\nfactored form of the equation of a polynomial is called themultiplicity. The zero associated with this factor, has"
	},
	{
		"chunk_id" : "00001341",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplicity 2 because the factor occurs twice.\nThex-intercept is the repeated solution of factor The graph passes through the axis at the\nintercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a\ncubicwith the same S-shape near the intercept as the toolkit function We call this a triple zero, or a zero\nwith multiplicity 3."
	},
	{
		"chunk_id" : "00001342",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with multiplicity 3.\nForzeroswith even multiplicities, the graphstouchor are tangent to thex-axis. For zeros with odd multiplicities, the\ngraphscrossor intersect thex-axis. SeeFigure 8for examples of graphs of polynomial functions with multiplicity 1, 2,\nand 3.\nFigure8\nFor higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each\nincreasing even power, the graph will appear flatter as it approaches and leaves thex-axis."
	},
	{
		"chunk_id" : "00001343",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing\nodd power, the graph will appear flatter as it approaches and leaves thex-axis.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 445\nGraphical Behavior of Polynomials atx-Intercepts\nIf a polynomial contains a factor of the form the behavior near the intercept is determined by the power\nWe say that is a zero ofmultiplicity"
	},
	{
		"chunk_id" : "00001344",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We say that is a zero ofmultiplicity\nThe graph of a polynomial function will touch thex-axis at zeros with even multiplicities. The graph will cross the\nx-axis at zeros with odd multiplicities.\nThe sum of the multiplicities is the degree of the polynomial function.\n...\nHOW TO\nGiven a graph of a polynomial function of degree identify the zeros and their multiplicities.\n1. If the graph crosses thex-axis and appears almost linear at the intercept, it is a single zero."
	},
	{
		"chunk_id" : "00001345",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. If the graph touches thex-axis and bounces off of the axis, it is a zero with even multiplicity.\n3. If the graph crosses thex-axis at a zero, it is a zero with odd multiplicity.\n4. The sum of the multiplicities is\nEXAMPLE6\nIdentifying Zeros and Their Multiplicities\nUse the graph of the function of degree 6 inFigure 9to identify the zeros of the function and their possible multiplicities.\nFigure9\nSolution\nThe polynomial function is of degree 6. The sum of the multiplicities must be 6."
	},
	{
		"chunk_id" : "00001346",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Starting from the left, the first zero occurs at The graph touches thex-axis, so the multiplicity of the zero must\nbe even. The zero of most likely has multiplicity\nThe next zero occurs at The graph looks almost linear at this point. This is a single zero of multiplicity 1.\nThe last zero occurs at The graph crosses thex-axis, so the multiplicity of the zero must be odd. We know that the\nmultiplicity is likely 3 and that the sum of the multiplicities is 6."
	},
	{
		"chunk_id" : "00001347",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Use the graph of the function of degree 9 inFigure 10to identify the zeros of the function and\ntheir multiplicities.\n446 5  Polynomial and Rational Functions\nFigure10\nDetermining End Behavior\nAs we have already learned, the behavior of a graph of apolynomial functionof the form\nwill either ultimately rise or fall as increases without bound and will either rise or fall as decreases without bound."
	},
	{
		"chunk_id" : "00001348",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is\ntrue for very small inputs, say 100 or 1,000.\nRecall that we call this behavior theend behaviorof a function. As we pointed out when discussing quadratic equations,\nwhen the leading term of a polynomial function, is an even power function, as increases or decreases without\nbound, increases without bound. When the leading term is an odd power function, as decreases without bound,"
	},
	{
		"chunk_id" : "00001349",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "also decreases without bound; as increases without bound, also increases without bound. If the leading\nterm is negative, it will change the direction of the end behavior.Figure 11summarizes all four cases.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 447\nFigure11\nUnderstanding the Relationship between Degree and Turning Points\nIn addition to the end behavior, recall that we can analyze a polynomial functions local behavior. It may have a turning"
	},
	{
		"chunk_id" : "00001350",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to\nrising). Look at the graph of the polynomial function inFigure 12. The graph has three\nturning points.\n448 5  Polynomial and Rational Functions\nFigure12\nThis function is a 4thdegree polynomial function and has 3 turning points. The maximum number of turning points of\na polynomial function is always one less than the degree of the function.\nInterpreting Turning Points"
	},
	{
		"chunk_id" : "00001351",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Interpreting Turning Points\nAturning pointis a point of the graph where the graph changes from increasing to decreasing (rising to falling) or\ndecreasing to increasing (falling to rising).\nA polynomial of degree will have at most turning points.\nEXAMPLE7\nFinding the Maximum Number of Turning Points Using the Degree of a Polynomial Function\nFind the maximum number of turning points of each polynomial function.\n \nSolution\n\nFirst, rewrite the polynomial function in descending order:"
	},
	{
		"chunk_id" : "00001352",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identify the degree of the polynomial function. This polynomial function is of degree 5.\nThe maximum number of turning points is\n\nFirst, identify the leading term of the polynomial function if the function were expanded.\nThen, identify the degree of the polynomial function. This polynomial function is of degree 4.\nThe maximum number of turning points is\nGraphing Polynomial Functions\nWe can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial"
	},
	{
		"chunk_id" : "00001353",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions. Let us put this all together and look at the steps required to graph polynomial functions.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 449\n...\nHOW TO\nGiven a polynomial function, sketch the graph.\n1. Find the intercepts.\n2. Check for symmetry. If the function is an even function, its graph is symmetrical about the axis, that is,\nIf a function is an odd function, its graph is symmetrical about the origin, that is,"
	},
	{
		"chunk_id" : "00001354",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the intercepts.\n4. Determine the end behavior by examining the leading term.\n5. Use the end behavior and the behavior at the intercepts to sketch a graph.\n6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial.\n7. Optionally, use technology to check the graph.\nEXAMPLE8\nSketching the Graph of a Polynomial Function\nSketch a graph of\nSolution"
	},
	{
		"chunk_id" : "00001355",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketch a graph of\nSolution\nThis graph has twox-intercepts. At the factor is squared, indicating a multiplicity of 2. The graph will bounce at\nthisx-intercept. At the function has a multiplicity of one, indicating the graph will cross through the axis at this\nintercept.\nThey-intercept is found by evaluating\nThey-intercept is\nAdditionally, we can see the leading term, if this polynomial were multiplied out, would be so the end behavior is"
	},
	{
		"chunk_id" : "00001356",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs\nincreasing as the inputs approach negative infinity. SeeFigure 13.\nFigure13\nTo sketch this, we consider that:\n As the function  so we know the graph starts in the second quadrant and is decreasing toward\nthe axis.\n450 5  Polynomial and Rational Functions\n Since is not equal to the graph does not display symmetry."
	},
	{
		"chunk_id" : "00001357",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " At the graph bounces off of thex-axis, so the function must start increasing.\nAt the graph crosses they-axis at they-intercept. SeeFigure 14.\nFigure14\nSomewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the\ngraph passes through the next intercept at SeeFigure 15.\nFigure15\nAs the function so we know the graph continues to decrease, and we can stop drawing the graph in\nthe fourth quadrant."
	},
	{
		"chunk_id" : "00001358",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the fourth quadrant.\nUsing technology, we can create the graph for the polynomial function, shown inFigure 16, and verify that the resulting\ngraph looks like our sketch inFigure 15.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 451\nFigure16 The complete graph of the polynomial function\nTRY IT #3 Sketch a graph of\nUsing the Intermediate Value Theorem\nIn some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of"
	},
	{
		"chunk_id" : "00001359",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thex-axis, we can confirm that there is a zero between them. Consider a polynomial function whose graph is smooth\nand continuous. TheIntermediate Value Theoremstates that for two numbers and in the domain of if and\nthen the function takes on every value between and (While the theorem is intuitive, the\nproof is actually quite complicated and requires higher mathematics.) We can apply this theorem to a special case that is"
	},
	{
		"chunk_id" : "00001360",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "useful in graphing polynomial functions. If a point on the graph of a continuous function at lies above the axis\nand another point at lies below the axis, there must exist a third point between and where the\ngraph crosses the axis. Call this point This means that we are assured there is a solution where\nIn other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value"
	},
	{
		"chunk_id" : "00001361",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to a positive value, the function must cross the axis.Figure 17shows that there is a zero between and\nFigure17 Using the Intermediate Value Theorem to show there exists a zero.\nIntermediate Value Theorem\nLet be a polynomial function. TheIntermediate Value Theoremstates that if and have opposite signs,\nthen there exists at least one value between and for which\n452 5  Polynomial and Rational Functions\nEXAMPLE9\nUsing the Intermediate Value Theorem"
	},
	{
		"chunk_id" : "00001362",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nUsing the Intermediate Value Theorem\nShow that the function has at least two real zeros between and\nSolution\nAs a start, evaluate at the integer values and SeeTable 2.\n1 2 3 4\n5 0 3 2\nTable2\nWe see that one zero occurs at Also, since is negative and is positive, by the Intermediate Value\nTheorem, there must be at least one real zero between 3 and 4.\nWe have shown that there are at least two real zeros between and\nAnalysis"
	},
	{
		"chunk_id" : "00001363",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe can also see on the graph of the function inFigure 18that there are two real zeros between and\nFigure18\nTRY IT #4 Show that the function has at least one real zero between and\nWriting Formulas for Polynomial Functions\nNow that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs.\nBecause apolynomial functionwritten in factored form will have anx-intercept where each factor is equal to zero, we"
	},
	{
		"chunk_id" : "00001364",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can form a function that will pass through a set ofx-intercepts by introducing a corresponding set of factors.\nFactored Form of Polynomials\nIf a polynomial of lowest degree has horizontal intercepts at then the polynomial can be written\nin the factored form: where the powers on each factor can be\ndetermined by the behavior of the graph at the corresponding intercept, and the stretch factor can be determined\ngiven a value of the function other than thex-intercept.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001365",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a graph of a polynomial function, write a formula for the function.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 453\n1. Identify thex-intercepts of the graph to find the factors of the polynomial.\n2. Examine the behavior of the graph at thex-intercepts to determine the multiplicity of each factor.\n3. Find the polynomial of least degree containing all the factors found in the previous step."
	},
	{
		"chunk_id" : "00001366",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Use any other point on the graph (they-intercept may be easiest) to determine the stretch factor.\nEXAMPLE10\nWriting a Formula for a Polynomial Function from the Graph\nWrite a formula for the polynomial function shown inFigure 19.\nFigure19\nSolution\nThis graph has threex-intercepts: and They-intercept is located at At and the graph\npasses through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At the"
	},
	{
		"chunk_id" : "00001367",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree\n(quadratic). Together, this gives us\nTo determine the stretch factor, we utilize another point on the graph. We will use the intercept to solve for\nThe graphed polynomial appears to represent the function\nTRY IT #5 Given the graph shown inFigure 20, write a formula for the function shown.\n454 5  Polynomial and Rational Functions\nFigure20\nUsing Local and Global Extrema"
	},
	{
		"chunk_id" : "00001368",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure20\nUsing Local and Global Extrema\nWith quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex.\nFor general polynomials, finding these turning points is not possible without more advanced techniques from calculus.\nEven then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of\nturning points using technology to generate a graph."
	},
	{
		"chunk_id" : "00001369",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point\non the entire graph. In these cases, we say that the turning point is aglobal maximumor aglobal minimum. These are\nalso referred to as the absolute maximum and absolute minimum values of the function.\nLocal and Global Extrema\nAlocal maximumorlocal minimumat (sometimes called the relative maximum or minimum, respectively) is"
	},
	{
		"chunk_id" : "00001370",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the output at the highest or lowest point on the graph in an open interval around If a function has a local\nmaximum at then for all in an open interval around If a function has a local minimum at\nthen for all in an open interval around\nAglobal maximumorglobal minimumis the output at the highest or lowest point of the function. If a function has\na global maximum at then for all If a function has a global minimum at then for all\nWe can see the difference between local and global extrema inFigure 21.\nFigure21"
	},
	{
		"chunk_id" : "00001371",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure21\nQ&A Do all polynomial functions have a global minimum or maximum?\nNo. Only polynomial functions of even degree have a global minimum or maximum. For example,\nhas neither a global maximum nor a global minimum.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 455\nEXAMPLE11\nUsing Local Extrema to Solve Applications\nAn open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic and"
	},
	{
		"chunk_id" : "00001372",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.\nSolution\nWe will start this problem by drawing a picture like that inFigure 22, labeling the width of the cut-out squares with a\nvariable,\nFigure22\nNotice that after a square is cut out from each end, it leaves a cm by cm rectangle for the base of\nthe box, and the box will be cm tall. This gives the volume"
	},
	{
		"chunk_id" : "00001373",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice, since the factors are and the three zeros are 10, 7, and 0, respectively. Because a height of 0\ncm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares,\nso values may take on are greater than zero or less than 7. This means we will restrict the domain of this function to\nUsing technology to sketch the graph of on this reasonable domain, we get a graph like that inFigure"
	},
	{
		"chunk_id" : "00001374",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. We can use this graph to estimate the maximum value for the volume, restricted to values for that are reasonable\nfor this problemvalues from 0 to 7.\nFigure23\nFrom this graph, we turn our focus to only the portion on the reasonable domain, We can estimate the\nmaximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve\nthis estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in"
	},
	{
		"chunk_id" : "00001375",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on our graph to produceFigure 24.\n456 5  Polynomial and Rational Functions\nFigure24\nFrom this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares\nmeasure approximately 2.7 cm on each side.\nTRY IT #6 Use technology to find the maximum and minimum values on the interval of the function\nMEDIA\nAccess the following online resource for additional instruction and practice with graphing polynomial functions."
	},
	{
		"chunk_id" : "00001376",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Intermediate Value Theorem(http://openstax.org/l/ivt)\n5.3 SECTION EXERCISES\nVerbal\n1. What is the difference 2. If a polynomial function of 3. Explain how the\nbetween an intercept and degree has distinct Intermediate Value\na zero of a polynomial zeros, what do you know Theorem can assist us in\nfunction about the graph of the finding a zero of a function.\nfunction?\n4. Explain how the factored 5. If the graph of a polynomial\nform of the polynomial just touches thex-axis and"
	},
	{
		"chunk_id" : "00001377",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form of the polynomial just touches thex-axis and\nhelps us in graphing it. then changes direction,\nwhat can we conclude about\nthe factored form of the\npolynomial?\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 457\nAlgebraic\nFor the following exercises, find the ort-intercepts of the polynomial functions.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23."
	},
	{
		"chunk_id" : "00001378",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23.\nFor the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one\nzero within the given interval.\n24. between 25. between 26. between\nand and and\n27. between 28. between 29.\nand . and between and\nFor the following exercises, find the zeros and give the multiplicity of each.\n30. 31. 32.\n33. 34. 35.\n36. 37.\n38. 39. 40.\n41.\nGraphical"
	},
	{
		"chunk_id" : "00001379",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "33. 34. 35.\n36. 37.\n38. 39. 40.\n41.\nGraphical\nFor the following exercises, graph the polynomial functions. Note and intercepts, multiplicity, and end behavior.\n42. 43. 44.\n45. 46. 47.\n458 5  Polynomial and Rational Functions\nFor the following exercises, use the graphs to write the formula for a polynomial function of least degree.\n48. 49. 50.\n51. 52.\nFor the following exercises, use the graph to identify zeros and multiplicity.\n53. 54. 55.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001380",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "53. 54. 55.\nAccess for free at openstax.org\n5.3  Graphs of Polynomial Functions 459\n56.\nFor the following exercises, use the given information about the polynomial graph to write the equation.\n57. Degree 3. Zeros at 58. Degree 3. Zeros at 59. Degree 5. Roots of\nand and multiplicity 2 at and\ny-intercept at y-intercept at , and a root of\nmultiplicity 1 at\ny-intercept at\n60. Degree 4. Root of 61. Degree 5. Double zero at 62. Degree 3. Zeros at\nmultiplicity 2 at and and triple zero at and"
	},
	{
		"chunk_id" : "00001381",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplicity 2 at and and triple zero at and\na roots of multiplicity 1 at Passes through the y-intercept at\nand point\ny-intercept at\n63. Degree 3. Zeros at 64. Degree 5. Roots of 65. Degree 4. Roots of\nand multiplicity 2 at and multiplicity 2 at and\ny-intercept at and a root of roots of multiplicity 1 at\nmultiplicity 1 at and\ny-intercept at y-intercept at\n66. Double zero at and\ntriple zero at Passes\nthrough the point\nTechnology"
	},
	{
		"chunk_id" : "00001382",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "through the point\nTechnology\nFor the following exercises, use a calculator to approximate local minima and maxima or the global minimum and\nmaximum.\n67. 68. 69.\n70. 71.\n460 5  Polynomial and Rational Functions\nExtensions\nFor the following exercises, use the graphs to write a polynomial function of least degree.\n72. 73. 74.\nReal-World Applications\nFor the following exercises, write the polynomial function that models the given situation."
	},
	{
		"chunk_id" : "00001383",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "75. A rectangle has a length of 76. Consider the same 77. A square has sides of 12\n10 units and a width of 8 rectangle of the preceding units. Squares by\nunits. Squares of by problem. Squares of by units are cut out of\nunits are cut out of each units are cut out of each each corner, and then the\ncorner, and then the sides corner. Express the volume sides are folded up to\nare folded up to create an of the box as a polynomial create an open box.\nopen box. Express the in terms of Express the volume of the"
	},
	{
		"chunk_id" : "00001384",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "volume of the box as a box as a function in terms\npolynomial function in of\nterms of\n78. A cylinder has a radius of 79. A right circular cone has a\nunits and a height of radius of and a\n3 units greater. Express the height 3 units less. Express\nvolume of the cylinder as a the volume of the cone as a\npolynomial function. polynomial function. The\nvolume of a cone is\nfor radius\nand height\n5.4 Dividing Polynomials\nLearning Objectives\nIn this section, you will:\nUse long division to divide polynomials."
	},
	{
		"chunk_id" : "00001385",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use long division to divide polynomials.\nUse synthetic division to divide polynomials.\nAccess for free at openstax.org\n5.4  Dividing Polynomials 461\nFigure1 Lincoln Memorial, Washington, D.C. (credit: Ron Cogswell, Flickr)\nThe exterior of the Lincoln Memorial in Washington, D.C., is a large rectangular solid with length 61.5 meters (m), width\n40 m, and height 30 m.1 We can easily find the volume using elementary geometry."
	},
	{
		"chunk_id" : "00001386",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So the volume is 73,800 cubic meters Suppose we knew the volume, length, and width. We could divide to find the\nheight.\nAs we can confirm from the dimensions above, the height is 30 m. We can use similar methods to find any of the missing\ndimensions. We can also use the same method if any, or all, of the measurements contain variable expressions. For\nexample, suppose the volume of a rectangular solid is given by the polynomial The length of"
	},
	{
		"chunk_id" : "00001387",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the solid is given by the width is given by To find the height of the solid, we can use polynomial division, which\nis the focus of this section.\nUsing Long Division to Divide Polynomials\nWe are familiar with thelong divisionalgorithm for ordinary arithmetic. We begin by dividing into the digits of the\ndividend that have the greatest place value. We divide, multiply, subtract, include the digit in the next place value\nposition, and repeat. For example, lets divide 178 by 3 using long division."
	},
	{
		"chunk_id" : "00001388",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Another way to look at the solution is as a sum of parts. This should look familiar, since it is the same method used to\ncheck division in elementary arithmetic.\nWe call this theDivision Algorithmand will discuss it more formally after looking at an example.\nDivision of polynomials that contain more than one term has similarities to long division of whole numbers. We can write"
	},
	{
		"chunk_id" : "00001389",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 National Park Service. \"Lincoln Memorial Building Statistics.\"\" http://www.nps.gov/linc/historyculture/lincoln-memorial-building-statistics.htm."
	},
	{
		"chunk_id" : "00001390",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "divide two polynomials. For example, if we were to divide by using the long division algorithm,\nit would look like this:\nWe have found\nor\nWe can identify thedividend, thedivisor, thequotient, and theremainder.\nWriting the result in this manner illustrates the Division Algorithm.\nThe Division Algorithm\nTheDivision Algorithmstates that, given a polynomial dividend and a non-zero polynomial divisor where\nthe degree of is less than or equal to the degree of , there exist unique polynomials and such that"
	},
	{
		"chunk_id" : "00001391",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the quotient and is the remainder. The remainder is either equal to zero or has degree strictly less than\nIf then divides evenly into This means that, in this case, both and are factors of\nAccess for free at openstax.org\n5.4  Dividing Polynomials 463\n...\nHOW TO\nGiven a polynomial and a binomial, use long division to divide the polynomial by the binomial.\n1. Set up the division problem.\n2. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the"
	},
	{
		"chunk_id" : "00001392",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "divisor.\n3. Multiply the answer by the divisor and write it below the like terms of the dividend.\n4. Subtract the bottombinomialfrom the top binomial.\n5. Bring down the next term of the dividend.\n6. Repeat steps 25 until reaching the last term of the dividend.\n7. If the remainder is non-zero, express as a fraction using the divisor as the denominator.\nEXAMPLE1\nUsing Long Division to Divide a Second-Degree Polynomial\nDivide by\nSolution\nThe quotient is The remainder is 0. We write the result as\nor\nAnalysis"
	},
	{
		"chunk_id" : "00001393",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or\nAnalysis\nThis division problem had a remainder of 0. This tells us that the dividend is divided evenly by the divisor, and that the\ndivisor is a factor of the dividend.\nEXAMPLE2\nUsing Long Division to Divide a Third-Degree Polynomial\nDivide by\n464 5  Polynomial and Rational Functions\nSolution\nThere is a remainder of 1. We can express the result as:\nAnalysis\nWe can check our work by using the Division Algorithm to rewrite the solution. Then multiply.\nNotice, as we write our result,\n the dividend is"
	},
	{
		"chunk_id" : "00001394",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice, as we write our result,\n the dividend is\n the divisor is\n the quotient is\n the remainder is\nTRY IT #1 Divide by\nUsing Synthetic Division to Divide Polynomials\nAs weve seen, long division of polynomials can involve many steps and be quite cumbersome.Synthetic divisionis a\nshorthand method of dividing polynomials for the special case of dividing by a linear factor whose leading coefficient is\n1.\nTo illustrate the process, recall the example at the beginning of the section."
	},
	{
		"chunk_id" : "00001395",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Divide by using the long division algorithm.\nThe final form of the process looked like this:\nThere is a lot of repetition in the table. If we dont write the variables but, instead, line up their coefficients in columns\nunder the division sign and also eliminate the partial products, we already have a simpler version of the entire problem.\nSynthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to"
	},
	{
		"chunk_id" : "00001396",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and\nsubtracting the middle product, we change the sign of the divisor to 2, multiply and add. The process starts by\nbringing down the leading coefficient.\nAccess for free at openstax.org\n5.4  Dividing Polynomials 465\nWe then multiply it by the divisor and add, repeating this process column by column, until there are no entries left. The"
	},
	{
		"chunk_id" : "00001397",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case,\nthe quotient is and the remainder is The process will be made more clear inExample 3.\nSynthetic Division\nSynthetic division is a shortcut that can be used when the divisor is a binomial in the form where is a real\nnumber. Insynthetic division, only the coefficients are used in the division process.\n...\nHOW TO\nGiven two polynomials, use synthetic division to divide."
	},
	{
		"chunk_id" : "00001398",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Write for the divisor.\n2. Write the coefficients of the dividend.\n3. Bring the lead coefficient down.\n4. Multiply the lead coefficient by Write the product in the next column.\n5. Add the terms of the second column.\n6. Multiply the result by Write the product in the next column.\n7. Repeat steps 5 and 6 for the remaining columns.\n8. Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree"
	},
	{
		"chunk_id" : "00001399",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.\nEXAMPLE3\nUsing Synthetic Division to Divide a Second-Degree Polynomial\nUse synthetic division to divide by\nSolution\nBegin by setting up the synthetic division. Write and the coefficients.\nBring down the lead coefficient. Multiply the lead coefficient by\nContinue by adding the numbers in the second column. Multiply the resulting number by Write the result in the next"
	},
	{
		"chunk_id" : "00001400",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "column. Then add the numbers in the third column.\nThe result is The remainder is 0. So is a factor of the original polynomial.\nAnalysis\nJust as with long division, we can check our work by multiplying the quotient by the divisor and adding the remainder.\n466 5  Polynomial and Rational Functions\nEXAMPLE4\nUsing Synthetic Division to Divide a Third-Degree Polynomial\nUse synthetic division to divide by\nSolution"
	},
	{
		"chunk_id" : "00001401",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use synthetic division to divide by\nSolution\nThe binomial divisor is so Add each column, multiply the result by 2, and repeat until the last column is\nreached.\nThe result is The remainder is 0. Thus, is a factor of\nAnalysis\nThe graph of the polynomial function inFigure 2shows a zero at This\nconfirms that is a factor of\nFigure2\nEXAMPLE5\nUsing Synthetic Division to Divide a Fourth-Degree Polynomial\nUse synthetic division to divide by\nSolution"
	},
	{
		"chunk_id" : "00001402",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use synthetic division to divide by\nSolution\nNotice there is nox-term. We will use a zero as the coefficient for that term.\nThe result is\nAccess for free at openstax.org\n5.4  Dividing Polynomials 467\nTRY IT #2 Use synthetic division to divide by\nUsing Polynomial Division to Solve Application Problems\nPolynomial division can be used to solve a variety of application problems involving expressions for area and volume. We"
	},
	{
		"chunk_id" : "00001403",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "looked at an application at the beginning of this section. Now we will solve that problem in the following example.\nEXAMPLE6\nUsing Polynomial Division in an Application Problem\nThe volume of a rectangular solid is given by the polynomial The length of the solid is given by\nand the width is given by Find the height, of the solid.\nSolution\nThere are a few ways to approach this problem. We need to divide the expression for the volume of the solid by the"
	},
	{
		"chunk_id" : "00001404",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expressions for the length and width. Let us create a sketch as inFigure 3.\nFigure3\nWe can now write an equation by substituting the known values into the formula for the volume of a rectangular solid.\nTo solve for first divide both sides by\nNow solve for using synthetic division.\nThe quotient is and the remainder is 0. The height of the solid is\nTRY IT #3 The area of a rectangle is given by The width of the rectangle is given by\nFind an expression for the length of the rectangle.\nMEDIA"
	},
	{
		"chunk_id" : "00001405",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with polynomial division.\nDividing a Trinomial by a Binomial Using Long Division(http://openstax.org/l/dividetribild)\nDividing a Polynomial by a Binomial Using Long Division(http://openstax.org/l/dividepolybild)\nEx 2: Dividing a Polynomial by a Binomial Using Synthetic Division(http://openstax.org/l/dividepolybisd2)\nEx 4: Dividing a Polynomial by a Binomial Using Synthetic Division(http://openstax.org/l/dividepolybisd4)"
	},
	{
		"chunk_id" : "00001406",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "468 5  Polynomial and Rational Functions\n5.4 SECTION EXERCISES\nVerbal\n1. If division of a polynomial by a binomial results in a 2. If a polynomial of degree is divided by a binomial\nremainder of zero, what can be conclude? of degree 1, what is the degree of the quotient?\nAlgebraic\nFor the following exercises, use long division to divide. Specify the quotient and the remainder.\n3. 4. 5.\n6. 7. 8.\n9. 10. 11.\n12. 13."
	},
	{
		"chunk_id" : "00001407",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. 4. 5.\n6. 7. 8.\n9. 10. 11.\n12. 13.\nFor the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by\nsynthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.)\n14. 15. 16.\n17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26.\n27. 28.\n29. 30.\n31. 32.\n33. 34.\n35. 36.\n37.\nAccess for free at openstax.org\n5.4  Dividing Polynomials 469"
	},
	{
		"chunk_id" : "00001408",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.4  Dividing Polynomials 469\nFor the following exercises, use synthetic division to determine whether the first expression is a factor of the second. If it\nis, indicate the factorization.\n38. 39. 40.\n41. 42. 43.\nGraphical\nFor the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of\nthe polynomial suggested by the graph. The leading coefficient is one.\n44. Factor is 45. Factor is 46. Factor is\n47. Factor is 48. Factor is"
	},
	{
		"chunk_id" : "00001409",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. Factor is 48. Factor is\nFor the following exercises, use synthetic division to find the quotient and remainder.\n49. 50. 51.\n52. 53.\nTechnology\nFor the following exercises, use a calculator with CAS to answer the questions.\n54. Consider with What do you 55. Consider for What do you\nexpect the result to be if expect the result to be if\n470 5  Polynomial and Rational Functions\n56. Consider for What do you 57. Consider with What do you\nexpect the result to be if expect the result to be if"
	},
	{
		"chunk_id" : "00001410",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "58. Consider with What do you\nexpect the result to be if\nExtensions\nFor the following exercises, use synthetic division to determine the quotient involving a complex number.\n59. 60. 61.\n62. 63.\nReal-World Applications\nFor the following exercises, use the given length and area of a rectangle to express the width algebraically.\n64. Length is area is 65. Length is area is 66. Length is area is\nFor the following exercises, use the given volume of a box and its length and width to express the height of the box"
	},
	{
		"chunk_id" : "00001411",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "algebraically.\n67. Volume is length is 68. Volume is length is\nwidth is width is\n69. Volume is length is 70. Volume is length is\nwidth is width is\nFor the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder\nalgebraically.\n71. Volume is radius is 72. Volume is radius is\n73. Volume is\nradius is\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 471\n5.5 Zeros of Polynomial Functions\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00001412",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nEvaluate a polynomial using the Remainder Theorem.\nUse the Factor Theorem to solve a polynomial equation.\nUse the Rational Zero Theorem to find rational zeros.\nFind zeros of a polynomial function.\nUse the Linear Factorization Theorem to find polynomials with given zeros.\nUse Descartes Rule of Signs.\nSolve real-world applications of polynomial equations"
	},
	{
		"chunk_id" : "00001413",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A new bakery offers decorated, multi-tiered cakes for display and cutting at Quinceaera and wedding celebrations, as\nwell as sheet cakes for childrens birthday parties and other special occasions to serve most of the guests. The bakery\nwants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want\nthe length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of"
	},
	{
		"chunk_id" : "00001414",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the width. What should the dimensions of the cake pan be?\nThis problem can be solved by writing a cubic function and solving a cubic equation for the volume of the cake. In this\nsection, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations.\nEvaluating a Polynomial Using the Remainder Theorem\nIn the last section, we learned how to divide polynomials. We can now use polynomial division to evaluate polynomials"
	},
	{
		"chunk_id" : "00001415",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "using theRemainder Theorem. If the polynomial is divided by the remainder may be found quickly by evaluating\nthe polynomial function at that is, Lets walk through the proof of the theorem.\nRecall that theDivision Algorithmstates that, given a polynomial dividend and a non-zero polynomial divisor ,\nthere exist unique polynomials and such that\nand either or the degree of is less than the degree of . In practice divisors, will have degrees less"
	},
	{
		"chunk_id" : "00001416",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "than or equal to the degree of . If the divisor, is this takes the form\nSince the divisor is linear, the remainder will be a constant, And, if we evaluate this for we have\nIn other words, is the remainder obtained by dividing by\nThe Remainder Theorem\nIf a polynomial is divided by then the remainder is the value\n...\nHOW TO\nGiven a polynomial function evaluate at using the Remainder Theorem.\n1. Use synthetic division to divide the polynomial by\n2. The remainder is the value\nEXAMPLE1"
	},
	{
		"chunk_id" : "00001417",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. The remainder is the value\nEXAMPLE1\nUsing the Remainder Theorem to Evaluate a Polynomial\nUse the Remainder Theorem to evaluate at\n472 5  Polynomial and Rational Functions\nSolution\nTo find the remainder using the Remainder Theorem, use synthetic division to divide the polynomial by\nThe remainder is 25. Therefore,\nAnalysis\nWe can check our answer by evaluating\nTRY IT #1 Use the Remainder Theorem to evaluate at\nUsing the Factor Theorem to Solve a Polynomial Equation"
	},
	{
		"chunk_id" : "00001418",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TheFactor Theoremis another theorem that helps us analyze polynomial equations. It tells us how the zeros of a\npolynomial are related to the factors. Recall that the Division Algorithm.\nIf is a zero, then the remainder is and or\nNotice, written in this form, is a factor of We can conclude if is a zero of then is a factor of\nSimilarly, if is a factor of then the remainder of the Division Algorithm is 0. This tells\nus that is a zero."
	},
	{
		"chunk_id" : "00001419",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "us that is a zero.\nThis pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree in the complex number\nsystem will have zeros. We can use the Factor Theorem to completely factor a polynomial into the product of factors.\nOnce the polynomial has been completely factored, we can easily determine the zeros of the polynomial.\nThe Factor Theorem\nAccording to theFactor Theorem, is a zero of if and only if is a factor of\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001420",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.\n1. Use synthetic division to divide the polynomial by\n2. Confirm that the remainder is 0.\n3. Write the polynomial as the product of and the quadratic quotient.\n4. If possible, factor the quadratic.\n5. Write the polynomial as the product of factors.\nEXAMPLE2\nUsing the Factor Theorem to Find the Zeros of a Polynomial Expression"
	},
	{
		"chunk_id" : "00001421",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Show that is a factor of Find the remaining factors. Use the factors to determine the zeros of\nthepolynomial.\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 473\nSolution\nWe can use synthetic division to show that is a factor of the polynomial.\nThe remainder is zero, so is a factor of the polynomial. We can use the Division Algorithm to write the polynomial\nas the product of the divisor and the quotient:\nWe can factor the quadratic factor to write the polynomial as"
	},
	{
		"chunk_id" : "00001422",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "By the Factor Theorem, the zeros of are 2, 3, and 5.\nTRY IT #2 Use the Factor Theorem to find the zeros of given that is a\nfactor of the polynomial.\nUsing the Rational Zero Theorem to Find Rational Zeros\nAnother use for the Remainder Theorem is to test whether a rational number is a zero for a given polynomial. But first\nwe need a pool of rational numbers to test. TheRational Zero Theoremhelps us to narrow down the number of"
	},
	{
		"chunk_id" : "00001423",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "possible rational zeros using the ratio of the factors of the constant term and factors of the leadingcoefficientof the\npolynomial\nConsider a quadratic function with two zeros, and By the Factor Theorem, these zeros have factors\nassociated with them. Let us set each factor equal to 0, and then construct the original quadratic function absent its\nstretching factor.\nNotice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3."
	},
	{
		"chunk_id" : "00001424",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, two of the factors from the leading coefficient, 20, are the two denominators from the original rational roots: 5\nand 4.\nWe can infer that the numerators of the rational roots will always be factors of the constant term and the denominators\nwill be factors of the leading coefficient. This is the essence of the Rational Zero Theorem; it is a means to give us a pool\nof possible rational zeros.\nThe Rational Zero Theorem\nTheRational Zero Theoremstates that, if the polynomial has integer"
	},
	{
		"chunk_id" : "00001425",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coefficients, then every rational zero of has the form where is a factor of the constant term and is a\nfactor of the leading coefficient\nWhen the leading coefficient is 1, the possible rational zeros are the factors of the constant term.\n474 5  Polynomial and Rational Functions\n...\nHOW TO\nGiven a polynomial function use the Rational Zero Theorem to find rational zeros.\n1. Determine all factors of the constant term and all factors of the leading coefficient."
	},
	{
		"chunk_id" : "00001426",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Determine all possible values of where is a factor of the constant term and is a factor of the leading\ncoefficient. Be sure to include both positive and negative candidates.\n3. Determine which possible zeros are actual zeros by evaluating each case of\nEXAMPLE3\nListing All Possible Rational Zeros\nList all possible rational zeros of\nSolution\nThe only possible rational zeros of are the quotients of the factors of the last term, 4, and the factors of the leading\ncoefficient, 2."
	},
	{
		"chunk_id" : "00001427",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coefficient, 2.\nThe constant term is 4; the factors of 4 are\nThe leading coefficient is 2; the factors of 2 are\nIf any of the four real zeros are rational zeros, then they will be of one of the following factors of 4 divided by one of the\nfactors of 2.\nNote that and which have already been listed. So we can shorten our list.\nEXAMPLE4\nUsing the Rational Zero Theorem to Find Rational Zeros\nUse the Rational Zero Theorem to find the rational zeros of\nSolution"
	},
	{
		"chunk_id" : "00001428",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe Rational Zero Theorem tells us that if is a zero of then is a factor of 1 and is a factor of 2.\nThe factors of 1 are and the factors of 2 are and The possible values for are and These are the\npossible rational zeros for the function. We can determine which of the possible zeros are actual zeros by substituting\nthese values for in\nOf those, are not zeros of 1 is the only rational zero of\nTRY IT #3 Use the Rational Zero Theorem to find the rational zeros of\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001429",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n5.5  Zeros of Polynomial Functions 475\nFinding the Zeros of Polynomial Functions\nThe Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once\nwe have done this, we can usesynthetic divisionrepeatedly to determine all of thezerosof a polynomial function.\n...\nHOW TO\nGiven a polynomial function use synthetic division to find its zeros.\n1. Use the Rational Zero Theorem to list all possible rational zeros of the function."
	},
	{
		"chunk_id" : "00001430",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the\npolynomial. If the remainder is 0, the candidate is a zero. If the remainder is not zero, discard the candidate.\n3. Repeat step two using the quotient found with synthetic division. If possible, continue until the quotient is a\nquadratic.\n4. Find the zeros of the quadratic function. Two possible methods for solving quadratics are factoring and using the\nquadratic formula.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00001431",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadratic formula.\nEXAMPLE5\nFinding the Zeros of a Polynomial Function with Repeated Real Zeros\nFind the zeros of\nSolution\nThe Rational Zero Theorem tells us that if is a zero of then is a factor of 1 and is a factor of 4.\nThe factors of are and the factors of are and The possible values for are and These\nare the possible rational zeros for the function. We will use synthetic division to evaluate each possible zero until we find\none that gives a remainder of 0. Lets begin with 1."
	},
	{
		"chunk_id" : "00001432",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Dividing by gives a remainder of 0, so 1 is a zero of the function. The polynomial can be written as\nThe quadratic is a perfect square. can be written as\nWe already know that 1 is a zero. The other zero will have a multiplicity of 2 because the factor is squared. To find the\nother zero, we can set the factor equal to 0.\nThe zeros of the function are 1 and with multiplicity 2.\nAnalysis\nLook at the graph of the function inFigure 1. Notice, at the graph bounces off thex-axis, indicating the even"
	},
	{
		"chunk_id" : "00001433",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplicity (2,4,6) for the zero At the graph crosses thex-axis, indicating the odd multiplicity (1,3,5) for\nthe zero\n476 5  Polynomial and Rational Functions\nFigure1\nUsing the Fundamental Theorem of Algebra\nNow that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of\ncomplex zeros of a polynomial function. TheFundamental Theorem of Algebratells us that every polynomial function"
	},
	{
		"chunk_id" : "00001434",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has at least one complex zero. This theorem forms the foundation for solving polynomial equations.\nSuppose is a polynomial function of degree four, and The Fundamental Theorem of Algebra states that there\nis at least one complex solution, call it By the Factor Theorem, we can write as a product of and a\npolynomial quotient. Since is linear, the polynomial quotient will be of degree three. Now we apply the"
	},
	{
		"chunk_id" : "00001435",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Fundamental Theorem of Algebra to the third-degree polynomial quotient. It will have at least one complex zero, call it\nSo we can write the polynomial quotient as a product of and a new polynomial quotient of degree two.\nContinue to apply the Fundamental Theorem of Algebra until all of the zeros are found. There will be four of them and\neach one will yield a factor of\nThe Fundamental Theorem of Algebra\nTheFundamental Theorem of Algebrastates that, if is a polynomial of degreen > 0, then has at least one"
	},
	{
		"chunk_id" : "00001436",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "complex zero.\nWe can use this theorem to argue that, if is a polynomial of degree and is a non-zero real number, then\nhas exactly linear factors\nwhere are complex numbers. Therefore, has roots if we allow for multiplicities.\nQ&A Does every polynomial have at least one imaginary zero?\nNo. Real numbers are a subset of complex numbers, but not the other way around. A complex number is\nnot necessarily imaginary. Real numbers are also complex numbers.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001437",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nFinding the Zeros of a Polynomial Function with Complex Zeros\nFind the zeros of\nSolution\nThe Rational Zero Theorem tells us that if is a zero of then is a factor of 3 and is a factor of 3.\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 477\nThe factors of 3 are and The possible values for and therefore the possible rational zeros for the function, are\nWe will use synthetic division to evaluate each possible zero until we find one that gives a remainder of"
	},
	{
		"chunk_id" : "00001438",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0. Lets begin with 3.\nDividing by gives a remainder of 0, so 3 is a zero of the function. The polynomial can be written as\nWe can then set the quadratic equal to 0 and solve to find the other zeros of the function.\nThe zeros of are 3 and\nAnalysis\nLook at the graph of the function inFigure 2. Notice that, at the graph crosses thex-axis, indicating an odd\nmultiplicity (1) for the zero Also note the presence of the two turning points. This means that, since there is a 3rd"
	},
	{
		"chunk_id" : "00001439",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "degree polynomial, we are looking at the maximum number of turning points. So, the end behavior of increasing\nwithout bound to the right and decreasing without bound to the left will continue. Thus, all thex-intercepts for the\nfunction are shown. So either the multiplicity of is 1 and there are two complex solutions, which is what we\nfound, or the multiplicity at is three. Either way, our result is correct.\nFigure2\nTRY IT #4 Find the zeros of"
	},
	{
		"chunk_id" : "00001440",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nTRY IT #4 Find the zeros of\nUsing the Linear Factorization Theorem to Find Polynomials with Given Zeros\nA vital implication of theFundamental Theorem of Algebra, as we stated above, is that a polynomial function of degree\nwill have zeros in the set of complex numbers, if we allow for multiplicities. This means that we can factor the\npolynomial function into factors. TheLinear Factorization Theoremtells us that a polynomial function will have the"
	},
	{
		"chunk_id" : "00001441",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same number of factors as its degree, and that each factor will be in the form where is a complex number.\nLet be a polynomial function with real coefficients, and suppose is a zero of Then, by the Factor\nTheorem, is a factor of For to have real coefficients, must also be a factor of This is\ntrue because any factor other than when multiplied by will leave imaginary components in the\nproduct. Only multiplication with conjugate pairs will eliminate the imaginary parts and result in real coefficients. In"
	},
	{
		"chunk_id" : "00001442",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "other words, if a polynomial function with real coefficients has a complex zero then the complex conjugate\nmust also be a zero of This is called theComplex Conjugate Theorem.\n478 5  Polynomial and Rational Functions\nComplex Conjugate Theorem\nAccording to theLinear Factorization Theorem,a polynomial function will have the same number of factors as its\ndegree, and each factor will be in the form , where is a complex number."
	},
	{
		"chunk_id" : "00001443",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If the polynomial function has real coefficients and a complex zero in the form then the complex conjugate\nof the zero, is also a zero.\n...\nHOW TO\nGiven the zeros of a polynomial function and a point (c,f(c)) on the graph of use the Linear Factorization\nTheorem to find the polynomial function.\n1. Use the zeros to construct the linear factors of the polynomial.\n2. Multiply the linear factors to expand the polynomial.\n3. Substitute into the function to determine the leading coefficient.\n4. Simplify.\nEXAMPLE7"
	},
	{
		"chunk_id" : "00001444",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Simplify.\nEXAMPLE7\nUsing the Linear Factorization Theorem to Find a Polynomial with Given Zeros\nFind a fourth degree polynomial with real coefficients that has zeros of 3, 2, such that\nSolution\nBecause is a zero, by the Complex Conjugate Theorem is also a zero. The polynomial must have factors of\nand Since we are looking for a degree 4 polynomial, and now have four zeros, we have\nall four factors. Lets begin by multiplying these factors.\nWe need to findato ensure Substitute and into"
	},
	{
		"chunk_id" : "00001445",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We need to findato ensure Substitute and into\nSo the polynomial function is\nor\nAnalysis\nWe found that both and were zeros, but only one of these zeros needed to be given. If is a zero of a polynomial\nwith real coefficients, then must also be a zero of the polynomial because is the complex conjugate of\nQ&A If were given as a zero of a polynomial with real coefficients, would also need to be a\nzero?\nYes. When any complex number with an imaginary component is given as a zero of a polynomial with real"
	},
	{
		"chunk_id" : "00001446",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coefficients, the conjugate must also be a zero of the polynomial.\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 479\nTRY IT #5 Find a third degree polynomial with real coefficients that has zeros of 5 and such that\nUsing Descartes Rule of Signs\nThere is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial\nfunction. If the polynomial is written in descending order,Descartes Rule of Signstells us of a relationship between the"
	},
	{
		"chunk_id" : "00001447",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of sign changes in and the number of positive real zeros. For example, the polynomial function below has\none sign change.\nThis tells us that the function must have 1 positive real zero.\nThere is a similar relationship between the number of sign changes in and the number of negative real zeros.\nIn this case, has 3 sign changes. This tells us that could have 3 or 1 negative real zeros.\nDescartes Rule of Signs\nAccording toDescartes Rule of Signs, if we let be a polynomial function"
	},
	{
		"chunk_id" : "00001448",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with real coefficients:\n The number of positive real zeros is either equal to the number of sign changes of or is less than the\nnumber of sign changes by an even integer.\n The number of negative real zeros is either equal to the number of sign changes of or is less than the\nnumber of sign changes by an even integer.\nEXAMPLE8\nUsing Descartes Rule of Signs\nUse Descartes Rule of Signs to determine the possible numbers of positive and negative real zeros for\nSolution"
	},
	{
		"chunk_id" : "00001449",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nBegin by determining the number of sign changes.\nFigure3\nThere are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine to determine the\nnumber of negative real roots.\nFigure4\nAgain, there are two sign changes, so there are either 2 or 0 negative real roots.\nThere are four possibilities, as we can see inTable 1.\n480 5  Polynomial and Rational Functions\nPositive Real Zeros Negative Real Zeros Complex Zeros Total Zeros\n2 2 0 4\n2 0 2 4\n0 2 2 4\n0 0 4 4\nTable1\nAnalysis"
	},
	{
		"chunk_id" : "00001450",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2 2 0 4\n2 0 2 4\n0 2 2 4\n0 0 4 4\nTable1\nAnalysis\nWe can confirm the numbers of positive and negative real roots by examining a graph of the function. SeeFigure 5. We\ncan see from the graph that the function has 0 positive real roots and 2 negative real roots.\nFigure5\nTRY IT #6 Use Descartes Rule of Signs to determine the maximum possible numbers of positive and\nnegative real zeros for Use a graph to verify the numbers\nof positive and negative real zeros for the function.\nSolving Real-World Applications"
	},
	{
		"chunk_id" : "00001451",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Real-World Applications\nWe have now introduced a variety of tools for solving polynomial equations. Lets use these tools to solve the bakery\nproblem from the beginning of the section.\nEXAMPLE9\nSolving Polynomial Equations\nA new bakery offers decorated, multi-tiered cakes for display and cutting at Quinceaera and wedding celebrations, as\nwell as sheet cakes for childrens birthday parties and other special occasions to serve most of the guests. The bakery"
	},
	{
		"chunk_id" : "00001452",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "wants the volume of a small sheet cake to be 351 cubic inches. The cake is in the shape of a rectangular solid. They want\nthe length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of\nthe width. What should the dimensions of the cake pan be?\nSolution\nBegin by writing an equation for the volume of the cake. The volume of a rectangular solid is given by We were"
	},
	{
		"chunk_id" : "00001453",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given that the length must be four inches longer than the width, so we can express the length of the cake as\nWe were given that the height of the cake is one-third of the width, so we can express the height of the cake as\nLets write the volume of the cake in terms of width of the cake.\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 481\nSubstitute the given volume into this equation."
	},
	{
		"chunk_id" : "00001454",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substitute the given volume into this equation.\nDescartes' rule of signs tells us there is one positive solution. The Rational Zero Theorem tells us that the possible\nrational zeros are and We can use synthetic division to\ntest these possible zeros. Only positive numbers make sense as dimensions for a cake, so we need not test any negative\nvalues. Lets begin by testing values that make the most sense as dimensions for a small sheet cake. Use synthetic\ndivision to check"
	},
	{
		"chunk_id" : "00001455",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "division to check\nSince 1 is not a solution, we will check\nSince 3 is not a solution either, we will test\nSynthetic division gives a remainder of 0, so 9 is a solution to the equation. We can use the relationships between the\nwidth and the other dimensions to determine the length and height of the sheet cake pan.\nThe sheet cake pan should have dimensions 13 inches by 9 inches by 3 inches.\nTRY IT #7 A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters."
	},
	{
		"chunk_id" : "00001456",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The client tells the manufacturer that, because of the contents, the length of the container must\nbe one meter longer than the width, and the height must be one meter greater than twice the\nwidth. What should the dimensions of the container be?\nMEDIA\nAccess these online resources for additional instruction and practice with zeros of polynomial functions.\nReal Zeros, Factors, and Graphs of Polynomial Functions(http://openstax.org/l/realzeros)"
	},
	{
		"chunk_id" : "00001457",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Complex Factorization Theorem(http://openstax.org/l/factortheorem)\nFind the Zeros of a Polynomial Function(http://openstax.org/l/findthezeros)\nFind the Zeros of a Polynomial Function 2(http://openstax.org/l/findthezeros2)\nFind the Zeros of a Polynomial Function 3(http://openstax.org/l/findthezeros3)\n482 5  Polynomial and Rational Functions\n5.5 SECTION EXERCISES\nVerbal\n1. Describe a use for the 2. Explain why the Rational 3. What is the difference"
	},
	{
		"chunk_id" : "00001458",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Remainder Theorem. Zero Theorem does not between rational and real\nguarantee finding zeros of a zeros?\npolynomial function.\n4. If Descartes Rule of Signs 5. If synthetic division reveals a\nreveals a no change of signs zero, why should we try that\nor one sign of changes, value again as a possible\nwhat specific conclusion can solution?\nbe drawn?\nAlgebraic\nFor the following exercises, use the Remainder Theorem to find the remainder.\n6. 7. 8.\n9. 10.\n11. 12.\n13."
	},
	{
		"chunk_id" : "00001459",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7. 8.\n9. 10.\n11. 12.\n13.\nFor the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one\nfactor.\n14. 15.\n16. 17.\n18. 19.\n20. 21.\nFor the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\nAccess for free at openstax.org\n5.5  Zeros of Polynomial Functions 483\n34. 35. 36.\n37. 38. 39.\nFor the following exercises, find all complex solutions (real and non-real)."
	},
	{
		"chunk_id" : "00001460",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "40. 41. 42.\n43. 44. 45.\nGraphical\nFor the following exercises, use Descartes Rule to determine the possible number of positive and negative solutions.\nConfirm with the given graph.\n46. 47. 48.\n49. 50. 51.\n52. 53.\n54. 55.\nNumeric\nFor the following exercises, list all possible rational zeros for the functions.\n56. 57. 58.\n59. 60.\nTechnology\nFor the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational\nzeros. All real solutions are rational."
	},
	{
		"chunk_id" : "00001461",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "zeros. All real solutions are rational.\n61. 62. 63.\n64. 65.\nExtensions\nFor the following exercises, construct a polynomial function of least degree possible using the given information.\n66. Real roots: 1, 1, 3 and 67. Real roots: 1, 1 (with 68. Real roots: 2, (with\nmultiplicity 2 and 1) and multiplicity 2) and\n484 5  Polynomial and Rational Functions\n69. Real roots: , 0, and 70. Real roots: 4, 1, 1, 4 and\nReal-World Applications\nFor the following exercises, find the dimensions of the box described."
	},
	{
		"chunk_id" : "00001462",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "71. The length is twice as long 72. The length, width, and 73. The length is one inch\nas the width. The height is height are consecutive more than the width, which\n2 inches greater than the whole numbers. The is one inch more than the\nwidth. The volume is 192 volume is 120 cubic inches. height. The volume is\ncubic inches. 86.625 cubic inches.\n74. The length is three times 75. The length is 3 inches more\nthe height and the height than the width. The width\nis one inch less than the is 2 inches more than the"
	},
	{
		"chunk_id" : "00001463",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "width. The volume is 108 height. The volume is 120\ncubic inches. cubic inches.\nFor the following exercises, find the dimensions of the right circular cylinder described.\n76. The radius is 3 inches more 77. The height is one less than 78. The radius and height\nthan the height. The one half the radius. The differ by one meter. The\nvolume is cubic volume is cubic radius is larger and the\nmeters. meters. volume is cubic\nmeters.\n79. The radius and height 80. The radius is meter"
	},
	{
		"chunk_id" : "00001464",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "79. The radius and height 80. The radius is meter\ndiffer by two meters. The greater than the height.\nheight is greater and the\nThe volume is cubic\nvolume is cubic\nmeters.\nmeters.\n5.6 Rational Functions\nLearning Objectives\nIn this section, you will:\nUse arrow notation.\nSolve applied problems involving rational functions.\nFind the domains of rational functions.\nIdentify vertical asymptotes.\nIdentify horizontal asymptotes.\nGraph rational functions."
	},
	{
		"chunk_id" : "00001465",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph rational functions.\nSuppose we know that the cost of making a product is dependent on the number of items, produced. This is given by\nthe equation If we want to know the average cost for producing items, we would\ndivide the cost function by the number of items,\nThe average cost function, which yields the average cost per item for items produced, is\nMany other application problems require finding an average value in a similar way, giving us variables in the"
	},
	{
		"chunk_id" : "00001466",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator. Written without a variable in the denominator, this function will contain a negative integer power.\nAccess for free at openstax.org\n5.6  Rational Functions 485\nIn the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for\nexponents. In this section, we explore rational functions, which have variables in the denominator.\nUsing Arrow Notation"
	},
	{
		"chunk_id" : "00001467",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Arrow Notation\nWe have seen the graphs of the basicreciprocal functionand the squared reciprocal function from our study of toolkit\nfunctions. Examine these graphs, as shown inFigure 1, and notice some of their features.\nFigure1\nSeveral things are apparent if we examine the graph of\n1. On the left branch of the graph, the curve approaches thex-axis \n2. As the graph approaches from the left, the curve drops, but as we approach zero from the right, the curve\nrises."
	},
	{
		"chunk_id" : "00001468",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rises.\n3. Finally, on the right branch of the graph, the curves approaches thex-axis \nTo summarize, we usearrow notationto show that or is approaching a particular value. SeeTable 1.\nSymbol Meaning\napproaches from the left ( but close to )\napproaches from the right ( but close to )\n approaches infinity ( increases without bound)\n approaches negative infinity ( decreases without bound)\n the output approaches infinity (the output increases without bound)"
	},
	{
		"chunk_id" : "00001469",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the output approaches negative infinity (the output decreases without bound)\nthe output approaches\nTable1\n486 5  Polynomial and Rational Functions\nLocal Behavior of\nLets begin by looking at the reciprocal function, We cannot divide by zero, which means the function is\nundefined at so zero is not in the domain.As the input values approach zero from the left side (becoming very\nsmall, negative values), the function values decrease without bound (in other words, they approach negative infinity). We"
	},
	{
		"chunk_id" : "00001470",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can see this behavior inTable 2.\n0.1 0.01 0.001 0.0001\n10 100 1000 10,000\nTable2\nWe write in arrow notation\n\nAs the input values approach zero from the right side (becoming very small, positive values), the function values increase\nwithout bound (approaching infinity). We can see this behavior inTable 3.\n0.1 0.01 0.001 0.0001\n10 100 1000 10,000\nTable3\nWe write in arrow notation\n\nSeeFigure 2.\nFigure2"
	},
	{
		"chunk_id" : "00001471",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We write in arrow notation\n\nSeeFigure 2.\nFigure2\nThis behavior creates avertical asymptote, which is a vertical line that the graph approaches but never crosses. In this\ncase, the graph is approaching the vertical line as the input becomes close to zero. SeeFigure 3.\nAccess for free at openstax.org\n5.6  Rational Functions 487\nFigure3\nVertical Asymptote\nAvertical asymptoteof a graph is a vertical line where the graph tends toward positive or negative infinity as\nthe inputs approach We write\n"
	},
	{
		"chunk_id" : "00001472",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inputs approach We write\n\nEnd Behavior of\nAs the values of approach infinity, the function values approach 0. As the values of approach negative infinity, the\nfunction values approach 0. SeeFigure 4. Symbolically, using arrow notation\n \nFigure4\nBased on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it\nseems to level off as the inputs become large. This behavior creates ahorizontal asymptote, a horizontal line that the"
	},
	{
		"chunk_id" : "00001473",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the\nhorizontal line SeeFigure 5.\n488 5  Polynomial and Rational Functions\nFigure5\nHorizontal Asymptote\nAhorizontal asymptoteof a graph is a horizontal line where the graph approaches the line as the inputs\nincrease or decrease without bound. We write\n \nEXAMPLE1\nUsing Arrow Notation\nUse arrow notation to describe the end behavior and local behavior of the function graphed inFigure 6.\nFigure6"
	},
	{
		"chunk_id" : "00001474",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nSolution\nNotice that the graph is showing a vertical asymptote at which tells us that the function is undefined at\n \nAnd as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a\nhorizontal asymptote at As the inputs increase without bound, the graph levels off at 4.\n \nAccess for free at openstax.org\n5.6  Rational Functions 489\nTRY IT #1 Use arrow notation to describe the end behavior and local behavior for the reciprocal squared\nfunction."
	},
	{
		"chunk_id" : "00001475",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\nEXAMPLE2\nUsing Transformations to Graph a Rational Function\nSketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and\nvertical asymptotes of the graph, if any.\nSolution\nShifting the graph left 2 and up 3 would result in the function\nor equivalently, by giving the terms a common denominator,\nThe graph of the shifted function is displayed inFigure 7.\nFigure7"
	},
	{
		"chunk_id" : "00001476",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure7\nNotice that this function is undefined at and the graph also is showing a vertical asymptote at\n \nAs the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating\na horizontal asymptote at\n\nAnalysis\nNotice that horizontal and vertical asymptotes are shifted left 2 and up 3 along with the function.\nTRY IT #2 Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared"
	},
	{
		"chunk_id" : "00001477",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function that has been shifted right 3 units and down 4 units.\nSolving Applied Problems Involving Rational Functions\nInExample 2, we shifted a toolkit function in a way that resulted in the function This is an example of a\nrational function. Arational functionis a function that can be written as the quotient of two polynomial functions. Many\nreal-world problems require us to find the ratio of two polynomial functions. Problems involving rates and\nconcentrations often involve rational functions."
	},
	{
		"chunk_id" : "00001478",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "concentrations often involve rational functions.\n490 5  Polynomial and Rational Functions\nRational Function\nArational functionis a function that can be written as the quotient of two polynomial functions\nEXAMPLE3\nSolving an Applied Problem Involving a Rational Function\nAfter running out of pre-packaged supplies, a nurse in a refugee camp is preparing an intravenous sugar solution for\npatients in the camp hospital. A large mixing tank currently contains 100 gallons of distilled water into which 5 pounds of"
	},
	{
		"chunk_id" : "00001479",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sugar have been mixed. A tap will open pouring 10 gallons per minute of water into the tank at the same time sugar is\npoured into the tank at a rate of 1 pound per minute. Find the ratio of sugar to water, in pounds per gallon in the tank\nafter 12 minutes. Is that a greater ratio of sugar to water, in pounds per gallon than at the beginning?\nSolution\nLet be the number of minutes since the tap opened. Since the water increases at 10 gallons per minute, and the sugar"
	},
	{
		"chunk_id" : "00001480",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increases at 1 pound per minute, these are constant rates of change. This tells us the amount of water in the tank is\nchanging linearly, as is the amount of sugar in the tank. We can write an equation independently for each:\nThe ratio of sugar to water, in pounds per gallon, , will be the ratio of pounds of sugar to gallons of water\nThe ratio of sugar to water, in pounds per gallon after 12 minutes is given by evaluating at"
	},
	{
		"chunk_id" : "00001481",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This means the ratio of sugar to water, in pounds per gallon is 17 pounds of sugar to 220 gallons of water.\nAt the beginning, the ratio of sugar to water, in pounds per gallon is\nSince the ratio of sugar to water, in pounds per gallon is greater after 12 minutes than at the\nbeginning.\nTRY IT #3 There are 1,200 first-year and 1,500 second-year students at a rally at noon. After 12 p.m., 20 first-\nyear students arrive at the rally every five minutes while 15 second-year students leave the rally."
	},
	{
		"chunk_id" : "00001482",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the ratio of first-year to second-year students at 1 p.m.\nFinding the Domains of Rational Functions\nAvertical asymptoterepresents a value at which a rational function is undefined, so that value is not in the domain of\nthe function. A reciprocal function cannot have values in its domain that cause the denominator to equal zero. In\ngeneral, to find the domain of a rational function, we need to determine which inputs would cause division by zero.\nDomain of a Rational Function"
	},
	{
		"chunk_id" : "00001483",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Domain of a Rational Function\nThe domain of a rational function includes all real numbers except those that cause the denominator to equal zero.\nAccess for free at openstax.org\n5.6  Rational Functions 491\n...\nHOW TO\nGiven a rational function, find the domain.\n1. Set the denominator equal to zero.\n2. Solve to find thex-values that cause the denominator to equal zero.\n3. The domain is all real numbers except those found in Step 2.\nEXAMPLE4\nFinding the Domain of a Rational Function\nFind the domain of"
	},
	{
		"chunk_id" : "00001484",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain of\nSolution\nBegin by setting the denominator equal to zero and solving.\nThe denominator is equal to zero when The domain of the function is all real numbers except\nAnalysis\nA graph of this function, as shown inFigure 8, confirms that the function is not defined when\nFigure8\nThere is a vertical asymptote at and a hole in the graph at We will discuss these types of holes in greater\ndetail later in this section.\nTRY IT #4 Find the domain of\nIdentifying Vertical Asymptotes of Rational Functions"
	},
	{
		"chunk_id" : "00001485",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are\nasymptotes. We may even be able to approximate their location. Even without the graph, however, we can still\ndetermine whether a given rational function has any asymptotes, and calculate their location.\nVertical Asymptotes\nThe vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not"
	},
	{
		"chunk_id" : "00001486",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.\n492 5  Polynomial and Rational Functions\n...\nHOW TO\nGiven a rational function, identify any vertical asymptotes of its graph.\n1. Factor the numerator and denominator.\n2. Note any restrictions in the domain of the function.\n3. Reduce the expression by canceling common factors in the numerator and the denominator."
	},
	{
		"chunk_id" : "00001487",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical\nasymptotes occur.\n5. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities, or\nholes.\nEXAMPLE5\nIdentifying Vertical Asymptotes\nFind the vertical asymptotes of the graph of\nSolution\nFirst, factor the numerator and denominator.\nTo find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to"
	},
	{
		"chunk_id" : "00001488",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "zero:\nNeither nor are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph in\nFigure 9confirms the location of the two vertical asymptotes.\nFigure9\nRemovable Discontinuities\nOccasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We\ncall such a hole aremovable discontinuity.\nFor example, the function may be re-written by factoring the numerator and the denominator.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001489",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n5.6  Rational Functions 493\nNotice that is a common factor to the numerator and the denominator. The zero of this factor, is the\nlocation of the removable discontinuity. Notice also that is not a factor in both the numerator and denominator. The\nzero of this factor, is the vertical asymptote. SeeFigure 10. [Note that removable discontinuities may not be visible\nwhen we use a graphing calculator, depending upon the window selected.]\nFigure10"
	},
	{
		"chunk_id" : "00001490",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure10\nRemovable Discontinuities of Rational Functions\nAremovable discontinuityoccurs in the graph of a rational function at if is a zero for a factor in the\ndenominator that is common with a factor in the numerator. We factor the numerator and denominator and check\nfor common factors. If we find any, we set the common factor equal to 0 and solve. This is the location of the\nremovable discontinuity. This is true if the multiplicity of this factor is greater than or equal to that in the"
	},
	{
		"chunk_id" : "00001491",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that\nvalue.\nEXAMPLE6\nIdentifying Vertical Asymptotes and Removable Discontinuities for a Graph\nFind the vertical asymptotes and removable discontinuities of the graph of\nSolution\nFactor the numerator and the denominator.\nNotice that there is a common factor in the numerator and the denominator, The zero for this factor is This\nis the location of the removable discontinuity."
	},
	{
		"chunk_id" : "00001492",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the location of the removable discontinuity.\nNotice that there is a factor in the denominator that is not in the numerator, The zero for this factor is The\nvertical asymptote is SeeFigure 11.\n494 5  Polynomial and Rational Functions\nFigure11\nThe graph of this function will have the vertical asymptote at but at the graph will have a hole.\nTRY IT #5 Find the vertical asymptotes and removable discontinuities of the graph of\nIdentifying Horizontal Asymptotes of Rational Functions"
	},
	{
		"chunk_id" : "00001493",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "While vertical asymptotes describe the behavior of a graph as theoutputgets very large or very small, horizontal\nasymptotes help describe the behavior of a graph as theinputgets very large or very small. Recall that a polynomials\nend behavior will mirror that of the leading term. Likewise, a rational functions end behavior will mirror that of the ratio\nof the function that is the ratio of the leading terms.\nThere are three distinct outcomes when checking for horizontal asymptotes:"
	},
	{
		"chunk_id" : "00001494",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Case 1:If the degree of the denominator > degree of the numerator, there is ahorizontal asymptoteat\nIn this case, the end behavior is This tells us that, as the inputs increase or decrease without bound, this\nfunction will behave similarly to the function and the outputs will approach zero, resulting in a horizontal\nasymptote at SeeFigure 12. Note that this graph crosses the horizontal asymptote.\nFigure12 Horizontal asymptote when"
	},
	{
		"chunk_id" : "00001495",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12 Horizontal asymptote when\nCase 2:If the degree of the denominator < degree of the numerator by one, we get a slant asymptote.\nAccess for free at openstax.org\n5.6  Rational Functions 495\nIn this case, the end behavior is This tells us that as the inputs increase or decrease without bound,\nthis function will behave similarly to the function As the inputs grow large, the outputs will grow and not level"
	},
	{
		"chunk_id" : "00001496",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "off, so this graph has no horizontal asymptote. However, the graph of looks like a diagonal line, and since\nwill behave similarly to it will approach a line close to This line is a slant asymptote.\nTo find the equation of the slant asymptote, divide The quotient is and the remainder is 2. The slant\nasymptote is the graph of the line SeeFigure 13.\nFigure13 Slant asymptote when where degree of\nCase 3:If the degree of the denominator = degree of the numerator, there is a horizontal asymptote at where"
	},
	{
		"chunk_id" : "00001497",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and are the leading coefficients of and for\nIn this case, the end behavior is This tells us that as the inputs grow large, this function will behave like\nthe function which is a horizontal line. As  resulting in a horizontal asymptote at See\nFigure 14. Note that this graph crosses the horizontal asymptote.\nFigure14 Horizontal asymptote when\n496 5  Polynomial and Rational Functions\nNotice that, while the graph of a rational function will never cross avertical asymptote, the graph may or may not cross a"
	},
	{
		"chunk_id" : "00001498",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal or slant asymptote. Also, although the graph of a rational function may have many vertical asymptotes, the\ngraph will have at most one horizontal (or slant) asymptote.\nIt should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one,\ntheend behaviorof the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the\nfunction\nwith end behavior"
	},
	{
		"chunk_id" : "00001499",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function\nwith end behavior\nthe end behavior of the graph would look similar to that of an even polynomial with a positive leading coefficient.\n \nHorizontal Asymptotes of Rational Functions\nThehorizontal asymptoteof a rational function can be determined by looking at the degrees of the numerator and\ndenominator.\n Degree of numeratoris less thandegree of denominator: horizontal asymptote at\n Degree of numeratoris greater than degree of denominator by one: no horizontal asymptote; slant asymptote."
	},
	{
		"chunk_id" : "00001500",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Degree of numeratoris equal todegree of denominator: horizontal asymptote at ratio of leading coefficients.\nEXAMPLE7\nIdentifying Horizontal and Slant Asymptotes\nFor the functions listed, identify the horizontal or slant asymptote.\n  \nSolution\nFor these solutions, we will use\n The degree of so we can find the horizontal asymptote by taking the ratio\nof the leading terms. There is a horizontal asymptote at or\n The degree of and degree of Since by 1, there is a slant asymptote found at"
	},
	{
		"chunk_id" : "00001501",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The quotient is and the remainder is 13. There is a slant asymptote at\n The degree of degree of so there is a horizontal asymptote\nEXAMPLE8\nIdentifying Horizontal Asymptotes\nIn the sugar concentration problem earlier, we created the equation\nFind the horizontal asymptote and interpret it in context of the problem.\nAccess for free at openstax.org\n5.6  Rational Functions 497\nSolution\nBoth the numerator and denominator are linear (degree 1). Because the degrees are equal, there will be a horizontal"
	},
	{
		"chunk_id" : "00001502",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptote at the ratio of the leading coefficients. In the numerator, the leading term is with coefficient 1. In the\ndenominator, the leading term is with coefficient 10. The horizontal asymptote will be at the ratio of these values:\n\nThis function will have a horizontal asymptote at\nThis tells us that as the values oftincrease, the values of will approach In context, this means that, as more time"
	},
	{
		"chunk_id" : "00001503",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "goes by, the concentration of sugar in the tank will approach one-tenth of a pound of sugar per gallon of water or\npounds per gallon.\nEXAMPLE9\nIdentifying Horizontal and Vertical Asymptotes\nFind the horizontal and vertical asymptotes of the function\nSolution\nFirst, note that this function has no common factors, so there are no potential removable discontinuities.\nThe function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The"
	},
	{
		"chunk_id" : "00001504",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator will be zero at indicating vertical asymptotes at these values.\nThe numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than\nthe degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards\nzero as the inputs get large, and so as  This function will have a horizontal asymptote at See\nFigure 15.\nFigure15\nTRY IT #6 Find the vertical and horizontal asymptotes of the function:"
	},
	{
		"chunk_id" : "00001505",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "498 5  Polynomial and Rational Functions\nIntercepts of Rational Functions\nArational functionwill have ay-intercept at , if the function is defined at zero. A rational function will not have a\ny-intercept if the function is not defined at zero.\nLikewise, a rational function will havex-intercepts at the inputs that cause the output to be zero. Since a fraction is\nonly equal to zero when the numerator is zero,x-intercepts can only occur when the numerator of the rational\nfunction is equal to zero.\nEXAMPLE10"
	},
	{
		"chunk_id" : "00001506",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function is equal to zero.\nEXAMPLE10\nFinding the Intercepts of a Rational Function\nFind the intercepts of\nSolution\nWe can find they-intercept by evaluating the function at zero\nThex-intercepts will occur when the function is equal to zero:\nThey-intercept is thex-intercepts are and SeeFigure 16.\nFigure16\nAccess for free at openstax.org\n5.6  Rational Functions 499\nTRY IT #7 Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a"
	},
	{
		"chunk_id" : "00001507",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rational function. Then, find thex- andy-intercepts and the horizontal and vertical asymptotes.\nGraphing Rational Functions\nInExample 9, we see that the numerator of a rational function reveals thex-intercepts of the graph, whereas the\ndenominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have\ninteger powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we\nsaw with polynomials."
	},
	{
		"chunk_id" : "00001508",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "saw with polynomials.\nThe vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal\nfunctions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side\nof the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards\nnegative infinity. SeeFigure 17.\nFigure17"
	},
	{
		"chunk_id" : "00001509",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "negative infinity. SeeFigure 17.\nFigure17\nWhen the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads\ntoward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See\nFigure 18.\nFigure18\nFor example, the graph of is shown inFigure 19.\n500 5  Polynomial and Rational Functions\nFigure19\n At thex-intercept corresponding to the factor of the numerator, the graph \"bounces\"\""
	},
	{
		"chunk_id" : "00001510",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with the quadratic nature of the factor.\n At thex-intercept corresponding to the factor of the numerator, the graph passes through the axis as\nwe would expect from a linear factor.\n At the vertical asymptote corresponding to the factor of the denominator, the graph heads towards\npositive infinity on both sides of the asymptote, consistent with the behavior of the function\n At the vertical asymptote corresponding to the factor of the denominator, the graph heads towards"
	},
	{
		"chunk_id" : "00001511",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "positive infinity on the left side of the asymptote and towards negative infinity on the right side.\n...\nHOW TO\nGiven a rational function, sketch a graph.\n1. Evaluate the function at 0 to find they-intercept.\n2. Factor the numerator and denominator.\n3. For factors in the numerator not common to the denominator, determine where each factor of the numerator is\nzero to find thex-intercepts.\n4. Find the multiplicities of thex-intercepts to determine the behavior of the graph at those points."
	},
	{
		"chunk_id" : "00001512",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. For factors in the denominator, note the multiplicities of the zeros to determine the local behavior. For those\nfactors not common to the numerator, find the vertical asymptotes by setting those factors equal to zero and\nthen solve.\n6. For factors in the denominator common to factors in the numerator, find the removable discontinuities by\nsetting those factors equal to 0 and then solve.\n7. Compare the degrees of the numerator and the denominator to determine the horizontal or slant asymptotes."
	},
	{
		"chunk_id" : "00001513",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8. Sketch the graph.\nAccess for free at openstax.org\n5.6  Rational Functions 501\nEXAMPLE11\nGraphing a Rational Function\nSketch a graph of\nSolution\nWe can start by noting that the function is already factored, saving us a step.\nNext, we will find the intercepts. Evaluating the function at zero gives they-intercept:\nTo find thex-intercepts, we determine when the numerator of the function is zero. Setting each factor equal to zero, we"
	},
	{
		"chunk_id" : "00001514",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "findx-intercepts at and At each, the behavior will be linear (multiplicity 1), with the graph passing through\nthe intercept.\nWe have ay-intercept at andx-intercepts at and\nTo find the vertical asymptotes, we determine when the denominator is equal to zero. This occurs when and\nwhen giving us vertical asymptotes at and\nThere are no common factors in the numerator and denominator. This means there are no removable discontinuities."
	},
	{
		"chunk_id" : "00001515",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finally, the degree of denominator is larger than the degree of the numerator, telling us this graph has a horizontal\nasymptote at\nTo sketch the graph, we might start by plotting the three intercepts. Since the graph has nox-intercepts between the\nvertical asymptotes, and they-intercept is positive, we know the function must remain positive between the asymptotes,\nletting us fill in the middle portion of the graph as shown inFigure 20.\nFigure20"
	},
	{
		"chunk_id" : "00001516",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure20\nThe factor associated with the vertical asymptote at was squared, so we know the behavior will be the same on\nboth sides of the asymptote. The graph heads toward positive infinity as the inputs approach the asymptote on the right,\nso the graph will head toward positive infinity on the left as well.\nFor the vertical asymptote at the factor was not squared, so the graph will have opposite behavior on either side"
	},
	{
		"chunk_id" : "00001517",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the asymptote. SeeFigure 21. After passing through thex-intercepts, the graph will then level off toward an output of\nzero, as indicated by the horizontal asymptote.\n502 5  Polynomial and Rational Functions\nFigure21\nTRY IT #8 Given the function use the characteristics of polynomials and rational\nfunctions to describe its behavior and sketch the function.\nWriting Rational Functions\nNow that we have analyzed the equations for rational functions and how they relate to a graph of the function, we can"
	},
	{
		"chunk_id" : "00001518",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "use information given by a graph to write the function. A rational function written in factored form will have an\nx-intercept where each factor of the numerator is equal to zero. (An exception occurs in the case of a removable\ndiscontinuity.) As a result, we can form a numerator of a function whose graph will pass through a set ofx-intercepts by\nintroducing a corresponding set of factors. Likewise, because the function will have a vertical asymptote where each"
	},
	{
		"chunk_id" : "00001519",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "factor of the denominator is equal to zero, we can form a denominator that will produce the vertical asymptotes by\nintroducing a corresponding set of factors.\nWriting Rational Functions from Intercepts and Asymptotes\nIf arational functionhasx-intercepts at vertical asymptotes at and no\nthen the function can be written in the form:\nwhere the powers or on each factor can be determined by the behavior of the graph at the corresponding"
	},
	{
		"chunk_id" : "00001520",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intercept or asymptote, and the stretch factor can be determined given a value of the function other than the\nx-intercept or by the horizontal asymptote if it is nonzero.\n...\nHOW TO\nGiven a graph of a rational function, write the function.\n1. Determine the factors of the numerator. Examine the behavior of the graph at thex-intercepts to determine the\nzeroes and their multiplicities. (This is easy to do when finding the simplest function with small"
	},
	{
		"chunk_id" : "00001521",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplicitiessuch as 1 or 3but may be difficult for larger multiplicitiessuch as 5 or 7, for example.)\n2. Determine the factors of the denominator. Examine the behavior on both sides of each vertical asymptote to\nAccess for free at openstax.org\n5.6  Rational Functions 503\ndetermine the factors and their powers.\n3. Use any clear point on the graph to find the stretch factor.\nEXAMPLE12\nWriting a Rational Function from Intercepts and Asymptotes"
	},
	{
		"chunk_id" : "00001522",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write an equation for the rational function shown inFigure 22.\nFigure22\nSolution\nThe graph appears to havex-intercepts at and At both, the graph passes through the intercept,\nsuggesting linear factors. The graph has two vertical asymptotes. The one at seems to exhibit the basic behavior\nsimilar to with the graph heading toward positive infinity on one side and heading toward negative infinity on the"
	},
	{
		"chunk_id" : "00001523",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "other. The asymptote at is exhibiting a behavior similar to with the graph heading toward negative infinity on\nboth sides of the asymptote. SeeFigure 23.\nFigure23\nWe can use this information to write a function of the form\nTo find the stretch factor, we can use another clear point on the graph, such as they-intercept\n504 5  Polynomial and Rational Functions\nThis gives us a final function of\nMEDIA\nAccess these online resources for additional instruction and practice with rational functions."
	},
	{
		"chunk_id" : "00001524",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Rational Functions(http://openstax.org/l/graphrational)\nFind the Equation of a Rational Function(http://openstax.org/l/equatrational)\nDetermining Vertical and Horizontal Asymptotes(http://openstax.org/l/asymptote)\nFind the Intercepts, Asymptotes, and Hole of a Rational Function(http://openstax.org/l/interasymptote)\n5.6 SECTION EXERCISES\nVerbal\n1. What is the fundamental 2. What is the fundamental 3. If the graph of a rational"
	},
	{
		"chunk_id" : "00001525",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference in the algebraic difference in the graphs of function has a removable\nrepresentation of a polynomial functions and discontinuity, what must be\npolynomial function and a rational functions? true of the functional rule?\nrational function?\n4. Can a graph of a rational 5. Can a graph of a rational\nfunction have no vertical function have no\nasymptote? If so, how? x-intercepts? If so, how?\nAlgebraic\nFor the following exercises, find the domain of the rational functions.\n6. 7. 8.\n9."
	},
	{
		"chunk_id" : "00001526",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7. 8.\n9.\nFor the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19.\nAccess for free at openstax.org\n5.6  Rational Functions 505\nFor the following exercises, find thex- andy-intercepts for the functions.\n20. 21. 22.\n23. 24.\nFor the following exercises, describe the local and end behavior of the functions.\n25. 26. 27.\n28. 29.\nFor the following exercises, find the slant asymptote of the functions.\n30. 31. 32."
	},
	{
		"chunk_id" : "00001527",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "30. 31. 32.\n33. 34.\nGraphical\nFor the following exercises, use the given transformation to graph the function. Note the vertical and horizontal\nasymptotes.\n35. The reciprocal function 36. The reciprocal function 37. The reciprocal squared\nshifted up two units. shifted down one unit and function shifted to the\nleft three units. right 2 units.\n38. The reciprocal squared\nfunction shifted down 2\nunits and right 1 unit."
	},
	{
		"chunk_id" : "00001528",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function shifted down 2\nunits and right 1 unit.\nFor the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the\nhorizontal or slant asymptote of the functions. Use that information to sketch a graph.\n39. 40. 41.\n42. 43. 44.\n45. 46. 47.\n48. 49. 50.\n506 5  Polynomial and Rational Functions\nFor the following exercises, write an equation for a rational function with the given characteristics.\n51. Vertical asymptotes at and 52. Vertical asymptotes at and"
	},
	{
		"chunk_id" : "00001529",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-intercepts at and y-intercept at x-intercepts at and y-intercept at\n53. Vertical asymptotes at and 54. Vertical asymptotes at and\nx-intercepts at and Horizontal x-intercepts at and Horizontal\nasymptote at asymptote at\n55. Vertical asymptote at Double zero at 56. Vertical asymptote at Double zero at\ny-intercept at y-intercept at\nFor the following exercises, use the graphs to write an equation for the function.\n57. 58. 59.\n60. 61. 62.\n63. 64.\nAccess for free at openstax.org\n5.6  Rational Functions 507"
	},
	{
		"chunk_id" : "00001530",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.6  Rational Functions 507\nNumeric\nFor the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting\nthe horizontal asymptote\n65. 66. 67.\n68. 69.\nTechnology\nFor the following exercises, use a calculator to graph Use the graph to solve\n70. 71. 72.\n73. 74.\nExtensions\nFor the following exercises, identify the removable discontinuity.\n75. 76. 77.\n78. 79.\nReal-World Applications"
	},
	{
		"chunk_id" : "00001531",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "75. 76. 77.\n78. 79.\nReal-World Applications\nFor the following exercises, express a rational function that describes the situation.\n80. In the refugee camp hospital, a large mixing tank 81. In the refugee camp hospital, a large mixing tank\ncurrently contains 200 gallons of water, into which currently contains 300 gallons of water, into which\n10 pounds of sugar have been mixed. A tap will 8 pounds of sugar have been mixed. A tap will"
	},
	{
		"chunk_id" : "00001532",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "open, pouring 10 gallons of water per minute into open, pouring 20 gallons of water per minute into\nthe tank at the same time sugar is poured into the the tank at the same time sugar is poured into the\ntank at a rate of 3 pounds per minute. Find the tank at a rate of 2 pounds per minute. Find the\nconcentration (pounds per gallon) of sugar in the concentration (pounds per gallon) of sugar in the\ntank after minutes. tank after minutes."
	},
	{
		"chunk_id" : "00001533",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tank after minutes. tank after minutes.\nFor the following exercises, use the given rational function to answer the question.\n82. The concentration of a drug in a patients 83. The concentration of a drug in a patients\nbloodstream hours after injection is given by bloodstream hours after injection is given by\nWhat happens to the concentration Use a calculator to approximate\nof the drug as increases? the time when the concentration is highest.\n508 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001534",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "508 5  Polynomial and Rational Functions\nFor the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to\nanswer the question.\n84. An open box with a square 85. A rectangular box with a 86. A right circular cylinder has\nbase is to have a volume of square base is to have a volume of 100 cubic inches.\n108 cubic inches. Find the volume of 20 cubic feet. Find the radius and height\ndimensions of the box that The material for the base that will yield minimum"
	},
	{
		"chunk_id" : "00001535",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will have minimum surface costs 30 cents/ square foot. surface area. Let =\narea. Let = length of the The material for the sides radius.\nside of the base. costs 10 cents/square foot.\nThe material for the top\ncosts 20 cents/square foot.\nDetermine the dimensions\nthat will yield minimum\ncost. Let = length of the\nside of the base.\n87. A right circular cylinder 88. A right circular cylinder is\nwith no top has a volume to have a volume of 40\nof 50 cubic meters. Find the cubic inches. It costs 4"
	},
	{
		"chunk_id" : "00001536",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radius that will yield cents/square inch to\nminimum surface area. Let construct the top and\n= radius. bottom and 1 cent/square\ninch to construct the rest of\nthe cylinder. Find the\nradius to yield minimum\ncost. Let = radius.\n5.7 Inverses and Radical Functions\nLearning Objectives\nIn this section, you will:\nFind the inverse of an invertible polynomial function.\nRestrict the domain to find the inverse of a polynomial function."
	},
	{
		"chunk_id" : "00001537",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Park rangers and other trail managers may construct rock piles, stacks, or other arrangements, usually called cairns, to\nmark trails or other landmarks. (Rangers and environmental scientists discourage hikers from doing the same, in order\nto avoid confusion and preserve the habitats of plants and animals.) A cairn in the form of a mound of gravel is in the\nshape of a cone with the height equal to twice the radius.\nFigure1\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 509"
	},
	{
		"chunk_id" : "00001538",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.7  Inverses and Radical Functions 509\nThe volume is found using a formula from elementary geometry.\nWe have written the volume in terms of the radius However, in some cases, we may start out with the volume and\nwant to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound\nwith a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the\nformula"
	},
	{
		"chunk_id" : "00001539",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "formula\nThis function is the inverse of the formula for in terms of\nIn this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we\nencounter in the process.\nFinding the Inverse of a Polynomial Function\nTwo functions and are inverse functions if for every coordinate pair in there exists a corresponding\ncoordinate pair in the inverse function, In other words, the coordinate pairs of the inverse functions have the"
	},
	{
		"chunk_id" : "00001540",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input and output interchanged. Only one-to-one functions have inverses. Recall that a one-to-one function has a unique\noutput value for each input value and passes the horizontal line test.\nFor example, suppose the Sustainability Club builds a water runoff collector in the shape of a parabolic trough as shown\ninFigure 2. We can use the information in the figure to find the surface area of the water in the trough as a function of\nthe depth of the water.\nFigure2"
	},
	{
		"chunk_id" : "00001541",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the depth of the water.\nFigure2\nBecause it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system\nat the cross section, with measured horizontally and measured vertically, with the origin at the vertex of the\nparabola. SeeFigure 3.\n510 5  Polynomial and Rational Functions\nFigure3\nFrom this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know"
	},
	{
		"chunk_id" : "00001542",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equation will have form Our equation will need to pass through the point (6, 18), from which we can\nsolve for the stretch factor\nOur parabolic cross section has the equation\nWe are interested in thesurface areaof the water, so we must determine the width at the top of the water as a function\nof the water depth. For any depth the width will be given by so we need to solve the equation above for and find"
	},
	{
		"chunk_id" : "00001543",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inverse function. However, notice that the original function is not one-to-one, and indeed, given any output there are\ntwo inputs that produce the same output, one positive and one negative.\nTo find an inverse, we can restrict our original function to a limited domain on which itisone-to-one. In this case, it\nmakes sense to restrict ourselves to positive values. On this domain, we can find an inverse by solving for the input\nvariable:"
	},
	{
		"chunk_id" : "00001544",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variable:\nThis is not a function as written. We are limiting ourselves to positive values, so we eliminate the negative solution,\ngiving us the inverse function were looking for.\nBecause is the distance from the center of the parabola to either side, the entire width of the water at the top will be\nThe trough is 3 feet (36 inches) long, so the surface area will then be:\nThis example illustrates two important points:"
	},
	{
		"chunk_id" : "00001545",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This example illustrates two important points:\n1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.\n2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 511\nfunctions.\nFunctions involving roots are often calledradical functions. While it is not possible to find an inverse of most polynomial"
	},
	{
		"chunk_id" : "00001546",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions, some basic polynomials do have inverses. Such functions are calledinvertible functions, and we use the\nnotation\nWarning: is not the same as the reciprocal of the function This use of 1 is reserved to denote inverse\nfunctions. To denote the reciprocal of a function we would need to write\nAn important relationship between inverse functions is that they undo each other. If is the inverse of a function"
	},
	{
		"chunk_id" : "00001547",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then is the inverse of the function In other words, whatever the function does to undoes itand vice-\nversa.\nand\nNote that the inverse switches the domain and range of the original function.\nVerifying Two Functions Are Inverses of One Another\nTwo functions, and are inverses of one another if for all in the domain of and\n...\nHOW TO\nGiven a polynomial function, find the inverse of the function by restricting the domain in such a way that the\nnew function is one-to-one.\n1. Replace with\n2. Interchange and"
	},
	{
		"chunk_id" : "00001548",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Replace with\n2. Interchange and\n3. Solve for and rename the function\nEXAMPLE1\nVerifying Inverse Functions\nShow that and are inverses, for .\nSolution\nWe must show that and\n512 5  Polynomial and Rational Functions\nTherefore, and are inverses.\nTRY IT #1 Show that and are inverses.\nEXAMPLE2\nFinding the Inverse of a Cubic Function\nFind the inverse of the function\nSolution\nThis is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is"
	},
	{
		"chunk_id" : "00001549",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one-to-one. Solving for the inverse by solving for\nAnalysis\nLook at the graph of and Notice that one graph is the reflection of the other about the line This is always\nthe case when graphing a function and its inverse function.\nAlso, since the method involved interchanging and notice corresponding points. If is on the graph of then\nis on the graph of Since is on the graph of then is on the graph of Similarly, since is\non the graph of then is on the graph of SeeFigure 4.\nFigure4"
	},
	{
		"chunk_id" : "00001550",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\nTRY IT #2 Find the inverse function of\nRestricting the Domain to Find the Inverse of a Polynomial Function\nSo far, we have been able to find the inverse functions ofcubic functionswithout having to restrict their domains.\nHowever, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their\ndomain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would"
	},
	{
		"chunk_id" : "00001551",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then have aninverse function. Since quadratic functions are not one-to-one, we must restrict their domain in order to\nfind their inverses.\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 513\nRestricting the Domain\nIf a function is not one-to-one, it cannot have an inverse. If we restrict the domain of the function so that it becomes\none-to-one, thus creating a new function, this new function will have an inverse.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001552",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse.\n1. Restrict the domain by determining a domain on which the original function is one-to-one.\n2. Replace with\n3. Interchange and\n4. Solve for and rename the function or pair of function\n5. Revise the formula for by ensuring that the outputs of the inverse function correspond to the restricted\ndomain of the original function.\nEXAMPLE3"
	},
	{
		"chunk_id" : "00001553",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "domain of the original function.\nEXAMPLE3\nRestricting the Domain to Find the Inverse of a Polynomial Function\nFind the inverse function of\n \nSolution\nThe original function is not one-to-one, but the function is restricted to a domain of or on\nwhich it is one-to-one. SeeFigure 5.\nFigure5\nTo find the inverse, start by replacing with the simple variable\nThis is not a function as written. We need to examine the restrictions on the domain of the original function to determine"
	},
	{
		"chunk_id" : "00001554",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inverse. Since we reversed the roles of and for the original we looked at the domain: the values could\nassume. When we reversed the roles of and this gave us the values could assume. For this function, so for\nthe inverse, we should have which is what our inverse function gives.\n The domain of the original function was restricted to so the outputs of the inverse need to be the same,\nand we must use the + case:\n514 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001555",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "514 5  Polynomial and Rational Functions\n The domain of the original function was restricted to so the outputs of the inverse need to be the same,\nand we must use the  case:\nAnalysis\nOn the graphs inFigure 6, we see the original function graphed on the same set of axes as its inverse function. Notice\nthat together the graphs show symmetry about the line The coordinate pair is on the graph of and the\ncoordinate pair is on the graph of For any coordinate pair, if is on the graph of then is on the"
	},
	{
		"chunk_id" : "00001556",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph of Finally, observe that the graph of intersects the graph of on the line Points of intersection for\nthe graphs of and will always lie on the line\nFigure6\nEXAMPLE4\nFinding the Inverse of a Quadratic Function When the Restriction Is Not Specified\nRestrict the domain and then find the inverse of\nSolution\nWe can see this is a parabola with vertex at that opens upward. Because the graph will be decreasing on one side"
	},
	{
		"chunk_id" : "00001557",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the vertex and increasing on the other side, we can restrict this function to a domain on which it will be one-to-one by\nlimiting the domain to\nTo find the inverse, we will use the vertex form of the quadratic. We start by replacing with a simple variable, then\nsolve for\nNow we need to determine which case to use. Because we restricted our original function to a domain of the\noutputs of the inverse should be the same, telling us to utilize the + case"
	},
	{
		"chunk_id" : "00001558",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If the quadratic had not been given in vertex form, rewriting it into vertex form would be the first step. This way we may\neasily observe the coordinates of the vertex to help us restrict the domain.\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 515\nAnalysis\nNotice that we arbitrarily decided to restrict the domain on We could just have easily opted to restrict the domain\non in which case Observe the original function graphed on the same set of axes as its"
	},
	{
		"chunk_id" : "00001559",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inverse function inFigure 7. Notice that both graphs show symmetry about the line The coordinate pair\nis on the graph of and the coordinate pair is on the graph of Observe from the graph of both functions\non the same set of axes that\n\nand\n\nFinally, observe that the graph of intersects the graph of along the line\nFigure7\nTRY IT #3 Find the inverse of the function on the domain\nSolving Applications of Radical Functions"
	},
	{
		"chunk_id" : "00001560",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Applications of Radical Functions\nNotice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we\nwant to find theinverse of a radical function, we will need to restrict the domain of the answer because the range of the\noriginal function is limited.\n...\nHOW TO\nGiven a radical function, find the inverse.\n1. Determine the range of the original function.\n2. Replace with then solve for"
	},
	{
		"chunk_id" : "00001561",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Replace with then solve for\n3. If necessary, restrict the domain of the inverse function to the range of the original function.\nEXAMPLE5\nFinding the Inverse of a Radical Function\nRestrict the domain of the function and then find the inverse.\nSolution\nNote that the original function has range Replace with then solve for\n516 5  Polynomial and Rational Functions\nRecall that the domain of this function must be limited to the range of the original function.\nAnalysis"
	},
	{
		"chunk_id" : "00001562",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNotice inFigure 8that the inverse is a reflection of the original function over the line Because the original function\nhas only positive outputs, the inverse function has only positive inputs.\nFigure8\nTRY IT #4 Restrict the domain and then find the inverse of the function\nSolving Applications of Radical Functions\nRadical functions are common in physical models, as we saw in the section opener. We now have enough tools to be\nable to solve the problem posed at the start of the section.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001563",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nSolving an Application with a Cubic Function\nPark rangers construct a mound of gravel in the shape of a cone with the height equal to twice the radius. The volume of\nthe cone in terms of the radius is given by\nFind the inverse of the function that determines the volume of a cone and is a function of the radius\nThen use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use\nSolution"
	},
	{
		"chunk_id" : "00001564",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nStart with the given function for Notice that the meaningful domain for the function is since negative radii\nwould not make sense in this context nor would a radius of 0. Also note the range of the function (hence, the domain of\nthe inverse function) is Solve for in terms of using the method outlined previously. Note that in real-world\napplications, we do not swap the variables when finding inverses. Instead, we change which variable is considered to be\nthe independent variable."
	},
	{
		"chunk_id" : "00001565",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the independent variable.\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 517\nThis is the result stated in the section opener. Now evaluate this for and\nTherefore, the radius is about 3.63 ft.\nDetermining the Domain of a Radical Function Composed with Other Functions\nWhen radical functions are composed with other functions, determining domain can become more complicated.\nEXAMPLE7\nFinding the Domain of a Radical Function Composed with a Rational Function\nFind the domain of the function"
	},
	{
		"chunk_id" : "00001566",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the domain of the function\nSolution\nBecause a square root is only defined when the quantity under the radical is non-negative, we need to determine where\nThe output of a rational function can change signs (change from positive to negative or vice versa) at\nx-intercepts and at vertical asymptotes. For this equation, the graph could change signs at\nTo determine the intervals on which the rational expression is positive, we could test some values in the expression or"
	},
	{
		"chunk_id" : "00001567",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sketch a graph. While both approaches work equally well, for this example we will use a graph as shown inFigure 9.\nFigure9\nThis function has twox-intercepts, both of which exhibit linear behavior near thex-intercepts. There is one vertical\nasymptote, corresponding to a linear factor; this behavior is similar to the basic reciprocal toolkit function, and there is\nno horizontal asymptote because the degree of the numerator is larger than the degree of the denominator. There is a\ny-intercept at"
	},
	{
		"chunk_id" : "00001568",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-intercept at\nFrom they-intercept andx-intercept at we can sketch the left side of the graph. From the behavior at the\nasymptote, we can sketch the right side of the graph.\n518 5  Polynomial and Rational Functions\nFrom the graph, we can now tell on which intervals the outputs will be non-negative, so that we can be sure that the\noriginal function will be defined. has domain or in interval notation, \nFinding Inverses of Rational Functions"
	},
	{
		"chunk_id" : "00001569",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Inverses of Rational Functions\nAs with finding inverses of quadratic functions, it is sometimes desirable to find theinverse of a rational function,\nparticularly of rational functions that are the ratio of linear functions, such as in concentration applications.\nEXAMPLE8\nFinding the Inverse of a Rational Function\nThe function represents the concentration of an acid solution after mL of 40% solution has been added"
	},
	{
		"chunk_id" : "00001570",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for in terms of Then\nuse your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.\nSolution\nWe first want the inverse of the function in order to determine how many mL we need for a given concentration. We will\nsolve for in terms of\nNow evaluate this function at 35%, which is\nWe can conclude that 300 mL of the 40% solution should be added."
	},
	{
		"chunk_id" : "00001571",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Find the inverse of the function\nMEDIA\nAccess these online resources for additional instruction and practice with inverses and radical functions.\nGraphing the Basic Square Root Function(http://openstax.org/l/graphsquareroot)\nFind the Inverse of a Square Root Function(http://openstax.org/l/inversesquare)\nFind the Inverse of a Rational Function(http://openstax.org/l/inverserational)\nFind the Inverse of a Rational Function and an Inverse Function Value(http://openstax.org/l/rationalinverse)"
	},
	{
		"chunk_id" : "00001572",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Inverse Functions(http://openstax.org/l/inversefunction)\nAccess for free at openstax.org\n5.7  Inverses and Radical Functions 519\n5.7 SECTION EXERCISES\nVerbal\n1. Explain why we cannot find 2. Why must we restrict the 3. When finding the inverse of\ninverse functions for all domain of a quadratic a radical function, what\npolynomial functions. function when finding its restriction will we need to\ninverse? make?\n4. The inverse of a quadratic\nfunction will always take\nwhat form?\nAlgebraic"
	},
	{
		"chunk_id" : "00001573",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function will always take\nwhat form?\nAlgebraic\nFor the following exercises, find the inverse of the function on the given domain.\n5.  6.  7. \n8.  9.  10. \n11. \nFor the following exercises, find the inverse of the functions.\n12. 13. 14.\n15.\nFor the following exercises, find the inverse of the functions.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28.  29.  30. \n520 5  Polynomial and Rational Functions\nGraphical"
	},
	{
		"chunk_id" : "00001574",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphical\nFor the following exercises, find the inverse of the function and graph both the function and its inverse.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40.\nFor the following exercises, use a graph to help determine the domain of the functions.\n41. 42. 43.\n44. 45.\nTechnology\nFor the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph\nof the inverse withy-coordinates given.\n46. 47. 48.\n49. 50.\nExtensions"
	},
	{
		"chunk_id" : "00001575",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47. 48.\n49. 50.\nExtensions\nFor the following exercises, find the inverse of the functions with positive real numbers.\n51. 52. 53.\n54. 55.\nReal-World Applications\nFor the following exercises, determine the function described and then use it to answer the question.\n56. An object dropped from a height of 200 meters 57. An object dropped from a height of 600 feet has a\nhas a height, in meters after seconds have height, in feet after seconds have elapsed,\nlapsed, such that Express as such that Express as a"
	},
	{
		"chunk_id" : "00001576",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a function of height, and find the time to reach function of height and find the time to reach a\na height of 50 meters. height of 400 feet.\nAccess for free at openstax.org\n5.8  Modeling Using Variation 521\n58. The volume, of a sphere in terms of its radius, 59. The surface area, of a sphere in terms of its\nis given by Express as a function radius, is given by Express as a\nof and find the radius of a sphere with volume function of and find the radius of a sphere with"
	},
	{
		"chunk_id" : "00001577",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 200 cubic feet. a surface area of 1000 square inches.\n60. A container holds 100 mL of a solution that is 25 61. The period in seconds, of a simple pendulum as\nmL acid. If mL of a solution that is 60% acid is a function of its length in feet, is given by\nadded, the function gives the . Express as a function of and\nconcentration, as a function of the number of\ndetermine the length of a pendulum with period\nmL added, Express as a function of and\nof 2 seconds.\ndetermine the number of mL that need to be"
	},
	{
		"chunk_id" : "00001578",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determine the number of mL that need to be\nadded to have a solution that is 50% acid.\n62. The volume of a cylinder , in terms of radius, 63. The surface area, of a cylinder in terms of its\nand height, is given by If a cylinder radius, and height, is given by\nhas a height of 6 meters, express the radius as a If the height of the cylinder is 4\nfunction of and find the radius of a cylinder with feet, express the radius as a function of and find"
	},
	{
		"chunk_id" : "00001579",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "volume of 300 cubic meters. the radius if the surface area is 200 square feet.\n64. The volume of a right circular cone, in terms of 65. Consider a cone with height of 30 feet. Express the\nits radius, and its height, is given by radius, in terms of the volume, and find the\nExpress in terms of if the height radius of a cone with volume of 1000 cubic feet.\nof the cone is 12 feet and find the radius of a cone\nwith volume of 50 cubic inches.\n5.8 Modeling Using Variation\nLearning Objectives"
	},
	{
		"chunk_id" : "00001580",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.8 Modeling Using Variation\nLearning Objectives\nIn this section, you will:\nSolve direct variation problems.\nSolve inverse variation problems.\nSolve problems involving joint variation.\nA pre-owned car dealer has just offered their best candidate, Nicole, a position in sales. The position offers 16%\ncommission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600,"
	},
	{
		"chunk_id" : "00001581",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "she will earn $736. As she considers the offer, she takes into account the typical price of the dealer's cars, the overall\nmarket, and how many she can reasonably expect to sell. In this section, we will look at relationships, such as this one,\nbetween earnings, sales, and commission rate.\nSolving Direct Variation Problems\nIn the example above, Nicoles earnings can be found by multiplying her sales by her commission. The formula"
	},
	{
		"chunk_id" : "00001582",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tells us her earnings, come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a\ntable, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. SeeTable 1.\n, sales price Interpretation\n$4,600 A sale of a $4,600 vehicle results in $736 earnings.\n$9,200 A sale of a $9,200 vehicle results in $1472 earnings.\n$18,400 A sale of a $18,400 vehicle results in $2944 earnings.\nTable1"
	},
	{
		"chunk_id" : "00001583",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nNotice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of\n522 5  Polynomial and Rational Functions\nthe vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output\nincreases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is\ncalleddirect variation. Each variable in this type of relationshipvaries directlywith the other."
	},
	{
		"chunk_id" : "00001584",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 1represents the data for Nicoles potential earnings. We say that earnings vary directly with the sales price of the\ncar. The formula is used for direct variation. The value is a nonzero constant greater than zero and is called\ntheconstant of variation. In this case, and We saw functions like this one when we discussed power\nfunctions.\nFigure1\nDirect Variation\nIf are related by an equation of the form"
	},
	{
		"chunk_id" : "00001585",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If are related by an equation of the form\nthen we say that the relationship isdirect variationand varies directlywith, or is proportional to, the th power of\nIn direct variation relationships, there is a nonzero constant ratio where is called theconstant of\nvariation, which help defines the relationship between the variables.\n...\nHOW TO\nGiven a description of a direct variation problem, solve for an unknown.\n1. Identify the input, and the output,"
	},
	{
		"chunk_id" : "00001586",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the input, and the output,\n2. Determine the constant of variation. You may need to divide by the specified power of to determine the\nconstant of variation.\n3. Use the constant of variation to write an equation for the relationship.\n4. Substitute known values into the equation to find the unknown.\nEXAMPLE1\nSolving a Direct Variation Problem\nThe quantity varies directly with the cube of If when find when is 6.\nAccess for free at openstax.org\n5.8  Modeling Using Variation 523\nSolution"
	},
	{
		"chunk_id" : "00001587",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.8  Modeling Using Variation 523\nSolution\nThe general formula for direct variation with a cube is The constant can be found by dividing by the cube of\nNow use the constant to write an equation that represents this relationship.\nSubstitute and solve for\nAnalysis\nThe graph of this equation is a simple cubic, as shown inFigure 2.\nFigure2\nQ&A Do the graphs of all direct variation equations look likeExample 1?"
	},
	{
		"chunk_id" : "00001588",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "No. Direct variation equations are power functionsthey may be linear, quadratic, cubic, quartic, radical,\netc. But all of the graphs pass through\nTRY IT #1 The quantity varies directly with the square of If when find when is 4.\nSolving Inverse Variation Problems\nWater temperature in an ocean varies inversely to the waters depth. The formula gives us the temperature\nin degrees Fahrenheit at a depth in feet below Earths surface. Consider the Atlantic Ocean, which covers 22% of Earths"
	},
	{
		"chunk_id" : "00001589",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "surface. At a certain location, at the depth of 500 feet, the temperature may be 28F.\nIf we createTable 2, we observe that, as the depth increases, the water temperature decreases.\ndepth Interpretation\n500 ft At a depth of 500 ft, the water temperature is 28 F.\nTable2\n524 5  Polynomial and Rational Functions\ndepth Interpretation\n1000 ft At a depth of 1,000 ft, the water temperature is 14 F.\n2000 ft At a depth of 2,000 ft, the water temperature is 7 F.\nTable2"
	},
	{
		"chunk_id" : "00001590",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table2\nWe notice in the relationship between these variables that, as one quantity increases, the other decreases. The two\nquantities are said to beinversely proportionaland each termvaries inverselywith the other. Inversely proportional\nrelationships are also calledinverse variations.\nFor our example,Figure 3depicts theinverse variation. We say the water temperature varies inversely with the depth of"
	},
	{
		"chunk_id" : "00001591",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the water because, as the depth increases, the temperature decreases. The formula for inverse variation in this\ncase uses\nFigure3\nInverse Variation\nIf and are related by an equation of the form\nwhere is a nonzero constant, then we say that varies inverselywith the power of Ininversely\nproportionalrelationships, orinverse variations, there is a constant multiple\nEXAMPLE2\nWriting a Formula for an Inversely Proportional Relationship"
	},
	{
		"chunk_id" : "00001592",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist\ndrives.\nSolution\nRecall that multiplying speed by time gives distance. If we let represent the drive time in hours, and represent the\nvelocity (speed or rate) at which the tourist drives, then Because the distance is fixed at 100 miles,\nso Because time is a function of velocity, we can write"
	},
	{
		"chunk_id" : "00001593",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can see that the constant of variation is 100 and, although we can write the relationship using the negative exponent,\nAccess for free at openstax.org\n5.8  Modeling Using Variation 525\nit is more common to see it written as a fraction. We say that time varies inversely with velocity.\n...\nHOW TO\nGiven a description of an indirect variation problem, solve for an unknown.\n1. Identify the input, and the output,"
	},
	{
		"chunk_id" : "00001594",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the input, and the output,\n2. Determine the constant of variation. You may need to multiply by the specified power of to determine the\nconstant of variation.\n3. Use the constant of variation to write an equation for the relationship.\n4. Substitute known values into the equation to find the unknown.\nEXAMPLE3\nSolving an Inverse Variation Problem\nA quantity varies inversely with the cube of If when find when is 6.\nSolution"
	},
	{
		"chunk_id" : "00001595",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe general formula for inverse variation with a cube is The constant can be found by multiplying by the cube\nof\nNow we use the constant to write an equation that represents this relationship.\nSubstitute and solve for\nAnalysis\nThe graph of this equation is a rational function, as shown inFigure 4.\nFigure4\nTRY IT #2 A quantity varies inversely with the square of If when find when is 4.\n526 5  Polynomial and Rational Functions\nSolving Problems Involving Joint Variation"
	},
	{
		"chunk_id" : "00001596",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Problems Involving Joint Variation\nMany situations are more complicated than a basic direct variation or inverse variation model. One variable often\ndepends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables,\nthis is calledjoint variation. For example, the cost of busing students for each school trip varies with the number of\nstudents attending and the distance from the school. The variable cost, varies jointly with the number of students,"
	},
	{
		"chunk_id" : "00001597",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and the distance,\nJoint Variation\nJoint variation occurs when a variable varies directly or inversely with multiple variables.\nFor instance, if varies directly with both and we have If varies directly with and inversely with we\nhave Notice that we only use one constant in a joint variation equation.\nEXAMPLE4\nSolving Problems Involving Joint Variation\nA quantity varies directly with the square of and inversely with the cube root of If when and find\nwhen and\nSolution"
	},
	{
		"chunk_id" : "00001598",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when and\nSolution\nBegin by writing an equation to show the relationship between the variables.\nSubstitute and to find the value of the constant\nNow we can substitute the value of the constant into the equation for the relationship.\nTo find when and we will substitute values for and into our equation.\nTRY IT #3 A quantity varies directly with the square of and inversely with If when and\nfind when and\nMEDIA"
	},
	{
		"chunk_id" : "00001599",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "find when and\nMEDIA\nAccess these online resources for additional instruction and practice with direct and inverse variation.\nDirect Variation(http://openstax.org/l/directvariation)\nInverse Variation(http://openstax.org/l/inversevariatio)\nDirect and Inverse Variation(http://openstax.org/l/directinverse)\nAccess for free at openstax.org\n5.8  Modeling Using Variation 527\n5.8 SECTION EXERCISES\nVerbal\n1. What is true of the 2. If two variables vary 3. Is there a limit to the"
	},
	{
		"chunk_id" : "00001600",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "appearance of graphs that inversely, what will an number of variables that\nreflect a direct variation equation representing their can vary jointly? Explain.\nbetween two variables? relationship look like?\nAlgebraic\nFor the following exercises, write an equation describing the relationship of the given variables.\n4. varies directly as and 5. varies directly as the 6. varies directly as the\nwhen square of and when square root of and when"
	},
	{
		"chunk_id" : "00001601",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when square of and when square root of and when\n7. varies directly as the cube 8. varies directly as the cube 9. varies directly as the\nof and when root of and when fourth power of and when\n10. varies inversely as and 11. varies inversely as the 12. varies inversely as the\nwhen square of and when cube of and when\n13. varies inversely as the 14. varies inversely as the 15. varies inversely as the\nfourth power of and square root of and when cube root of and when\nwhen"
	},
	{
		"chunk_id" : "00001602",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when\n16. varies jointly with and 17. varies jointly as and 18. varies jointly as the\nand when and and when square of and the square\nof and when and\nthen then\n19. varies jointly as and the 20. varies jointly as the 21. varies jointly as and\nsquare root of and when square of the cube of and inversely as . When\nand then and the square root of , and ,\nWhen and then\nthen\n22. varies jointly as the 23. varies jointly as and\nsquare of and the square and inversely as the square"
	},
	{
		"chunk_id" : "00001603",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "root of and inversely as root of and the square of\nthe cube of When When\nand\nand\nthen\nthen\n528 5  Polynomial and Rational Functions\nNumeric\nFor the following exercises, use the given information to find the unknown value.\n24. varies directly as When then 25. varies directly as the square of When\nFind wneh then Find when\n26. varies directly as the cube of When 27. varies directly as the square root of When\nthen Find when then Find when"
	},
	{
		"chunk_id" : "00001604",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then Find when then Find when\n28. varies directly as the cube root of When 29. varies inversely with When then\nthen Find when Find when\n30. varies inversely with the square of When 31. varies inversely with the cube of When\nthen Find when then Find when\n32. varies inversely with the square root of When 33. varies inversely with the cube root of When\nthen Find when then Find when\n34. varies jointly as When and 35. varies jointly as When\nthen Find when and and then Find when\nand"
	},
	{
		"chunk_id" : "00001605",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then Find when and and then Find when\nand\n36. varies jointly as and the square of When 37. varies jointly as the square of and the square\nand then Find when root of When and then\nand Find when and\n38. varies jointly as and and inversely as 39. varies jointly as the square of and the cube of\nWhen and then Find and inversely as the square root of When\nwhen and and\nand then Find\nwhen and\n40. varies jointly as the square of and of and\ninversely as the square root of and of When\nand then Find\nwhen and"
	},
	{
		"chunk_id" : "00001606",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and then Find\nwhen and\nTechnology\nFor the following exercises, use a calculator to graph the equation implied by the given variation.\n41. varies directly with the 42. varies directly as the 43. varies directly as the\nsquare of and when cube of and when square root of and when\n44. varies inversely with 45. varies inversely as the\nand when square of and when\nAccess for free at openstax.org\n5.8  Modeling Using Variation 529\nExtensions"
	},
	{
		"chunk_id" : "00001607",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5.8  Modeling Using Variation 529\nExtensions\nFor the following exercises, use Keplers Law, which states that the square of the time, required for a planet to orbit\nthe Sun varies directly with the cube of the mean distance, that the planet is from the Sun.\n46. Using Earths time of 1 47. Use the result from the 48. Using Earths distance of\nyear and mean distance of previous exercise to 150 million kilometers, find\n93 million miles, find the determine the time the equation relating"
	},
	{
		"chunk_id" : "00001608",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation relating and required for Mars to orbit and\nthe Sun if its mean distance\nis 142 million miles.\n49. Use the result from the 50. Using Earths distance of 1\nprevious exercise to astronomical unit (A.U.),\ndetermine the time determine the time for\nrequired for Venus to orbit Saturn to orbit the Sun if its\nthe Sun if its mean distance mean distance is 9.54 A.U.\nis 108 million kilometers.\nReal-World Applications\nFor the following exercises, use the given information to answer the questions."
	},
	{
		"chunk_id" : "00001609",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "51. The distance that an 52. The velocity of a falling 53. The rate of vibration of a\nobject falls varies directly object varies directly to the string under constant\nwith the square of the time, , of the fall. If after 2 tension varies inversely\ntime, of the fall. If an seconds, the velocity of the with the length of the\nobject falls 16 feet in one object is 64 feet per string. If a string is 24\nsecond, how long for it to second, what is the velocity inches long and vibrates"
	},
	{
		"chunk_id" : "00001610",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fall 144 feet? after 5 seconds? 128 times per second, what\nis the length of a string\nthat vibrates 64 times per\nsecond?\n54. The volume of a gas held at 55. The weight of an object 56. The intensity of light\nconstant temperature above the surface of Earth measured in foot-candles\nvaries indirectly as the varies inversely with the varies inversely with the\npressure of the gas. If the square of the distance square of the distance\nvolume of a gas is 1200 from the center of Earth. If from the light source."
	},
	{
		"chunk_id" : "00001611",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cubic centimeters when a body weighs 50 pounds Suppose the intensity of a\nthe pressure is 200 when it is 3960 miles from light bulb is 0.08 foot-\nmillimeters of mercury, Earths center, what would candles at a distance of 3\nwhat is the volume when it weigh it were 3970 miles meters. Find the intensity\nthe pressure is 300 from Earths center? level at 8 meters.\nmillimeters of mercury?\n530 5  Polynomial and Rational Functions"
	},
	{
		"chunk_id" : "00001612",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "530 5  Polynomial and Rational Functions\n57. The current in a circuit 58. The force exerted by the 59. The horsepower (hp) that a\nvaries inversely with its wind on a plane surface shaft can safely transmit\nresistance measured in varies jointly with the varies jointly with its speed\nohms. When the current in square of the velocity of (in revolutions per minute\na circuit is 40 amperes, the the wind and with the area (rpm) and the cube of the"
	},
	{
		"chunk_id" : "00001613",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "resistance is 10 ohms. Find of the plane surface. If the diameter. If the shaft of a\nthe current if the resistance area of the surface is 40 certain material 3 inches in\nis 12 ohms. square feet surface and the diameter can transmit 45\nwind velocity is 20 miles hp at 100 rpm, what must\nper hour, the resulting the diameter be in order to\nforce is 15 pounds. Find the transmit 60 hp at 150 rpm?\nforce on a surface of 65\nsquare feet with a velocity\nof 30 miles per hour.\n60. The kinetic energy of a"
	},
	{
		"chunk_id" : "00001614",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 30 miles per hour.\n60. The kinetic energy of a\nmoving object varies jointly\nwith its mass and the\nsquare of its velocity If\nan object weighing 40\nkilograms with a velocity of\n15 meters per second has a\nkinetic energy of 1000\njoules, find the kinetic\nenergy if the velocity is\nincreased to 20 meters per\nsecond.\nAccess for free at openstax.org\n5  Chapter Review 531\nChapter Review\nKey Terms\narrow notation a way to represent symbolically the local and end behavior of a function by using arrows to indicate"
	},
	{
		"chunk_id" : "00001615",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that an input or output approaches a value\naxis of symmetry a vertical line drawn through the vertex of a parabola, that opens up or down, around which the\nparabola is symmetric; it is defined by\ncoefficient a nonzero real number multiplied by a variable raised to an exponent\nconstant of variation the non-zero value that helps define the relationship between variables in direct or inverse\nvariation"
	},
	{
		"chunk_id" : "00001616",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variation\ncontinuous function a function whose graph can be drawn without lifting the pen from the paper because there are\nno breaks in the graph\ndegree the highest power of the variable that occurs in a polynomial\nDescartes Rule of Signs a rule that determines the maximum possible numbers of positive and negative real zeros\nbased on the number of sign changes of and\ndirect variation the relationship between two variables that are a constant multiple of each other; as one quantity"
	},
	{
		"chunk_id" : "00001617",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increases, so does the other\nDivision Algorithm given a polynomial dividend and a non-zero polynomial divisor where the degree of\nis less than or equal to the degree of , there exist unique polynomials and such that\nwhere is the quotient and is the remainder. The remainder is either equal to zero or\nhas degree strictly less than\nend behavior the behavior of the graph of a function as the input decreases without bound and increases without\nbound"
	},
	{
		"chunk_id" : "00001618",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bound\nFactor Theorem is a zero of polynomial function if and only if is a factor of\nFundamental Theorem of Algebra a polynomial function with degree greater than 0 has at least one complex zero\ngeneral form of a quadratic function the function that describes a parabola, written in the form ,\nwhere and are real numbers and\nglobal maximum highest turning point on a graph; where for all\nglobal minimum lowest turning point on a graph; where for all"
	},
	{
		"chunk_id" : "00001619",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal asymptote a horizontal line where the graph approaches the line as the inputs increase or decrease\nwithout bound.\nIntermediate Value Theorem for two numbers and in the domain of if and then the\nfunction takes on every value between and ; specifically, when a polynomial function changes from a\nnegative value to a positive value, the function must cross the axis\ninverse variation the relationship between two variables in which the product of the variables is a constant"
	},
	{
		"chunk_id" : "00001620",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inversely proportional a relationship where one quantity is a constant divided by the other quantity; as one quantity\nincreases, the other decreases\ninvertible function any function that has an inverse function\njoint variation a relationship where a variable varies directly or inversely with multiple variables\nleading coefficient the coefficient of the leading term\nleading term the term containing the highest power of the variable"
	},
	{
		"chunk_id" : "00001621",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Linear Factorization Theorem allowing for multiplicities, a polynomial function will have the same number of factors\nas its degree, and each factor will be in the form , where is a complex number\nmultiplicity the number of times a given factor appears in the factored form of the equation of a polynomial; if a\npolynomial contains a factor of the form , is a zero of multiplicity\npolynomial function a function that consists of either zero or the sum of a finite number of non-zeroterms, each of"
	},
	{
		"chunk_id" : "00001622",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "which is a product of a number, called thecoefficientof the term, and a variable raised to a non-negative integer\npower.\npower function a function that can be represented in the form where is a constant, the base is a\nvariable, and the exponent, , is a constant\nrational function a function that can be written as the ratio of two polynomials\nRational Zero Theorem the possible rational zeros of a polynomial function have the form where is a factor of the"
	},
	{
		"chunk_id" : "00001623",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "constant term and is a factor of the leading coefficient.\nRemainder Theorem if a polynomial is divided by , then the remainder is equal to the value\nremovable discontinuity a single point at which a function is undefined that, if filled in, would make the function\ncontinuous; it appears as a hole on the graph of a function\nroots in a given function, the values of at which , also called zeros\n532 5  Chapter Review\nsmooth curve a graph with no sharp corners"
	},
	{
		"chunk_id" : "00001624",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "smooth curve a graph with no sharp corners\nstandard form of a quadratic function the function that describes a parabola, written in the form\n, where is the vertex\nsynthetic division a shortcut method that can be used to divide a polynomial by a binomial of the form\nterm of a polynomial function any of a polynomial function in the form\nturning point the location at which the graph of a function changes direction\nvaries directly a relationship where one quantity is a constant multiplied by the other quantity"
	},
	{
		"chunk_id" : "00001625",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "varies inversely a relationship where one quantity is a constant divided by the other quantity\nvertex the point at which a parabola changes direction, corresponding to the minimum or maximum value of the\nquadratic function\nvertex form of a quadratic function another name for the standard form of a quadratic function\nvertical asymptote a vertical line where the graph tends toward positive or negative infinity as the inputs\napproach\nzeros in a given function, the values of at which , also called roots"
	},
	{
		"chunk_id" : "00001626",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Key Equations\ngeneral form of a quadratic function\nstandard form of a quadratic function\ngeneral form of a polynomial function\nDivision Algorithm\nRational Function\nDirect variation is a nonzero constant.\nInverse variation is a nonzero constant.\nKey Concepts\n5.1Quadratic Functions\n A polynomial function of degree two is called a quadratic function.\n The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down."
	},
	{
		"chunk_id" : "00001627",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The axis of symmetry is the vertical line passing through the vertex. The zeros, or intercepts, are the points at\nwhich the parabola crosses the axis. The intercept is the point at which the parabola crosses the axis. See\nExample 1,Example 7, andExample 8.\n Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex\nof a parabola. Either form can be written from a graph. SeeExample 2."
	},
	{
		"chunk_id" : "00001628",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The vertex can be found from an equation representing a quadratic function. SeeExample 3.\n The domain of a quadratic function is all real numbers. The range varies with the function. SeeExample 4.\n A quadratic functions minimum or maximum value is given by the value of the vertex.\n The minimum or maximum value of a quadratic function can be used to determine the range of the function and to\nsolve many kinds of real-world problems, including problems involving area and revenue. SeeExample 5and"
	},
	{
		"chunk_id" : "00001629",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 6.\n The vertex and the intercepts can be identified and interpreted to solve real-world problems. SeeExample 9.\nAccess for free at openstax.org\n5  Chapter Review 533\n5.2Power Functions and Polynomial Functions\n A power function is a variable base raised to a number power. SeeExample 1.\n The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end\nbehavior.\n The end behavior depends on whether the power is even or odd. SeeExample 2andExample 3."
	},
	{
		"chunk_id" : "00001630",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A polynomial function is the sum of terms, each of which consists of a transformed power function with positive\nwhole number power. SeeExample 4.\n The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term\ncontaining the highest power of the variable is called the leading term. The coefficient of the leading term is called\nthe leading coefficient. SeeExample 5."
	},
	{
		"chunk_id" : "00001631",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the leading coefficient. SeeExample 5.\n The end behavior of a polynomial function is the same as the end behavior of the power function represented by\nthe leading term of the function. SeeExample 6andExample 7.\n A polynomial of degree will have at most x-intercepts and at most turning points. SeeExample 8,Example\n9,Example 10,Example 11, andExample 12.\n5.3Graphs of Polynomial Functions\n Polynomial functions of degree 2 or more are smooth, continuous functions. SeeExample 1."
	},
	{
		"chunk_id" : "00001632",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.\nSeeExample 2,Example 3,andExample 4.\n Another way to find the intercepts of a polynomial function is to graph the function and identify the points at\nwhich the graph crosses the axis. SeeExample 5.\n The multiplicity of a zero determines how the graph behaves at the intercepts. SeeExample 6.\n The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity."
	},
	{
		"chunk_id" : "00001633",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.\n The end behavior of a polynomial function depends on the leading term.\n The graph of a polynomial function changes direction at its turning points.\n A polynomial function of degree has at most turning points. SeeExample 7.\n To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that\nthe final graph has at most turning points. SeeExample 8andExample 10."
	},
	{
		"chunk_id" : "00001634",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Graphing a polynomial function helps to estimate local and global extremas. SeeExample 11.\n The Intermediate Value Theorem tells us that if have opposite signs, then there exists at least one\nvalue between and for which SeeExample 9.\n5.4Dividing Polynomials\n Polynomial long division can be used to divide a polynomial by any polynomial with equal or lower degree. See\nExample 1andExample 2.\n The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the"
	},
	{
		"chunk_id" : "00001635",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quotient added to the remainder.\n Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form SeeExample\n3,Example 4,andExample 5.\n Polynomial division can be used to solve application problems, including area and volume. SeeExample 6.\n5.5Zeros of Polynomial Functions\n To find determine the remainder of the polynomial when it is divided by This is known as the\nRemainder Theorem. SeeExample 1."
	},
	{
		"chunk_id" : "00001636",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Remainder Theorem. SeeExample 1.\n According to the Factor Theorem, is a zero of if and only if is a factor of SeeExample 2.\n According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be\nequal to a factor of the constant term divided by a factor of the leading coefficient. SeeExample 3andExample 4.\n When the leading coefficient is 1, the possible rational zeros are the factors of the constant term."
	},
	{
		"chunk_id" : "00001637",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Synthetic division can be used to find the zeros of a polynomial function. SeeExample 5.\n According to the Fundamental Theorem, every polynomial function has at least one complex zero. SeeExample 6.\n Every polynomial function with degree greater than 0 has at least one complex zero.\n Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will\nbe in the form where is a complex number. SeeExample 7."
	},
	{
		"chunk_id" : "00001638",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The number of positive real zeros of a polynomial function is either the number of sign changes of the function or\nless than the number of sign changes by an even integer.\n The number of negative real zeros of a polynomial function is either the number of sign changes of or less\n534 5  Chapter Review\nthan the number of sign changes by an even integer. SeeExample 8.\n Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division.\nSeeExample 9."
	},
	{
		"chunk_id" : "00001639",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 9.\n5.6Rational Functions\n We can use arrow notation to describe local behavior and end behavior of the toolkit functions and\nSeeExample 1.\n A function that levels off at a horizontal value has a horizontal asymptote. A function can have more than one\nvertical asymptote. SeeExample 2.\n Application problems involving rates and concentrations often involve rational functions. SeeExample 3."
	},
	{
		"chunk_id" : "00001640",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The domain of a rational function includes all real numbers except those that cause the denominator to equal zero.\nSeeExample 4.\n The vertical asymptotes of a rational function will occur where the denominator of the function is equal to zero and\nthe numerator is not zero. SeeExample 5.\n A removable discontinuity might occur in the graph of a rational function if an input causes both numerator and\ndenominator to be zero. SeeExample 6."
	},
	{
		"chunk_id" : "00001641",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "denominator to be zero. SeeExample 6.\n A rational functions end behavior will mirror that of the ratio of the leading terms of the numerator and\ndenominator functions. SeeExample 7,Example 8,Example 9, andExample 10.\n Graph rational functions by finding the intercepts, behavior at the intercepts and asymptotes, and end behavior. See\nExample 11.\n If a rational function hasx-intercepts at vertical asymptotes at and no\nthen the function can be written in the form\nSeeExample 12."
	},
	{
		"chunk_id" : "00001642",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 12.\n5.7Inverses and Radical Functions\n The inverse of a quadratic function is a square root function.\n If is the inverse of a function then is the inverse of the function SeeExample 1.\n While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See\nExample 2.\n To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See\nExample 3andExample 4."
	},
	{
		"chunk_id" : "00001643",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 3andExample 4.\n When finding the inverse of a radical function, we need a restriction on the domain of the answer. SeeExample 5\nandExample 7.\n Inverse and radical and functions can be used to solve application problems. SeeExample 6andExample 8.\n5.8Modeling Using Variation\n A relationship where one quantity is a constant multiplied by another quantity is called direct variation. SeeExample\n1.\n Two variables that are directly proportional to one another will have a constant ratio."
	},
	{
		"chunk_id" : "00001644",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A relationship where one quantity is a constant divided by another quantity is called inverse variation. SeeExample\n2.\n Two variables that are inversely proportional to one another will have a constant multiple. SeeExample 3.\n In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint\nvariation. SeeExample 4.\nAccess for free at openstax.org\n5  Exercises 535\nExercises\nReview Exercises\nQuadratic Functions"
	},
	{
		"chunk_id" : "00001645",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Exercises\nReview Exercises\nQuadratic Functions\nFor the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts.\nFinally, graph the function.\n1. 2.\nFor the following exercises, find the equation of the quadratic function using the given information.\n3. The vertex is and a 4. The vertex is and a\npoint on the graph is point on the graph is\nFor the following exercises, complete the task."
	},
	{
		"chunk_id" : "00001646",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, complete the task.\n5. A rectangular plot of land is to be enclosed by 6. An object projected from the ground at a 45 degree\nfencing. One side is along a river and so needs no angle with initial velocity of 120 feet per second has\nfence. If the total fencing available is 600 meters, height, in terms of horizontal distance traveled,\nfind the dimensions of the plot to have maximum given by Find the\narea.\nmaximum height the object attains.\nPower Functions and Polynomial Functions"
	},
	{
		"chunk_id" : "00001647",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Power Functions and Polynomial Functions\nFor the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading\ncoefficient.\n7. 8. 9.\nFor the following exercises, determine end behavior of the polynomial function.\n10. 11. 12.\nGraphs of Polynomial Functions\nFor the following exercises, find all zeros of the polynomial function, noting multiplicities.\n13. 14. 15.\n536 5  Exercises"
	},
	{
		"chunk_id" : "00001648",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13. 14. 15.\n536 5  Exercises\nFor the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.\n16. 17. 18. Use the Intermediate Value\nTheorem to show that at\nleast one zero lies between\n2 and 3 for the function\nDividing Polynomials\nFor the following exercises, use long division to find the quotient and remainder.\n19. 20.\nFor the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored\nform."
	},
	{
		"chunk_id" : "00001649",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form.\n21. 22. 23.\n24.\nZeros of Polynomial Functions\nFor the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.\n25. 26. 27.\n28.\nFor the following exercises, use Descartes Rule of Signs to find the possible number of positive and negative solutions.\n29. 30.\nRational Functions\nFor the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a\ngraph of the function.\n31. 32. 33.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001650",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "31. 32. 33.\nAccess for free at openstax.org\n5  Exercises 537\n34.\nFor the following exercises, find the slant asymptote.\n35. 36.\nInverses and Radical Functions\nFor the following exercises, find the inverse of the function with the domain given.\n37. 38. 39.\n40. 41. 42.\nModeling Using Variation\nFor the following exercises, find the unknown value.\n43. varies directly as the 44. varies inversely as the 45. varies jointly as the cube\nsquare of If when square root of If when of and as If when\nfind if find if and"
	},
	{
		"chunk_id" : "00001651",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "find if find if and\nfind if and\n46. varies jointly as and the\nsquare of and inversely\nas the cube of If when\nand\nfind if\nand\nFor the following exercises, solve the application problem.\n47. The weight of an object above the surface of the 48. The volume of an ideal gas varies directly with\nearth varies inversely with the distance from the the temperature and inversely with the\ncenter of the earth. If a person weighs 150 pounds pressure P. A cylinder contains oxygen at a"
	},
	{
		"chunk_id" : "00001652",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when he is on the surface of the earth (3,960 miles temperature of 310 degrees K and a pressure of\nfrom center), find the weight of the person if he is 18 atmospheres in a volume of 120 liters. Find the\n20 miles above the surface. pressure if the volume is decreased to 100 liters\nand the temperature is increased to 320 degrees\nK.\nPractice Test\nGive the degree and leading coefficient of the following polynomial function.\n1.\nDetermine the end behavior of the polynomial function.\n2. 3.\n538 5  Exercises"
	},
	{
		"chunk_id" : "00001653",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. 3.\n538 5  Exercises\nWrite the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.\n4.\nGiven information about the graph of a quadratic function, find its equation.\n5. Vertex and point on\ngraph\nSolve the following application problem.\n6. A rectangular field is to be\nenclosed by fencing. In\naddition to the enclosing\nfence, another fence is to\ndivide the field into two\nparts, running parallel to\ntwo sides. If 1,200 feet of\nfencing is available, find the"
	},
	{
		"chunk_id" : "00001654",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fencing is available, find the\nmaximum area that can be\nenclosed.\nFind all zeros of the following polynomial functions, noting multiplicities.\n7. 8.\nBased on the graph, determine the zeros of the function and multiplicities.\n9.\nUse long division to find the quotient.\n10.\nUse synthetic division to find the quotient. If the divisor is a factor, write the factored form.\n11. 12.\nUse the Rational Zero Theorem to help you find the zeros of the polynomial functions.\n13. 14.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001655",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13. 14.\nAccess for free at openstax.org\n5  Exercises 539\n15. 16.\nGiven the following information about a polynomial function, find the function.\n17. It has a double zero at 18. It has a zero of multiplicity\nand zeros at 3 at and another\nand . Itsy-intercept zero at . It contains\nis the point\nUse Descartes Rule of Signs to determine the possible number of positive and negative solutions.\n19."
	},
	{
		"chunk_id" : "00001656",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "19.\nFor the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.\n20. 21.\nFind the slant asymptote of the rational function.\n22.\nFind the inverse of the function.\n23. 24. 25.\nFind the unknown value.\n26. varies inversely as the 27. varies jointly with and\nsquare of and when the cube root of If when\nFind if and\nfind if and\nSolve the following application problem.\n28. The distance a body falls\nvaries directly as the\nsquare of the time it falls. If"
	},
	{
		"chunk_id" : "00001657",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "square of the time it falls. If\nan object falls 64 feet in 2\nseconds, how long will it\ntake to fall 256 feet?\n540 5  Exercises\nAccess for free at openstax.org\n6  Introduction 541\n6 EXPONENTIAL AND LOGARITHMIC FUNCTIONS\nElectron micrograph ofE.Colibacteria (credit: Mattosaurus, Wikimedia Commons)\nChapter Outline\n6.1Exponential Functions\n6.2Graphs of Exponential Functions\n6.3Logarithmic Functions\n6.4Graphs of Logarithmic Functions\n6.5Logarithmic Properties\n6.6Exponential and Logarithmic Equations"
	},
	{
		"chunk_id" : "00001658",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.6Exponential and Logarithmic Equations\n6.7Exponential and Logarithmic Models\n6.8Fitting Exponential Models to Data\nIntroduction to Exponential and Logarithmic Functions\nFocus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see\nhundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your"
	},
	{
		"chunk_id" : "00001659",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your\nown cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and\neven essential to the body.\nBacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two.\nWhen conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time"
	},
	{
		"chunk_id" : "00001660",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.1\nFor simplicitys sake, suppose we begin with a culture of one bacterial cell that can divide every hour.Table 1shows the\nnumber of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one\nthousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have\nover 16 million!"
	},
	{
		"chunk_id" : "00001661",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "over 16 million!\nHour 0 1 2 3 4 5 6 7 8 9 10\nBacteria 1 2 4 8 16 32 64 128 256 512 1024\nTable1\n1 Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.\n542 6  Exponential and Logarithmic Functions\nIn this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth\npatterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to"
	},
	{
		"chunk_id" : "00001662",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and\ninterpreting data.\n6.1 Exponential Functions\nLearning Objectives\nIn this section, you will:\nEvaluate exponential functions.\nFind the equation of an exponential function.\nUse compound interest formulas.\nEvaluate exponential functions with base .\nIndia is the second most populous country in the world with a population of about billion people in 2021. The"
	},
	{
		"chunk_id" : "00001663",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "population is growing at a rate of about each year2. If this rate continues, the population of India will exceed\nChinas population by the year When populations grow rapidly, we often say that the growth is exponential,\nmeaning that something is growing very rapidly. To a mathematician, however, the termexponential growthhas a very\nspecific meaning. In this section, we will take a look atexponential functions, which model this kind of rapid growth.\nIdentifying Exponential Functions"
	},
	{
		"chunk_id" : "00001664",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Exponential Functions\nWhen exploring linear growth, we observed a constant rate of changea constant number by which the output\nincreased for each unit increase in input. For example, in the equation the slope tells us the output\nincreases by 3 each time the input increases by 1. The scenario in the India population example is different because we\nhave apercentchange per unit time (rather than a constant change) in the number of people.\nDefining an Exponential Function"
	},
	{
		"chunk_id" : "00001665",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Defining an Exponential Function\nA study found that the percent of the population who are vegans in the United States doubled from 2009 to 2011. In\n2011, 2.5% of the population was vegan, adhering to a diet that does not include any animal productsno meat, poultry,\nfish, dairy, or eggs. If this rate continues, vegans will make up 10% of the U.S. population in 2015, 40% in 2019, and 80%\nin 2021.\nWhat exactly does it mean togrow exponentially? What does the worddoublehave in common withpercent increase?"
	},
	{
		"chunk_id" : "00001666",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "People toss these words around errantly. Are these words used correctly? The words certainly appear frequently in the\nmedia.\n Percent changerefers to achangebased on apercentof the original amount.\n Exponential growthrefers to anincreasebased on a constant multiplicative rate of change over equal increments\nof time, that is, apercentincrease of the original amount over time.\n Exponential decayrefers to adecreasebased on a constant multiplicative rate of change over equal increments of"
	},
	{
		"chunk_id" : "00001667",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "time, that is, apercentdecrease of the original amount over time.\nFor us to gain a clear understanding ofexponential growth, let us contrast exponential growth withlinear growth. We\nwill construct two functions. The first function is exponential. We will start with an input of 0, and increase each input by\n1. We will double the corresponding consecutive outputs. The second function is linear. We will start with an input of 0,"
	},
	{
		"chunk_id" : "00001668",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and increase each input by 1. We will add 2 to the corresponding consecutive outputs. SeeTable 1.\n0 1 0\n1 2 2\n2 4 4\n3 8 6\nTable1\n2 http://www.worldometers.info/world-population/. Accessed February 24, 2014.\nAccess for free at openstax.org\n6.1  Exponential Functions 543\n4 16 8\n5 32 10\n6 64 12\nTable1\nFromTable 1we can infer that for these two functions, exponential growth dwarfs linear growth.\n Exponential growthrefers to the original value from the range increases by thesame percentageover equal"
	},
	{
		"chunk_id" : "00001669",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "increments found in the domain.\n Linear growthrefers to the original value from the range increases by thesame amountover equal increments\nfound in the domain.\nApparently, the difference between the same percentage and the same amount is quite significant. For exponential\ngrowth, over equal increments, the constant multiplicative rate of change resulted in doubling the output whenever the"
	},
	{
		"chunk_id" : "00001670",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input increased by one. For linear growth, the constant additive rate of change over equal increments resulted in adding\n2 to the output whenever the input was increased by one.\nThe general form of theexponential functionis where is any nonzero number, is a positive real number\nnot equal to 1.\n If the function grows at a rate proportional to its size.\n If the function decays at a rate proportional to its size."
	},
	{
		"chunk_id" : "00001671",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lets look at the function from our example. We will create a table (Table 2) to determine the corresponding\noutputs over an interval in the domain from to\nTable2\nLet us examine the graph of by plotting the ordered pairs we observe on the table inFigure 1, and then make a few\nobservations.\nFigure1\nLets define the behavior of the graph of the exponential function and highlight some its key characteristics.\n the domain is  \n the range is \n as  \n544 6  Exponential and Logarithmic Functions\n as "
	},
	{
		"chunk_id" : "00001672",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " as \n is always increasing,\n the graph of will never touch thex-axis because base two raised to any exponent never has the result of zero.\n is the horizontal asymptote.\n they-intercept is 1.\nExponential Function\nFor any real number an exponential function is a function with the form\nwhere\n is a non-zero real number called the initial value and\n is any positive real number such that\n The domain of is all real numbers.\n The range of is all positive real numbers if"
	},
	{
		"chunk_id" : "00001673",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The range of is all positive real numbers if\n The range of is all negative real numbers if\n They-intercept is and the horizontal asymptote is\nEXAMPLE1\nIdentifying Exponential Functions\nWhich of the following equations arenotexponential functions?\n\n\n\n\nSolution\nBy definition, an exponential function has a constant as a base and an independent variable as an exponent. Thus,\ndoes not represent an exponential function because the base is an independent variable. In fact, is\na power function."
	},
	{
		"chunk_id" : "00001674",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a power function.\nRecall that the basebof an exponential function is always a positive constant, and Thus, does not\nrepresent an exponential function because the base, is less than\nTRY IT #1 Which of the following equations represent exponential functions?\n\n\n\n\nEvaluating Exponential Functions\nRecall that the base of an exponential function must be a positive real number other than Why do we limit the base"
	},
	{
		"chunk_id" : "00001675",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to positive values? To ensure that the outputs will be real numbers. Observe what happens if the base is not positive:\n Let and Then which is not a real number.\nWhy do we limit the base to positive values other than Because base results in the constant function. Observe what\nhappens if the base is\n Let Then for any value of\nTo evaluate an exponential function with the form we simply substitute with the given value, and calculate\nAccess for free at openstax.org\n6.1  Exponential Functions 545"
	},
	{
		"chunk_id" : "00001676",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.1  Exponential Functions 545\nthe resulting power. For example:\nLet What is\nTo evaluate an exponential function with a form other than the basic form, it is important to follow the order of\noperations. For example:\nLet What is\nNote that if the order of operations were not followed, the result would be incorrect:\nEXAMPLE2\nEvaluating Exponential Functions\nLet Evaluate without using a calculator.\nSolution\nFollow the order of operations. Be sure to pay attention to the parentheses."
	},
	{
		"chunk_id" : "00001677",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Let Evaluate using a calculator. Round to four decimal places.\nDefining Exponential Growth\nBecause the output of exponential functions increases very rapidly, the term exponential growth is often used in\neveryday language to describe anything that grows or increases rapidly. However, exponential growth can be defined\nmore precisely in a mathematical sense. If the growth rate is proportional to the amount present, the function models\nexponential growth.\nExponential Growth"
	},
	{
		"chunk_id" : "00001678",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential growth.\nExponential Growth\nA function that modelsexponential growthgrows by a rate proportional to the amount present. For any real\nnumber and any positive real numbers and such that an exponential growth function has the form\nwhere\n is the initial or starting value of the function.\n is the growth factor or growth multiplier per unit .\nIn more general terms, we have anexponential function, in which a constant base is raised to a variable exponent. To"
	},
	{
		"chunk_id" : "00001679",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "differentiate between linear and exponential functions, lets consider two companies, A and B. Company A has 100\nstores and expands by opening 50 new stores a year, so its growth can be represented by the function\n546 6  Exponential and Logarithmic Functions\nCompany B has 100 stores and expands by increasing the number of stores by 50% each year, so its\ngrowth can be represented by the function\nA few years of growth for these companies are illustrated inTable 3.\nYear, Stores, Company A Stores, Company B"
	},
	{
		"chunk_id" : "00001680",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Year, Stores, Company A Stores, Company B\nTable3\nThe graphs comparing the number of stores for each company over a five-year period are shown inFigure 2.We can see\nthat, with exponential growth, the number of stores increases much more rapidly than with linear growth.\nFigure2 The graph shows the numbers of stores Companies A and B opened over a five-year period.\nAccess for free at openstax.org\n6.1  Exponential Functions 547"
	},
	{
		"chunk_id" : "00001681",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.1  Exponential Functions 547\nNotice that the domain for both functions is  and the range for both functions is  After year 1, Company B\nalways has more stores than Company A.\nNow we will turn our attention to the function representing the number of stores for Company B,\nIn this exponential function, 100 represents the initial number of stores, 0.50 represents the growth rate, and\nrepresents the growth factor. Generalizing further, we can write this function as where"
	},
	{
		"chunk_id" : "00001682",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "100 is the initial value, is called thebase, and is called theexponent.\nEXAMPLE3\nEvaluating a Real-World Exponential Model\nAt the beginning of this section, we learned that the population of India was about billion in the year 2013, with an\nannual growth rate of about This situation is represented by the growth function where is\nthe number of years since To the nearest thousandth, what will the population of India be in\nSolution"
	},
	{
		"chunk_id" : "00001683",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nTo estimate the population in 2031, we evaluate the models for because 2031 is years after 2013. Rounding to\nthe nearest thousandth,\nThere will be about 1.549 billion people in India in the year 2031.\nTRY IT #3 The population of China was about 1.39 billion in the year 2013, with an annual growth rate of\nabout This situation is represented by the growth function where is\nthe number of years since To the nearest thousandth, what will the population of China be"
	},
	{
		"chunk_id" : "00001684",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for the year 2031? How does this compare to the population prediction we made for India in\nExample 3?\nFinding Equations of Exponential Functions\nIn the previous examples, we were given an exponential function, which we then evaluated for a given input. Sometimes\nwe are given information about an exponential function without knowing the function explicitly. We must use the\ninformation to first write the form of the function, then determine the constants and and evaluate the function.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001685",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven two data points, write an exponential model.\n1. If one of the data points has the form then is the initial value. Using substitute the second point into\nthe equation and solve for\n2. If neither of the data points have the form substitute both points into two equations with the form\nSolve the resulting system of two equations in two unknowns to find and\n3. Using the and found in the steps above, write the exponential function in the form\nEXAMPLE4"
	},
	{
		"chunk_id" : "00001686",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nWriting an Exponential Model When the Initial Value Is Known\nIn 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population\nwas growing exponentially. Write an exponential function representing the population of deer over time\nSolution\nWe let our independent variable be the number of years after 2006. Thus, the information given in the problem can be"
	},
	{
		"chunk_id" : "00001687",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "written as input-output pairs: (0, 80) and (6, 180). Notice that by choosing our input variable to be measured as years\nafter 2006, we have given ourselves the initial value for the function, We can now substitute the second point\ninto the equation to find\n548 6  Exponential and Logarithmic Functions\nNOTE:Unless otherwise stated, do not round any intermediate calculations. Then round the final answer to four places\nfor the remainder of this section."
	},
	{
		"chunk_id" : "00001688",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for the remainder of this section.\nThe exponential model for the population of deer is (Note that this exponential function models\nshort-term growth. As the inputs gets large, the output will get increasingly larger, so much so that the model may not\nbe useful in the long term.)\nWe can graph our model to observe the population growth of deer in the refuge over time. Notice that the graph in\nFigure 3passes through the initial points given in the problem, and We can also see that the domain for"
	},
	{
		"chunk_id" : "00001689",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the function is  and the range for the function is \nFigure3 Graph showing the population of deer over time, years after 2006\nTRY IT #4 A wolf population is growing exponentially. In 2011, wolves were counted. By the\npopulation had reached 236 wolves. What two points can be used to derive an exponential\nequation modeling this situation? Write the equation representing the population of wolves\nover time\nEXAMPLE5\nWriting an Exponential Model When the Initial Value is Not Known"
	},
	{
		"chunk_id" : "00001690",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find an exponential function that passes through the points and\nSolution\nBecause we dont have the initial value, we substitute both points into an equation of the form and then\nsolve the system for and\n Substituting gives\n Substituting gives\nUse the first equation to solve for in terms of\nAccess for free at openstax.org\n6.1  Exponential Functions 549\nSubstitute in the second equation, and solve for\nUse the value of in the first equation to solve for the value of\nThus, the equation is"
	},
	{
		"chunk_id" : "00001691",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thus, the equation is\nWe can graph our model to check our work. Notice that the graph inFigure 4passes through the initial points given in\nthe problem, and The graph is an example of anexponential decayfunction.\nFigure4 The graph of models exponential decay.\nTRY IT #5 Given the two points and find the equation of the exponential function that passes\nthrough these two points.\nQ&A Do two points always determine a unique exponential function?"
	},
	{
		"chunk_id" : "00001692",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Yes, provided the two points are either both above the x-axis or both below the x-axis and have different\nx-coordinates. But keep in mind that we also need to know that the graph is, in fact, an exponential\nfunction. Not every graph that looks exponential really is exponential. We need to know the graph is\nbased on a model that shows the same percent growth with each unit increase in which in many real\nworld cases involves time.\n550 6  Exponential and Logarithmic Functions\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001693",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the graph of an exponential function, write its equation.\n1. First, identify two points on the graph. Choose they-intercept as one of the two points whenever possible. Try to\nchoose points that are as far apart as possible to reduce round-off error.\n2. If one of the data points is they-intercept , then is the initial value. Using substitute the second point\ninto the equation and solve for\n3. If neither of the data points have the form substitute both points into two equations with the form"
	},
	{
		"chunk_id" : "00001694",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the resulting system of two equations in two unknowns to find and\n4. Write the exponential function,\nEXAMPLE6\nWriting an Exponential Function Given Its Graph\nFind an equation for the exponential function graphed inFigure 5.\nFigure5\nSolution\nWe can choose they-intercept of the graph, as our first point. This gives us the initial value, Next, choose a\npoint on the curve some distance away from that has integer coordinates. One such point is"
	},
	{
		"chunk_id" : "00001695",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because we restrict ourselves to positive values of we will use Substitute and into the standard form to yield\nthe equation\nTRY IT #6 Find an equation for the exponential function graphed inFigure 6.\nAccess for free at openstax.org\n6.1  Exponential Functions 551\nFigure6\n...\nHOW TO\nGiven two points on the curve of an exponential function, use a graphing calculator to find the equation.\n1. Press[STAT].\n2. Clear any existing entries in columnsL1orL2.\n3. InL1, enter thex-coordinates given."
	},
	{
		"chunk_id" : "00001696",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. InL1, enter thex-coordinates given.\n4. InL2, enter the correspondingy-coordinates.\n5. Press[STAT]again. Cursor right toCALC, scroll down toExpReg (Exponential Regression), and press[ENTER].\n6. The screen displays the values ofaandbin the exponential equation .\nEXAMPLE7\nUsing a Graphing Calculator to Find an Exponential Function\nUse a graphing calculator to find the exponential equation that includes the points and\nSolution"
	},
	{
		"chunk_id" : "00001697",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nFollow the guidelines above. First press[STAT],[EDIT],[1: Edit],and clear the listsL1andL2. Next, in theL1column,\nenter thex-coordinates, 2 and 5. Do the same in theL2column for they-coordinates, 24.8 and 198.4.\nNow press[STAT],[CALC],[0: ExpReg]and press[ENTER]. The values and will be displayed. The\nexponential equation is\nTRY IT #7 Use a graphing calculator to find the exponential equation that includes the points (3, 75.98) and\n(6, 481.07).\nApplying the Compound-Interest Formula"
	},
	{
		"chunk_id" : "00001698",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Applying the Compound-Interest Formula\nSavings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use\ncompound interest. The termcompoundingrefers to interest earned not only on the original value, but on the\naccumulated value of the account.\nTheannual percentage rate (APR)of an account, also called thenominal rate, is the yearly interest rate earned by an"
	},
	{
		"chunk_id" : "00001699",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "investment account. The termnominalis used when the compounding occurs a number of times other than once per\nyear. In fact, when interest is compounded more than once a year, the effective interest rate ends up beinggreaterthan\nthe nominal rate! This is a powerful tool for investing.\n552 6  Exponential and Logarithmic Functions\nWe can calculate the compound interest using the compound interest formula, which is an exponential function of the"
	},
	{
		"chunk_id" : "00001700",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variables time principal APR and number of compounding periods in a year\nFor example, observeTable 4, which shows the result of investing $1,000 at 10% for one year. Notice how the value of the\naccount increases as the compounding frequency increases.\nFrequency Value after 1 year\nAnnually $1100\nSemiannually $1102.50\nQuarterly $1103.81\nMonthly $1104.71\nDaily $1105.16\nTable4\nThe Compound Interest Formula\nCompound interestcan be calculated using the formula\nwhere\n is the account value,"
	},
	{
		"chunk_id" : "00001701",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where\n is the account value,\n is measured in years,\n is the starting amount of the account, often called the principal, or more generally present value,\n is the annual percentage rate (APR) expressed as a decimal, and\n is the number of compounding periods in one year.\nEXAMPLE8\nCalculating Compound Interest\nIf we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be\nworth in 10 years?\nSolution"
	},
	{
		"chunk_id" : "00001702",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "worth in 10 years?\nSolution\nBecause we are starting with $3,000, Our interest rate is 3%, so Because we are compounding\nquarterly, we are compounding 4 times per year, so We want to know the value of the account in 10 years, so we\nare looking for the value when\nThe account will be worth about $4,045.05 in 10 years.\nTRY IT #8 An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year).\nWhat will the investment be worth in 30 years?\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001703",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.1  Exponential Functions 553\nEXAMPLE9\nUsing the Compound Interest Formula to Solve for the Principal\nA 529 Plan is a college-savings plan that allows relatives to invest money to pay for a childs future college tuition; the\naccount grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to\n$40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest"
	},
	{
		"chunk_id" : "00001704",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "dollar, how much will Lily need to invest in the account now?\nSolution\nThe nominal interest rate is 6%, so Interest is compounded twice a year, so\nWe want to find the initial investment, needed so that the value of the account will be worth $40,000 in years.\nSubstitute the given values into the compound interest formula, and solve for\nLily will need to invest $13,801 to have $40,000 in 18 years.\nTRY IT #9 Refer toExample 9. To the nearest dollar, how much would Lily need to invest if the account is"
	},
	{
		"chunk_id" : "00001705",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compounded quarterly?\nEvaluating Functions with Basee\nAs we saw earlier, the amount earned on an account increases as the compounding frequency increases.Table 5shows\nthat the increase from annual to semi-annual compounding is larger than the increase from monthly to daily\ncompounding. This might lead us to ask whether this pattern will continue.\nExamine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies, listed inTable 5.\nFrequency Value\nAnnually $2\nSemiannually $2.25"
	},
	{
		"chunk_id" : "00001706",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Frequency Value\nAnnually $2\nSemiannually $2.25\nQuarterly $2.441406\nMonthly $2.613035\nDaily $2.714567\nHourly $2.718127\nOnce per minute $2.718279\nOnce per second $2.718282\nTable5\n554 6  Exponential and Logarithmic Functions\nThese values appear to be approaching a limit as increases without bound. In fact, as gets larger and larger, the\nexpression approaches a number used so frequently in mathematics that it has its own name: the letter This"
	},
	{
		"chunk_id" : "00001707",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its\napproximation to six decimal places is shown below.\nThe Number\nThe lettererepresents the irrational number\nThe lettereis used as a base for many real-world exponential models. To work with basee, we use the\napproximation, The constant was named by the Swiss mathematician Leonhard Euler (17071783) who\nfirst investigated and discovered many of its properties.\nEXAMPLE10"
	},
	{
		"chunk_id" : "00001708",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE10\nUsing a Calculator to Find Powers ofe\nCalculate Round to five decimal places.\nSolution\nOn a calculator, press the button labeled The window shows Type and then close parenthesis,\nPress [ENTER]. Rounding to decimal places, Caution: Many scientific calculators have an Exp\nbutton, which is used to enter numbers in scientific notation. It is not used to find powers of\nTRY IT #10 Use a calculator to find Round to five decimal places.\nInvestigating Continuous Growth"
	},
	{
		"chunk_id" : "00001709",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Investigating Continuous Growth\nSo far we have worked with rational bases for exponential functions. For most real-world phenomena, however,eis used\nas the base for exponential functions. Exponential models that use as the base are calledcontinuous growth or decay\nmodels. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and\nfluid dynamics.\nThe Continuous Growth/Decay Formula"
	},
	{
		"chunk_id" : "00001710",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Continuous Growth/Decay Formula\nFor all real numbers and all positive numbers and continuous growth or decay is represented by the formula\nwhere\n is the initial value,\n is the continuous growth rate per unit time,\n and is the elapsed time.\nIf , then the formula represents continuous growth. If , then the formula represents continuous decay.\nFor business applications, the continuous growth formula is called the continuous compounding formula and takes\nthe form\nwhere"
	},
	{
		"chunk_id" : "00001711",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the form\nwhere\n is the principal or the initial invested,\n is the growth or interest rate per unit time,\nAccess for free at openstax.org\n6.1  Exponential Functions 555\n and is the period or term of the investment.\n...\nHOW TO\nGiven the initial value, rate of growth or decay, and time solve a continuous growth or decay function.\n1. Use the information in the problem to determine , the initial value of the function.\n2. Use the information in the problem to determine the growth rate"
	},
	{
		"chunk_id" : "00001712",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If the problem refers to continuous growth, then\nb. If the problem refers to continuous decay, then\n3. Use the information in the problem to determine the time\n4. Substitute the given information into the continuous growth formula and solve for\nEXAMPLE11\nCalculating Continuous Growth\nA person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in\nthe account at the end of one year?\nSolution"
	},
	{
		"chunk_id" : "00001713",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the account at the end of one year?\nSolution\nSince the account is growing in value, this is a continuous compounding problem with growth rate The initial\ninvestment was $1,000, so We use the continuous compounding formula to find the value after year:\nThe account is worth $1,105.17 after one year.\nTRY IT #11 A person invests $100,000 at a nominal 12% interest per year compounded continuously. What\nwill be the value of the investment in 30 years?\nEXAMPLE12\nCalculating Continuous Decay"
	},
	{
		"chunk_id" : "00001714",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE12\nCalculating Continuous Decay\nRadon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?\nSolution\nSince the substance is decaying, the rate, , is negative. So, The initial amount of radon-222 was\nmg, so We use the continuous decay formula to find the value after days:\nSo 59.5115 mg of radon-222 will remain.\nTRY IT #12 Using the data inExample 12, how much radon-222 will remain after one year?\nMEDIA"
	},
	{
		"chunk_id" : "00001715",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with exponential functions.\n556 6  Exponential and Logarithmic Functions\nExponential Growth Function(http://openstax.org/l/expgrowth)\nCompound Interest(http://openstax.org/l/compoundint)\n6.1 SECTION EXERCISES\nVerbal\n1. Explain why the values of an 2. Given a formula for an 3. The Oxford Dictionary\nincreasing exponential exponential function, is it defines the wordnominalas"
	},
	{
		"chunk_id" : "00001716",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function will eventually possible to determine a value that is stated or\novertake the values of an whether the function grows expressed but not\nincreasing linear function. or decays exponentially just necessarily corresponding\nby looking at the formula? exactly to the real value.3\nExplain. Develop a reasonable\nargument for why the term\nnominal rateis used to\ndescribe the annual\npercentage rate of an\ninvestment account that\ncompounds interest.\nAlgebraic"
	},
	{
		"chunk_id" : "00001717",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compounds interest.\nAlgebraic\nFor the following exercises, identify whether the statement represents an exponential function. Explain.\n4. The average annual 5. A population of bacteria 6. The value of a coin collection\npopulation increase of a decreases by a factor of has increased by\npack of wolves is 25. every hours. annually over the last\nyears.\n7. For each training session, a 8. The height of a projectile at\npersonal trainer charges his time is represented by the\nclients less than the function"
	},
	{
		"chunk_id" : "00001718",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "clients less than the function\nprevious training session.\nFor the following exercises, consider this scenario: For each year the population of a forest of trees is represented by\nthe function In a neighboring forest, the population of the same type of tree is represented by the\nfunction (Round answers to the nearest whole number.)\n9. Which forests population is 10. Which forest had a greater 11. Assuming the population\ngrowing at a faster rate? number of trees initially? growth models continue to"
	},
	{
		"chunk_id" : "00001719",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "By how many? represent the growth of\nthe forests, which forest\nwill have a greater number\nof trees after years? By\nhow many?\n3 Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina.\nAccess for free at openstax.org\n6.1  Exponential Functions 557\n12. Assuming the population 13. Discuss the above results\ngrowth models continue to from the previous four\nrepresent the growth of exercises. Assuming the\nthe forests, which forest population growth models"
	},
	{
		"chunk_id" : "00001720",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will have a greater number continue to represent the\nof trees after years? By growth of the forests,\nhow many? which forest will have the\ngreater number of trees in\nthe long run? Why? What\nare some factors that\nmight influence the long-\nterm validity of the\nexponential growth model?\nFor the following exercises, determine whether the equation represents exponential growth, exponential decay, or\nneither. Explain.\n14. 15. 16.\n17."
	},
	{
		"chunk_id" : "00001721",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "neither. Explain.\n14. 15. 16.\n17.\nFor the following exercises, find the formula for an exponential function that passes through the two points given.\n18. and 19. and 20. and\n21. and 22. and\nFor the following exercises, determine whether the table could represent a function that is linear, exponential, or\nneither. If it appears to be exponential, find a function that passes through the points.\n23. 24.\n1 2 3 4 1 2 3 4\n70 40 10 -20 70 49 34.3 24.01\n25. 26. 27.\n1 2 3 4 1 2 3 4 1 2 3 4"
	},
	{
		"chunk_id" : "00001722",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "25. 26. 27.\n1 2 3 4 1 2 3 4 1 2 3 4\n80 61 42.9 25.61 10 20 40 80 -3.25 2 7.25 12.5\n558 6  Exponential and Logarithmic Functions\nFor the following exercises, use the compound interest formula,\n28. After a certain number of 29. What was the initial deposit 30. How many years had the\nyears, the value of an made to the account in the account from the previous\ninvestment account is previous exercise? exercise been\nrepresented by the accumulating interest?\nequation\nWhat is the value of the\naccount?"
	},
	{
		"chunk_id" : "00001723",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation\nWhat is the value of the\naccount?\n31. An account is opened with 32. How much more would the 33. Solve the compound\nan initial deposit of $6,500 account in the previous interest formula for the\nand earns interest exercise have been worth if principal, .\ncompounded semi- the interest were\nannually. What will the compounding weekly?\naccount be worth in\nyears?\n34. Use the formula found in 35. How much more would the 36. Use properties of rational"
	},
	{
		"chunk_id" : "00001724",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the previous exercise to account in the previous exponents to solve the\ncalculate the initial deposit two exercises be worth if it compound interest formula\nof an account that is worth were earning interest for for the interest rate,\nafter earning more years?\ninterest compounded\nmonthly for years.\n(Round to the nearest\ndollar.)\n37. Use the formula found in 38. Use the formula found in\nthe previous exercise to the previous exercise to\ncalculate the interest rate calculate the interest rate"
	},
	{
		"chunk_id" : "00001725",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for an account that was for an account that was\ncompounded semi- compounded monthly, had\nannually, had an initial an initial deposit of $5,500,\ndeposit of $9,000 and was and was worth $38,455\nworth $13,373.53 after 10 after 30 years.\nyears.\nFor the following exercises, determine whether the equation represents continuous growth, continuous decay, or\nneither. Explain.\n39. 40. 41.\n42. Suppose an investment 43. How much less would the\naccount is opened with an account from Exercise 42"
	},
	{
		"chunk_id" : "00001726",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "initial deposit of be worth after years if it\nearning interest were compounded\ncompounded continuously. monthly instead?\nHow much will the account\nbe worth after years?\nAccess for free at openstax.org\n6.1  Exponential Functions 559\nNumeric\nFor the following exercises, evaluate each function. Round answers to four decimal places, if necessary.\n44. for 45. for 46. for\n47. for 48. 49. for\nfor\n50. for\nTechnology"
	},
	{
		"chunk_id" : "00001727",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. for 48. 49. for\nfor\n50. for\nTechnology\nFor the following exercises, use a graphing calculator to find the equation of an exponential function given the points on\nthe curve.\n51. and 52. and 53. and\n54. and 55. and\nExtensions\n56. Theannual percentage 57. Repeat the previous 58. Recall that an exponential\nyield(APY) of an exercise to find the formula function is any equation\ninvestment account is a for the APY of an account written in the form\nrepresentation of the that compounds daily. Use such that"
	},
	{
		"chunk_id" : "00001728",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "actual interest rate earned the results from this and and are positive\non a compounding the previous exercise to numbers and Any\naccount. It is based on a develop a function for positive number can be\ncompounding period of the APY of any account that written as for\none year. Show that the compounds times per some value of . Use this\nAPY of an account that year. fact to rewrite the formula\ncompounds monthly can for an exponential function\nbe found with the formula that uses the number as\na base."
	},
	{
		"chunk_id" : "00001729",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a base.\n560 6  Exponential and Logarithmic Functions\n59. In an exponential decay 60. The formula for the\nfunction, the base of the amount in an investment\nexponent is a value account with a nominal\nbetween 0 and 1. Thus, for interest rate at any time\nsome number the is given by\nexponential decay function where is the amount of\ncan be written as principal initially deposited\nUse this into an account that\nformula, along with the compounds continuously.\nfact that to show Prove that the percentage"
	},
	{
		"chunk_id" : "00001730",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fact that to show Prove that the percentage\nthat an exponential decay of interest earned to\nfunction takes the form principal at any time can\nfor some be calculated with the\npositive number . formula\nReal-World Applications\n61. The fox population in a 62. A scientist begins with 100 63. In the year 1985, a house\ncertain region has an milligrams of a radioactive was valued at $110,000. By\nannual growth rate of 9% substance that decays the year 2005, the value"
	},
	{
		"chunk_id" : "00001731",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "per year. In the year 2012, exponentially. After 35 had appreciated to\nthere were 23,900 fox hours, 50mg of the $145,000. What was the\ncounted in the area. What substance remains. How annual growth rate\nis the fox population many milligrams will between 1985 and 2005?\npredicted to be in the year remain after 54 hours? Assume that the value\n2020? continued to grow by the\nsame percentage. What\nwas the value of the house\nin the year 2010?"
	},
	{
		"chunk_id" : "00001732",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "was the value of the house\nin the year 2010?\n64. A car was valued at $38,000 65. Jaylen wants to save 66. Kyoko has $10,000 that she\nin the year 2007. By 2013, $54,000 for a down wants to invest. Her bank\nthe value had depreciated payment on a home. How has several investment\nto $11,000 If the cars value much will he need to invest accounts to choose from,\ncontinues to drop by the in an account with 8.2% all compounding daily. Her"
	},
	{
		"chunk_id" : "00001733",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same percentage, what will APR, compounding daily, in goal is to have $15,000 by\nit be worth by 2017? order to reach his goal in 5 the time she finishes\nyears? graduate school in 6 years.\nTo the nearest hundredth\nof a percent, what should\nher minimum annual\ninterest rate be in order to\nreach her goal? (Hint: solve\nthe compound interest\nformula for the interest\nrate.)\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 561\n67. Alyssa opened a retirement 68. An investment account"
	},
	{
		"chunk_id" : "00001734",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "account with 7.25% APR in with an annual interest\nthe year 2000. Her initial rate of 7% was opened with\ndeposit was $13,500. How an initial deposit of $4,000\nmuch will the account be Compare the values of the\nworth in 2025 if interest account after 9 years when\ncompounds monthly? How the interest is compounded\nmuch more would she annually, quarterly,\nmake if interest monthly, and continuously.\ncompounded\ncontinuously?\n6.2 Graphs of Exponential Functions\nLearning Objectives\nGraph exponential functions."
	},
	{
		"chunk_id" : "00001735",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nGraph exponential functions.\nGraph exponential functions using transformations.\nAs we discussed in the previous section, exponential functions are used for many real-world applications such as finance,\nforensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation\ngives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot"
	},
	{
		"chunk_id" : "00001736",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a\npowerful tool. It gives us another layer of insight for predicting future events.\nGraphing Exponential Functions\nBefore we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a\nfunction of the form whose base is greater than one. Well use the function Observe how the\noutput values inTable 1change as the input increases by\nTable1"
	},
	{
		"chunk_id" : "00001737",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nEach output value is the product of the previous output and the base, We call the base theconstant ratio. In fact, for\nany exponential function with the form is the constant ratio of the function. This means that as the input\nincreases by 1, the output value will be the product of the base and the previous output, regardless of the value of\nNotice from the table that\n the output values are positive for all values of\n as increases, the output values increase without bound; and"
	},
	{
		"chunk_id" : "00001738",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " as decreases, the output values grow smaller, approaching zero.\nFigure 1shows the exponential growth function\n562 6  Exponential and Logarithmic Functions\nFigure1 Notice that the graph gets close to thex-axis, but never touches it.\nThe domain of is all real numbers, the range is  and the horizontal asymptote is\nTo get a sense of the behavior ofexponential decay, we can create a table of values for a function of the form"
	},
	{
		"chunk_id" : "00001739",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "whose base is between zero and one. Well use the function Observe how the output values in\nTable 2change as the input increases by\nTable2\nAgain, because the input is increasing by 1, each output value is the product of the previous output and the base, or\nconstant ratio\nNotice from the table that\n the output values are positive for all values of\n as increases, the output values grow smaller, approaching zero; and\n as decreases, the output values grow without bound."
	},
	{
		"chunk_id" : "00001740",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 2shows the exponential decay function,\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 563\nFigure2\nThe domain of is all real numbers, the range is  and the horizontal asymptote is\nCharacteristics of the Graph of the Parent Function\nAn exponential function with the form has these characteristics:\n one-to-onefunction\n horizontal asymptote:\n domain:  \n range: \n x-intercept: none\n y-intercept:\n increasing if\n decreasing if"
	},
	{
		"chunk_id" : "00001741",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " y-intercept:\n increasing if\n decreasing if\nFigure 3compares the graphs ofexponential growthand decay functions.\nFigure3\n564 6  Exponential and Logarithmic Functions\n...\nHOW TO\nGiven an exponential function of the form graph the function.\n1. Create a table of points.\n2. Plot at least point from the table, including they-intercept\n3. Draw a smooth curve through the points.\n4. State the domain,   the range,  and the horizontal asymptote,\nEXAMPLE1"
	},
	{
		"chunk_id" : "00001742",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nSketching the Graph of an Exponential Function of the Formf(x) =bx\nSketch a graph of State the domain, range, and asymptote.\nSolution\nBefore graphing, identify the behavior and create a table of points for the graph.\n Since is between zero and one, we know the function is decreasing. The left tail of the graph will increase\nwithout bound, and the right tail will approach the asymptote\n Create a table of points as inTable 3.\nTable3"
	},
	{
		"chunk_id" : "00001743",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Create a table of points as inTable 3.\nTable3\n Plot they-intercept, along with two other points. We can use and\nDraw a smooth curve connecting the points as inFigure 4.\nFigure4\nThe domain is   the range is  the horizontal asymptote is\nTRY IT #1 Sketch the graph of State the domain, range, and asymptote.\nGraphing Transformations of Exponential Functions\nTransformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions,"
	},
	{
		"chunk_id" : "00001744",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we can apply the four types of transformationsshifts, reflections, stretches, and compressionsto the parent function\nwithout loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted,\nreflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the\ntransformations applied.\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 565\nGraphing a Vertical Shift"
	},
	{
		"chunk_id" : "00001745",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing a Vertical Shift\nThe first transformation occurs when we add a constant to the parent function giving us avertical shift\nunits in the same direction as the sign. For example, if we begin by graphing a parent function, we can then\ngraph two vertical shifts alongside it, using the upward shift, and the downward shift,\nBoth vertical shifts are shown inFigure 5.\nFigure5\nObserve the results of shifting vertically:\n The domain,   remains unchanged.\n When the function is shifted up units to"
	},
	{
		"chunk_id" : "00001746",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " When the function is shifted up units to\n They-intercept shifts up units to\n The asymptote shifts up units to\n The range becomes \n When the function is shifted down units to\n They-intercept shifts down units to\n The asymptote also shifts down units to\n The range becomes \nGraphing a Horizontal Shift\nThe next transformation occurs when we add a constant to the input of the parent function giving us a"
	},
	{
		"chunk_id" : "00001747",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal shift units in theoppositedirection of the sign. For example, if we begin by graphing the parent function\nwe can then graph two horizontal shifts alongside it, using the shift left, and the shift\nright, Both horizontal shifts are shown inFigure 6.\nFigure6\nObserve the results of shifting horizontally:\n The domain,   remains unchanged.\n The asymptote, remains unchanged.\n They-intercept shifts such that:\n When the function is shifted left units to they-intercept becomes This is because"
	},
	{
		"chunk_id" : "00001748",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "so the initial value of the function is\n When the function is shifted right units to they-intercept becomes Again, see that\nso the initial value of the function is\n566 6  Exponential and Logarithmic Functions\nShifts of the Parent Functionf(x) =bx\nFor any constants and the function shifts the parent function\n vertically units, in thesamedirection of the sign of\n horizontally units, in theoppositedirection of the sign of\n They-intercept becomes\n The horizontal asymptote becomes\n The range becomes "
	},
	{
		"chunk_id" : "00001749",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The range becomes \n The domain,   remains unchanged.\n...\nHOW TO\nGiven an exponential function with the form graph the translation.\n1. Draw the horizontal asymptote\n2. Identify the shift as Shift the graph of left units if is positive, and right units if is\nnegative.\n3. Shift the graph of up units if is positive, and down units if is negative.\n4. State the domain,   the range,  and the horizontal asymptote\nEXAMPLE2\nGraphing a Shift of an Exponential Function"
	},
	{
		"chunk_id" : "00001750",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing a Shift of an Exponential Function\nGraph State the domain, range, and asymptote.\nSolution\nWe have an exponential equation of the form with and\nDraw the horizontal asymptote , so draw\nIdentify the shift as so the shift is\nShift the graph of left 1 units and down 3 units.\nFigure7\nThe domain is   the range is  the horizontal asymptote is\nTRY IT #2 Graph State domain, range, and asymptote.\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 567\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001751",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an equation of the form for use a graphing calculator to approximate the solution.\n Press[Y=]. Enter the given exponential equation in the line headed Y =.\n1\n Enter the given value for in the line headed Y =.\n2\n Press[WINDOW]. Adjust they-axis so that it includes the value entered for Y =.\n2\n Press[GRAPH]to observe the graph of the exponential function along with the line for the specified value of"
	},
	{
		"chunk_id" : "00001752",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To find the value of we compute the point of intersection. Press[2ND]then[CALC]. Select intersect and\npress[ENTER]three times. The point of intersection gives the value ofxfor the indicated value of the function.\nEXAMPLE3\nApproximating the Solution of an Exponential Equation\nSolve graphically. Round to the nearest thousandth.\nSolution\nPress[Y=]and enter next toY =. Then enter 42 next toY2=. For a window, use the values 3 to 3 for and\n1"
	},
	{
		"chunk_id" : "00001753",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1\n5 to 55 for Press[GRAPH]. The graphs should intersect somewhere near\nFor a better approximation, press[2ND]then[CALC]. Select[5: intersect]and press[ENTER]three times. The\nx-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different\nwindow or use a different value forGuess?) To the nearest thousandth,\nTRY IT #3 Solve graphically. Round to the nearest thousandth.\nGraphing a Stretch or Compression"
	},
	{
		"chunk_id" : "00001754",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing a Stretch or Compression\nWhile horizontal and vertical shifts involve adding constants to the input or to the function itself, astretchor\ncompressionoccurs when we multiply the parent function by a constant For example, if we begin by\ngraphing the parent function we can then graph the stretch, using to get as shown on\nthe left inFigure 8, and the compression, using to get as shown on the right inFigure 8.\nFigure8 (a) stretches the graph of vertically by a factor of (b) compresses the"
	},
	{
		"chunk_id" : "00001755",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph of vertically by a factor of\n568 6  Exponential and Logarithmic Functions\nStretches and Compressions of the Parent Function\nFor any factor the function\n is stretched vertically by a factor of if\n is compressed vertically by a factor of if\n has ay-intercept of\n has a horizontal asymptote at a range of  and a domain of   which are unchanged from\nthe parent function.\nEXAMPLE4\nGraphing the Stretch of an Exponential Function\nSketch a graph of State the domain, range, and asymptote.\nSolution"
	},
	{
		"chunk_id" : "00001756",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nBefore graphing, identify the behavior and key points on the graph.\n Since is between zero and one, the left tail of the graph will increase without bound as decreases, and the\nright tail will approach thex-axis as increases.\n Since the graph of will be stretched by a factor of\n Create a table of points as shown inTable 4.\nTable4\n Plot they-intercept, along with two other points. We can use and\nDraw a smooth curve connecting the points, as shown inFigure 9.\nFigure9"
	},
	{
		"chunk_id" : "00001757",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nThe domain is   the range is  the horizontal asymptote is\nTRY IT #4 Sketch the graph of State the domain, range, and asymptote.\nGraphing Reflections\nIn addition to shifting, compressing, and stretching a graph, we can also reflect it about thex-axis or they-axis. When we\nmultiply the parent function by we get a reflection about thex-axis. When we multiply the input by we\nget areflectionabout they-axis. For example, if we begin by graphing the parent function we can then graph"
	},
	{
		"chunk_id" : "00001758",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the two reflections alongside it. The reflection about thex-axis, is shown on the left side ofFigure 10, and\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 569\nthe reflection about they-axis is shown on the right side ofFigure 10.\nFigure10 (a) reflects the graph of about the x-axis. (b) reflects the graph of\nabout they-axis.\nReflections of the Parent Function\nThe function\n reflects the parent function about thex-axis.\n has ay-intercept of\n has a range of "
	},
	{
		"chunk_id" : "00001759",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " has ay-intercept of\n has a range of \n has a horizontal asymptote at and domain of   which are unchanged from the parent function.\nThe function\n reflects the parent function about they-axis.\n has ay-intercept of a horizontal asymptote at a range of  and a domain of   which\nare unchanged from the parent function.\nEXAMPLE5\nWriting and Graphing the Reflection of an Exponential Function\nFind and graph the equation for a function, that reflects about thex-axis. State its domain, range, and\nasymptote."
	},
	{
		"chunk_id" : "00001760",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptote.\nSolution\nSince we want to reflect the parent function about thex-axis, we multiply by to get,\nNext we create a table of points as inTable 5.\nTable5\nPlot they-intercept, along with two other points. We can use and\nDraw a smooth curve connecting the points:\n570 6  Exponential and Logarithmic Functions\nFigure11\nThe domain is   the range is  the horizontal asymptote is\nTRY IT #5 Find and graph the equation for a function, that reflects about they-axis. State\nits domain, range, and asymptote."
	},
	{
		"chunk_id" : "00001761",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "its domain, range, and asymptote.\nSummarizing Translations of the Exponential Function\nNow that we have worked with each type of translation for the exponential function, we can summarize them inTable 6\nto arrive at the general equation for translating exponential functions.\nTranslations of the Parent Function\nTranslation Form\nShift\n Horizontally units to the left\n Vertically units up\nStretch and Compress\n Stretch if\n Compression if\nReflect about thex-axis\nReflect about they-axis"
	},
	{
		"chunk_id" : "00001762",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Reflect about thex-axis\nReflect about they-axis\nGeneral equation for all translations\nTable6\nTranslations of Exponential Functions\nA translation of an exponential function has the form\nWhere the parent function, is\n shifted horizontally units to the left.\n stretched vertically by a factor of if\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 571\n compressed vertically by a factor of if\n shifted vertically units.\n reflected about thex-axis when"
	},
	{
		"chunk_id" : "00001763",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " reflected about thex-axis when\nNote the order of the shifts, transformations, and reflections follow the order of operations.\nEXAMPLE6\nWriting a Function from a Description\nWrite the equation for the function described below. Give the horizontal asymptote, the domain, and the range.\n is vertically stretched by a factor of , reflected across they-axis, and then shifted up units.\nSolution\nWe want to find an equation of the general form We use the description provided to find and"
	},
	{
		"chunk_id" : "00001764",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " We are given the parent function so\n The function is stretched by a factor of , so\n The function is reflected about they-axis. We replace with to get:\n The graph is shifted vertically 4 units, so\nSubstituting in the general form we get,\nThe domain is   the range is  the horizontal asymptote is\nTRY IT #6 Write the equation for function described below. Give the horizontal asymptote, the domain, and\nthe range.\n is compressed vertically by a factor of reflected across thex-axis and then"
	},
	{
		"chunk_id" : "00001765",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shifted down units.\nMEDIA\nAccess this online resource for additional instruction and practice with graphing exponential functions.\nGraph Exponential Functions(http://openstax.org/l/graphexpfunc)\n6.2 SECTION EXERCISES\nVerbal\n1. What role does the 2. What is the advantage of\nhorizontal asymptote of an knowing how to recognize\nexponential function play in transformations of the\ntelling us about the end graph of a parent function\nbehavior of the graph? algebraically?"
	},
	{
		"chunk_id" : "00001766",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "behavior of the graph? algebraically?\n572 6  Exponential and Logarithmic Functions\nAlgebraic\n3. The graph of is 4. The graph of 5. The graph of is\nreflected about they-axis is reflected about they-axis reflected about thex-axis\nand stretched vertically by a and compressed vertically and shifted upward units.\nfactor of What is the by a factor of What is the What is the equation of the\nequation of the new new function, State its\nequation of the new\nfunction, State its y-intercept, domain, and"
	},
	{
		"chunk_id" : "00001767",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, State its y-intercept, domain, and\nfunction, State its\ny-intercept, domain, and range.\ny-intercept, domain, and\nrange.\nrange.\n6. The graph of 7. The graph of\nis shifted right units, is\nstretched vertically by a\nshifted downward units,\nfactor of reflected about\nand then shifted left units,\nthex-axis, and then shifted\nstretched vertically by a\ndownward units. What is\nfactor of and reflected\nthe equation of the new\nabout thex-axis. What is the\nfunction, State its\nequation of the new"
	},
	{
		"chunk_id" : "00001768",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, State its\nequation of the new\ny-intercept (to the nearest\nfunction, State its\nthousandth), domain, and\ny-intercept, domain, and\nrange.\nrange.\nGraphical\nFor the following exercises, graph the function and its reflection about they-axis on the same axes, and give the\ny-intercept.\n8. 9. 10.\nFor the following exercises, graph each set of functions on the same axes.\n11. 12. and\nand\nFor the following exercises, match each function with one of the graphs inFigure 12.\nFigure12"
	},
	{
		"chunk_id" : "00001769",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12\nAccess for free at openstax.org\n6.2  Graphs of Exponential Functions 573\n13. 14. 15.\n16. 17. 18.\nFor the following exercises, use the graphs shown inFigure 13. All have the form\nFigure13\n19. Which graph has the 20. Which graph has the 21. Which graph has the\nlargest value for smallest value for largest value for\n22. Which graph has the\nsmallest value for\nFor the following exercises, graph the function and its reflection about thex-axis on the same axes.\n23. 24. 25."
	},
	{
		"chunk_id" : "00001770",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. 24. 25.\nFor the following exercises, graph the transformation of Give the horizontal asymptote, the domain, and the\nrange.\n26. 27. 28.\nFor the following exercises, describe the end behavior of the graphs of the functions.\n29. 30. 31.\nFor the following exercises, start with the graph of Then write a function that results from the given\ntransformation.\n32. Shift 4 units upward 33. Shift 3 units 34. Shift 2 units left\ndownward\n35. Shift 5 units right 36. Reflect about the 37. Reflect about the"
	},
	{
		"chunk_id" : "00001771",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-axis y-axis\n574 6  Exponential and Logarithmic Functions\nFor the following exercises, each graph is a transformation of Write an equation describing the transformation.\n38. 39. 40.\nFor the following exercises, find an exponential equation for the graph.\n41. 42.\nNumeric\nFor the following exercises, evaluate the exponential functions for the indicated value of\n43. for 44. for 45. for\nTechnology"
	},
	{
		"chunk_id" : "00001772",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "43. for 44. for 45. for\nTechnology\nFor the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest\nthousandth.\n46. 47. 48.\n49. 50.\nAccess for free at openstax.org\n6.3  Logarithmic Functions 575\nExtensions\n51. Explore and discuss the graphs of 52. Prove the conjecture made in the previous\nand Then make a conjecture about exercise.\nthe relationship between the graphs of the\nfunctions and for any real number"
	},
	{
		"chunk_id" : "00001773",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions and for any real number\n53. Explore and discuss the graphs of 54. Prove the conjecture made in the previous\nand Then make a exercise.\nconjecture about the relationship between the\ngraphs of the functions and for any\nreal numbernand real number\n6.3 Logarithmic Functions\nLearning Objectives\nIn this section, you will:\nConvert from logarithmic to exponential form.\nConvert from exponential to logarithmic form.\nEvaluate logarithms.\nUse common logarithms.\nUse natural logarithms."
	},
	{
		"chunk_id" : "00001774",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use common logarithms.\nUse natural logarithms.\nFigure1 Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)\nIn 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes4. One year later, another,\nstronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,5 like those shown in\nFigure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the"
	},
	{
		"chunk_id" : "00001775",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter\nScale. The Haitian earthquake registered a 7.0 on the Richter Scale6 whereas the Japanese earthquake registered a 9.0.7\nThe Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as\nan earthquake of magnitude 4. It is times as great! In this lesson, we will investigate the nature of"
	},
	{
		"chunk_id" : "00001776",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the Richter Scale and the base-ten function upon which it depends.\nConverting from Logarithmic to Exponential Form\nIn order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to\nbe able to convert between logarithmic and exponential form. For example, suppose the amount of energy released\n4 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/#summary. Accessed 3/4/2013."
	},
	{
		"chunk_id" : "00001777",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#summary. Accessed 3/4/2013.\n6 http://earthquake.usgs.gov/earthquakes/eqinthenews/2010/us2010rja6/. Accessed 3/4/2013.\n7 http://earthquake.usgs.gov/earthquakes/eqinthenews/2011/usc0001xgp/#details. Accessed 3/4/2013.\n576 6  Exponential and Logarithmic Functions\nfrom one earthquake were 500 times greater than the amount of energy released from another. We want to calculate"
	},
	{
		"chunk_id" : "00001778",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the difference in magnitude. The equation that represents this problem is where represents the difference\nin magnitudes on theRichter Scale. How would we solve for\nWe have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is\nsufficient to solve We know that and so it is clear that must be some value between\n2 and 3, since is increasing. We can examine a graph, as inFigure 2,to better estimate the solution.\nFigure2"
	},
	{
		"chunk_id" : "00001779",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nEstimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function.\nObserve that the graph inFigure 2passes the horizontal line test. The exponential function isone-to-one, so its\ninverse, is also a function. As is the case with all inverse functions, we simply interchange and and solve for\nto find the inverse function. To represent as a function of we use a logarithmic function of the form"
	},
	{
		"chunk_id" : "00001780",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The base logarithmof a number is the exponent by which we must raise to get that number.\nWe read a logarithmic expression as, The logarithm with base of is equal to  or, simplified, log base of is \nWe can also say,  raised to the power of is  because logs are exponents. For example, the base 2 logarithm of 32\nis 5, because 5 is the exponent we must apply to 2 to get 32. Since we can write We read this as log\nbase 2 of 32 is 5."
	},
	{
		"chunk_id" : "00001781",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "base 2 of 32 is 5.\nWe can express the relationship between logarithmic form and its corresponding exponential form as follows:\nNote that the base is always positive.\nBecause logarithm is a function, it is most correctly written as using parentheses to denote function evaluation,\njust as we would with However, when the input is a single variable or number, it is common to see the parentheses\ndropped and the expression written without parentheses, as Note that many calculators require parentheses\naround the"
	},
	{
		"chunk_id" : "00001782",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "around the\nWe can illustrate the notation of logarithms as follows:\nNotice that, comparing the logarithm function and the exponential function, the input and the output are switched. This\nAccess for free at openstax.org\n6.3  Logarithmic Functions 577\nmeans and are inverse functions.\nDefinition of the Logarithmic Function\nAlogarithmbase of a positive number satisfies the following definition.\nFor\nwhere,\n we read as, the logarithm with base of  or the log base of"
	},
	{
		"chunk_id" : "00001783",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the logarithm is the exponent to which must be raised to get\nAlso, since the logarithmic and exponential functions switch the and values, the domain and range of the\nexponential function are interchanged for the logarithmic function. Therefore,\n the domain of the logarithm function with base \n the range of the logarithm function with base  \nQ&A Can we take the logarithm of a negative number?\nNo. Because the base of an exponential function is always positive, no power of that base can ever be"
	},
	{
		"chunk_id" : "00001784",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of\nzero. Calculators may output a log of a negative number when in complex mode, but the log of a negative\nnumber is not a real number.\n...\nHOW TO\nGiven an equation in logarithmic form convert it to exponential form.\n1. Examine the equation and identify\n2. Rewrite as\nEXAMPLE1\nConverting from Logarithmic Form to Exponential Form\nWrite the following logarithmic equations in exponential form.\n \nSolution"
	},
	{
		"chunk_id" : "00001785",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nSolution\nFirst, identify the values of Then, write the equation in the form\n\nHere, Therefore, the equation is equivalent to\n\nHere, Therefore, the equation is equivalent to\nTRY IT #1 Write the following logarithmic equations in exponential form.\n \n578 6  Exponential and Logarithmic Functions\nConverting from Exponential to Logarithmic Form\nTo convert from exponents to logarithms, we follow the same steps in reverse. We identify the base exponent and\noutput Then we write\nEXAMPLE2"
	},
	{
		"chunk_id" : "00001786",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "output Then we write\nEXAMPLE2\nConverting from Exponential Form to Logarithmic Form\nWrite the following exponential equations in logarithmic form.\na.\nb.\nc.\nSolution\nFirst, identify the values of Then, write the equation in the form\na.\nHere, and Therefore, the equation is equivalent to\nb.\nHere, and Therefore, the equation is equivalent to\nc.\nHere, and Therefore, the equation is equivalent to\nTRY IT #2 Write the following exponential equations in logarithmic form.\n  \nEvaluating Logarithms"
	},
	{
		"chunk_id" : "00001787",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "  \nEvaluating Logarithms\nKnowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example,\nconsider We ask, To what exponent must be raised in order to get 8? Because we already know it\nfollows that\nNow consider solving and mentally.\n We ask, To what exponent must 7 be raised in order to get 49? We know Therefore,\n We ask, To what exponent must 3 be raised in order to get 27? We know Therefore,"
	},
	{
		"chunk_id" : "00001788",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, lets evaluate\nmentally.\n We ask, To what exponent must be raised in order to get  We know and so\nTherefore,\n...\nHOW TO\nGiven a logarithm of the form evaluate it mentally.\n1. Rewrite the argument as a power of\n2. Use previous knowledge of powers of identify by asking, To what exponent should be raised in order to get\n\nAccess for free at openstax.org\n6.3  Logarithmic Functions 579\nEXAMPLE3"
	},
	{
		"chunk_id" : "00001789",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.3  Logarithmic Functions 579\nEXAMPLE3\nSolving Logarithms Mentally\nSolve without using a calculator.\nSolution\nFirst we rewrite the logarithm in exponential form: Next, we ask, To what exponent must 4 be raised in order\nto get 64?\nWe know\nTherefore,\nTRY IT #3 Solve without using a calculator.\nEXAMPLE4\nEvaluating the Logarithm of a Reciprocal\nEvaluate without using a calculator.\nSolution\nFirst we rewrite the logarithm in exponential form: Next, we ask, To what exponent must 3 be raised in order\nto get "
	},
	{
		"chunk_id" : "00001790",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to get \nWe know but what must we do to get the reciprocal, Recall from working with exponents that\nWe use this information to write\nTherefore,\nTRY IT #4 Evaluate without using a calculator.\nUsing Common Logarithms\nSometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words,\nthe expression means We call a base-10 logarithm acommon logarithm. Common logarithms are"
	},
	{
		"chunk_id" : "00001791",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of\nstars and the pH of acids and bases also use common logarithms.\nDefinition of the Common Logarithm\nAcommon logarithmis a logarithm with base We write simply as The common logarithm of a\npositive number satisfies the following definition.\nFor\nWe read as, the logarithm with base of  or log base 10 of \nThe logarithm is the exponent to which must be raised to get"
	},
	{
		"chunk_id" : "00001792",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "580 6  Exponential and Logarithmic Functions\n...\nHOW TO\nGiven a common logarithm of the form evaluate it mentally.\n1. Rewrite the argument as a power of\n2. Use previous knowledge of powers of to identify by asking, To what exponent must be raised in order to\nget \nEXAMPLE5\nFinding the Value of a Common Logarithm Mentally\nEvaluate without using a calculator.\nSolution\nFirst we rewrite the logarithm in exponential form: Next, we ask, To what exponent must be raised in\norder to get 1000? We know\nTherefore,"
	},
	{
		"chunk_id" : "00001793",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "order to get 1000? We know\nTherefore,\nTRY IT #5 Evaluate\n...\nHOW TO\nGiven a common logarithm with the form evaluate it using a calculator.\n1. Press[LOG].\n2. Enter the value given for followed by[ ) ].\n3. Press[ENTER].\nEXAMPLE6\nFinding the Value of a Common Logarithm Using a Calculator\nEvaluate to four decimal places using a calculator.\nSolution\n Press[LOG].\n Enter 321,followed by[ ) ].\n Press[ENTER].\nRounding to four decimal places,\nAnalysis"
	},
	{
		"chunk_id" : "00001794",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Rounding to four decimal places,\nAnalysis\nNote that and that Since 321 is between 100 and 1000, we know that must be between\nand This gives us the following:\nTRY IT #6 Evaluate to four decimal places using a calculator.\nAccess for free at openstax.org\n6.3  Logarithmic Functions 581\nEXAMPLE7\nRewriting and Solving a Real-World Exponential Model\nThe amount of energy released from one earthquake was 500 times greater than the amount of energy released from"
	},
	{
		"chunk_id" : "00001795",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "another. The equation represents this situation, where is the difference in magnitudes on the Richter Scale.\nTo the nearest thousandth, what was the difference in magnitudes?\nSolution\nWe begin by rewriting the exponential equation in logarithmic form.\nNext we evaluate the logarithm using a calculator:\n Press[LOG].\n Enter followed by[ ) ].\n Press[ENTER].\n To the nearest thousandth,\nThe difference in magnitudes was about"
	},
	{
		"chunk_id" : "00001796",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The difference in magnitudes was about\nTRY IT #7 The amount of energy released from one earthquake was times greater than the amount of\nenergy released from another. The equation represents this situation, where is the\ndifference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference\nin magnitudes?\nUsing Natural Logarithms\nThe most frequently used base for logarithms is Base logarithms are important in calculus and some scientific"
	},
	{
		"chunk_id" : "00001797",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "applications; they are callednatural logarithms. The base logarithm, has its own notation,\nMost values of can be found only using a calculator. The major exception is that, because the logarithm of 1 is\nalways 0 in any base, For other natural logarithms, we can use the key that can be found on most scientific\ncalculators. We can also find the natural logarithm of any power of using the inverse property of logarithms.\nDefinition of the Natural Logarithm"
	},
	{
		"chunk_id" : "00001798",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Definition of the Natural Logarithm\nAnatural logarithmis a logarithm with base We write simply as The natural logarithm of a positive\nnumber satisfies the following definition.\nFor\nWe read as, the logarithm with base of  or the natural logarithm of \nThe logarithm is the exponent to which must be raised to get\nSince the functions and are inverse functions, for all and for\n...\nHOW TO\nGiven a natural logarithm with the form evaluate it using a calculator.\n1. Press[LN]."
	},
	{
		"chunk_id" : "00001799",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Press[LN].\n2. Enter the value given for followed by[ ) ].\n3. Press[ENTER].\n582 6  Exponential and Logarithmic Functions\nEXAMPLE8\nEvaluating a Natural Logarithm Using a Calculator\nEvaluate to four decimal places using a calculator.\nSolution\n Press[LN].\n Enter followed by[ ) ].\n Press[ENTER].\nRounding to four decimal places,\nTRY IT #8 Evaluate\nMEDIA\nAccess this online resource for additional instruction and practice with logarithms.\nIntroduction to Logarithms(http://openstax.org/l/intrologarithms)"
	},
	{
		"chunk_id" : "00001800",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.3 SECTION EXERCISES\nVerbal\n1. What is a base logarithm? 2. How is the logarithmic 3. How can the logarithmic\nDiscuss the meaning by function equation be\ninterpreting each part of the related to the exponential solved for using the\nequivalent equations function What is properties of exponents?\nand for the result of composing\nthese two functions?\n4. Discuss the meaning of the 5. Discuss the meaning of the\ncommon logarithm. What is natural logarithm. What is\nits relationship to a its relationship to a"
	},
	{
		"chunk_id" : "00001801",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "its relationship to a its relationship to a\nlogarithm with base and logarithm with base and\nhow does the notation how does the notation\ndiffer? differ?\nAlgebraic\nFor the following exercises, rewrite each equation in exponential form.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15.\nAccess for free at openstax.org\n6.3  Logarithmic Functions 583\nFor the following exercises, rewrite each equation in logarithmic form.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25."
	},
	{
		"chunk_id" : "00001802",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25.\nFor the following exercises, solve for by converting the logarithmic equation to exponential form.\n26. 27. 28.\n29. 30. 31.\n32. 33. 34.\n35.\nFor the following exercises, use the definition of common and natural logarithms to simplify.\n36. 37. 38.\n39. 40. 41.\nNumeric\nFor the following exercises, evaluate the base logarithmic expression without using a calculator.\n42. 43. 44.\n45."
	},
	{
		"chunk_id" : "00001803",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "42. 43. 44.\n45.\nFor the following exercises, evaluate the common logarithmic expression without using a calculator.\n46. 47. 48.\n49.\nFor the following exercises, evaluate the natural logarithmic expression without using a calculator.\n50. 51. 52.\n53.\n584 6  Exponential and Logarithmic Functions\nTechnology\nFor the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.\n54. 55. 56.\n57. 58.\nExtensions"
	},
	{
		"chunk_id" : "00001804",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "54. 55. 56.\n57. 58.\nExtensions\n59. Is in the domain of 60. Is in the range of 61. Is there a number such\nthe function the function that If so, what is\nIf so, what is the value of If so, for what value of that number? Verify the\nthe function when Verify the result. result.\nVerify the result.\n62. Is the following true: 63. Is the following true:\nVerify the\nVerify\nresult. the result.\nReal-World Applications\n64. The exposure index for 65. Refer to the previous 66. The intensity levelsIof two"
	},
	{
		"chunk_id" : "00001805",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a camera is a exercise. Suppose the light earthquakes measured on\nmeasurement of the meter on a camera a seismograph can be\namount of light that hits indicates an of and compared by the formula\nthe image receptor. It is the desired exposure time where\ndetermined by the is 16 seconds. What should\nis the magnitude given\nequation the f-stop setting be?\nby the Richter Scale. In\nwhere August 2009, an\nearthquake of magnitude\nis the f-stop setting on\n6.1 hit Honshu, Japan. In\nthe camera, and is the"
	},
	{
		"chunk_id" : "00001806",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.1 hit Honshu, Japan. In\nthe camera, and is the\nMarch 2011, that same\nexposure time in seconds.\nregion experienced yet\nSuppose the f-stop setting\nanother, more devastating\nis and the desired\nearthquake, this time with\nexposure time is seconds. a magnitude of 9.0.8 How\nWhat will the resulting\nmany times greater was\nexposure index be?\nthe intensity of the 2011\nearthquake? Round to the\nnearest whole number.\n6.4 Graphs of Logarithmic Functions\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00001807",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nIdentify the domain of a logarithmic function.\nGraph logarithmic functions.\n8 http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 585\nInGraphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us"
	},
	{
		"chunk_id" : "00001808",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because\nevery logarithmic function is the inverse function of an exponential function, we can think of every output on a\nlogarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the\ncausefor aneffect.\nTo illustrate, suppose we invest in an account that offers an annual interest rate of compounded"
	},
	{
		"chunk_id" : "00001809",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "continuously. We already know that the balance in our account for any year can be found with the equation\nBut what if we wanted to know the year for any balance? We would need to create a corresponding new function by\ninterchanging the input and the output; thus we would need to create a logarithmic model for this situation. By\ngraphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to"
	},
	{
		"chunk_id" : "00001810",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "know how many years it would take for our initial investment to double?Figure 1shows this point on the logarithmic\ngraph.\nFigure1\nIn this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to\ngraphing the family of logarithmic functions.\nFinding the Domain of a Logarithmic Function\nBefore working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function\nis defined."
	},
	{
		"chunk_id" : "00001811",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is defined.\nRecall that the exponential function is defined as for any real number and constant where\n The domain of is  \n The range of is \nIn the last section we learned that the logarithmic function is the inverse of the exponential function\nSo, as inverse functions:\n The domain of is the range of \n The range of is the domain of  \nTransformations of the parent function behave similarly to those of other functions. Just as with other"
	},
	{
		"chunk_id" : "00001812",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parent functions, we can apply the four types of transformationsshifts, stretches, compressions, and reflectionsto\nthe parent function without loss of shape.\nInGraphs of Exponential Functionswe saw that certain transformations can change therangeof Similarly,\napplying transformations to the parent function can change thedomain. When finding the domain of a\nlogarithmic function, therefore, it is important to remember that the domain consistsonly of positive real numbers. That"
	},
	{
		"chunk_id" : "00001813",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is, the argument of the logarithmic function must be greater than zero.\n586 6  Exponential and Logarithmic Functions\nFor example, consider This function is defined for any values of such that the argument, in this\ncase is greater than zero. To find the domain, we set up an inequality and solve for\nIn interval notation, the domain of is \n...\nHOW TO\nGiven a logarithmic function, identify the domain.\n1. Set up an inequality showing the argument greater than zero.\n2. Solve for"
	},
	{
		"chunk_id" : "00001814",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Solve for\n3. Write the domain in interval notation.\nEXAMPLE1\nIdentifying the Domain of a Logarithmic Shift\nWhat is the domain of\nSolution\nThe logarithmic function is defined only when the input is positive, so this function is defined when Solving\nthis inequality,\nThe domain of is \nTRY IT #1 What is the domain of\nEXAMPLE2\nIdentifying the Domain of a Logarithmic Shift and Reflection\nWhat is the domain of\nSolution"
	},
	{
		"chunk_id" : "00001815",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "What is the domain of\nSolution\nThe logarithmic function is defined only when the input is positive, so this function is defined when Solving\nthis inequality,\nThe domain of is \nTRY IT #2 What is the domain of\nGraphing Logarithmic Functions\nNow that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing\nlogarithmic functions. The family of logarithmic functions includes the parent function along with all its\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001816",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.4  Graphs of Logarithmic Functions 587\ntransformations: shifts, stretches, compressions, and reflections.\nWe begin with the parent function Because every logarithmic function of this form is the inverse of an\nexponential function with the form their graphs will be reflections of each other across the line To\nillustrate this, we can observe the relationship between the input and output values of and its equivalent\ninTable 1.\nTable1"
	},
	{
		"chunk_id" : "00001817",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inTable 1.\nTable1\nUsing the inputs and outputs fromTable 1, we can build another table to observe the relationship between points on the\ngraphs of the inverse functions and SeeTable 2.\nTable2\nAs wed expect, thex- andy-coordinates are reversed for the inverse functions.Figure 2shows the graph of and\nFigure2 Notice that the graphs of and are reflections about the line\nObserve the following from the graph:\n has ay-intercept at and has anx- intercept at\n The domain of   is the same as the range of"
	},
	{
		"chunk_id" : "00001818",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The domain of   is the same as the range of\n The range of  is the same as the domain of\nCharacteristics of the Graph of the Parent Function,\nFor any real number and constant we can see the following characteristics in the graph of\n588 6  Exponential and Logarithmic Functions\n one-to-one function\n vertical asymptote:\n domain: \n range:  \n x-intercept: and key point\n y-intercept: none\n increasing if\n decreasing if\nSeeFigure 3.\nFigure3"
	},
	{
		"chunk_id" : "00001819",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " decreasing if\nSeeFigure 3.\nFigure3\nFigure 4shows how changing the base in can affect the graphs. Observe that the graphs compress\nvertically as the value of the base increases. (Note:recall that the function has base\nFigure4 The graphs of three logarithmic functions with different bases, all greater than 1.\n...\nHOW TO\nGiven a logarithmic function with the form graph the function.\n1. Draw and label the vertical asymptote,\n2. Plot thex-intercept,\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001820",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.4  Graphs of Logarithmic Functions 589\n3. Plot the key point\n4. Draw a smooth curve through the points.\n5. State the domain,  the range,  and the vertical asymptote,\nEXAMPLE3\nGraphing a Logarithmic Function with the Formf(x) = log (x).\nb\nGraph State the domain, range, and asymptote.\nSolution\nBefore graphing, identify the behavior and key points for the graph.\n Since is greater than one, we know the function is increasing. The left tail of the graph will approach the"
	},
	{
		"chunk_id" : "00001821",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertical asymptote and the right tail will increase slowly without bound.\n Thex-intercept is\n The key point is on the graph.\n We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see\nFigure 5).\nFigure5\nThe domain is  the range is   and the vertical asymptote is\nTRY IT #3 Graph State the domain, range, and asymptote.\nGraphing Transformations of Logarithmic Functions"
	},
	{
		"chunk_id" : "00001822",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Transformations of Logarithmic Functions\nAs we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of\nother parent functions. We can shift, stretch, compress, and reflect theparent function without loss of\nshape.\nGraphing a Horizontal Shift off(x) = log (x)\nb\nWhen a constant is added to the input of the parent function the result is ahorizontal shift units in"
	},
	{
		"chunk_id" : "00001823",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "theoppositedirection of the sign on To visualize horizontal shifts, we can observe the general graph of the parent\nfunction and for alongside the shift left, and the shift right,\nSeeFigure 6.\n590 6  Exponential and Logarithmic Functions\nFigure6\nHorizontal Shifts of the Parent Function\nFor any constant the function\n shifts the parent function left units if\n shifts the parent function right units if\n has the vertical asymptote\n has domain \n has range  \n...\nHOW TO"
	},
	{
		"chunk_id" : "00001824",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " has domain \n has range  \n...\nHOW TO\nGiven a logarithmic function with the form graph the translation.\n1. Identify the horizontal shift:\na. If shift the graph of left units.\nb. If shift the graph of right units.\n2. Draw the vertical asymptote\n3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting\nfrom the coordinate.\n4. Label the three points.\n5. The Domain is  the range is   and the vertical asymptote is\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001825",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.4  Graphs of Logarithmic Functions 591\nEXAMPLE4\nGraphing a Horizontal Shift of the Parent Functiony= log (x)\nb\nSketch the horizontal shift alongside its parent function. Include the key points and asymptotes on\nthe graph. State the domain, range, and asymptote.\nSolution\nSince the function is we notice\nThus so This means we will shift the function right 2 units.\nThe vertical asymptote is or\nConsider the three key points from the parent function, and"
	},
	{
		"chunk_id" : "00001826",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The new coordinates are found by adding 2 to the coordinates.\nLabel the points and\nThe domain is  the range is   and the vertical asymptote is\nFigure7\nTRY IT #4 Sketch a graph of alongside its parent function. Include the key points and\nasymptotes on the graph. State the domain, range, and asymptote.\nGraphing a Vertical Shift ofy= log (x)\nb\nWhen a constant is added to the parent function the result is avertical shift units in the direction of"
	},
	{
		"chunk_id" : "00001827",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the sign on To visualize vertical shifts, we can observe the general graph of the parent function\nalongside the shift up, and the shift down, SeeFigure 8.\n592 6  Exponential and Logarithmic Functions\nFigure8\nVertical Shifts of the Parent Function\nFor any constant the function\n shifts the parent function up units if\n shifts the parent function down units if\n has the vertical asymptote\n has domain \n has range  \n...\nHOW TO\nGiven a logarithmic function with the form graph the translation."
	},
	{
		"chunk_id" : "00001828",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the vertical shift:\n If shift the graph of up units.\n If shift the graph of down units.\n2. Draw the vertical asymptote\n3. Identify three key points from the parent function. Find new coordinates for the shifted functions by adding to\nthe coordinate.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 593\n4. Label the three points.\n5. The domain is  the range is  and the vertical asymptote is\nEXAMPLE5\nGraphing a Vertical Shift of the Parent Functiony= log (x)\nb"
	},
	{
		"chunk_id" : "00001829",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "b\nSketch a graph of alongside its parent function. Include the key points and asymptote on the graph.\nState the domain, range, and asymptote.\nSolution\nSince the function is we will notice Thus\nThis means we will shift the function down 2 units.\nThe vertical asymptote is\nConsider the three key points from the parent function, and\nThe new coordinates are found by subtracting 2 from theycoordinates.\nLabel the points and\nThe domain is  the range is   and the vertical asymptote is\nFigure9"
	},
	{
		"chunk_id" : "00001830",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nThe domain is  the range is   and the vertical asymptote is\nTRY IT #5 Sketch a graph of alongside its parent function. Include the key points and\nasymptote on the graph. State the domain, range, and asymptote.\nGraphing Stretches and Compressions ofy= log (x)\nb\nWhen the parent function is multiplied by a constant the result is avertical stretchorcompression\nof the original graph. To visualize stretches and compressions, we set and observe the general graph of the parent"
	},
	{
		"chunk_id" : "00001831",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "594 6  Exponential and Logarithmic Functions\nfunction alongside the vertical stretch, and the vertical compression,\nSeeFigure 10.\nFigure10\nVertical Stretches and Compressions of the Parent Function\nFor any constant the function\n stretches the parent function vertically by a factor of if\n compresses the parent function vertically by a factor of if\n has the vertical asymptote\n has thex-intercept\n has domain \n has range  \n...\nHOW TO\nGiven a logarithmic function with the form graph the translation."
	},
	{
		"chunk_id" : "00001832",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify the vertical stretch or compressions:\n If the graph of is stretched by a factor of units.\n If the graph of is compressed by a factor of units.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 595\n2. Draw the vertical asymptote\n3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying\nthe coordinates by\n4. Label the three points.\n5. The domain is  the range is   and the vertical asymptote is\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001833",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nGraphing a Stretch or Compression of the Parent Functiony= log (x)\nb\nSketch a graph of alongside its parent function. Include the key points and asymptote on the graph.\nState the domain, range, and asymptote.\nSolution\nSince the function is we will notice\nThis means we will stretch the function by a factor of 2.\nThe vertical asymptote is\nConsider the three key points from the parent function, and\nThe new coordinates are found by multiplying the coordinates by 2.\nLabel the points and"
	},
	{
		"chunk_id" : "00001834",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Label the points and\nThe domain is  the range is   and the vertical asymptote is SeeFigure 11.\nFigure11\nThe domain is  the range is   and the vertical asymptote is\nTRY IT #6 Sketch a graph of alongside its parent function. Include the key points and\nasymptote on the graph. State the domain, range, and asymptote.\nEXAMPLE7\nCombining a Shift and a Stretch\nSketch a graph of State the domain, range, and asymptote.\n596 6  Exponential and Logarithmic Functions\nSolution"
	},
	{
		"chunk_id" : "00001835",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nRemember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the\nfunction vertically by a factor of 5, as inFigure 12. The vertical asymptote will be shifted to Thex-intercept will be\nThe domain will be  Two points will help give the shape of the graph: and We chose\nas thex-coordinate of one point to graph because when the base of the common logarithm.\nFigure12\nThe domain is  the range is   and the vertical asymptote is"
	},
	{
		"chunk_id" : "00001836",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Sketch a graph of the function State the domain, range, and asymptote.\nGraphing Reflections off(x) = log (x)\nb\nWhen the parent function is multiplied by the result is areflectionabout thex-axis. When theinputis\nmultiplied by the result is a reflection about they-axis. To visualize reflections, we restrict and observe the\ngeneral graph of the parent function alongside the reflection about thex-axis, and the\nreflection about they-axis,\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001837",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.4  Graphs of Logarithmic Functions 597\nFigure13\nReflections of the Parent Function\nThe function\n reflects the parent function about thex-axis.\n has domain,  range,   and vertical asymptote, which are unchanged from the parent\nfunction.\nThe function\n reflects the parent function about they-axis.\n has domain \n has range,   and vertical asymptote, which are unchanged from the parent function.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001838",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a logarithmic function with the parent function graph a translation.\n598 6  Exponential and Logarithmic Functions\n1. Draw the vertical asymptote, 1. Draw the vertical asymptote,\n2. Plot thex-intercept, 2. Plot thex-intercept,\n3. Reflect the graph of the parent function 3. Reflect the graph of the parent function\nabout thex-axis. about they-axis.\n4. Draw a smooth curve through the points. 4. Draw a smooth curve through the points."
	},
	{
		"chunk_id" : "00001839",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. State the domain, (0, ), the range, (, ), and the 5. State the domain, (, 0) the range, (, ) and the\nvertical asymptote . vertical asymptote\nTable3\nEXAMPLE8\nGraphing a Reflection of a Logarithmic Function\nSketch a graph of alongside its parent function. Include the key points and asymptote on the graph.\nState the domain, range, and asymptote.\nSolution\nBefore graphing identify the behavior and key points for the graph."
	},
	{
		"chunk_id" : "00001840",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Since is greater than one, we know that the parent function is increasing. Since theinputvalue is multiplied\nby is a reflection of the parent graph about they-axis. Thus, will be decreasing as moves\nfrom negative infinity to zero, and the right tail of the graph will approach the vertical asymptote\n Thex-intercept is\n We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.\nFigure14\nThe domain is  the range is   and the vertical asymptote is"
	},
	{
		"chunk_id" : "00001841",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #8 Graph State the domain, range, and asymptote.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 599\n...\nHOW TO\nGiven a logarithmic equation, use a graphing calculator to approximate solutions.\n1. Press[Y=]. Enter the given logarithm equation or equations asY =and, if needed,Y =.\n1 2\n2. Press[GRAPH]to observe the graphs of the curves and use[WINDOW]to find an appropriate view of the\ngraphs, including their point(s) of intersection."
	},
	{
		"chunk_id" : "00001842",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graphs, including their point(s) of intersection.\n3. To find the value of we compute the point of intersection. Press[2ND]then[CALC]. Select intersect and\npress[ENTER]three times. The point of intersection gives the value of for the point(s) of intersection.\nEXAMPLE9\nApproximating the Solution of a Logarithmic Equation\nSolve graphically. Round to the nearest thousandth.\nSolution\nPress[Y=]and enter next toY =. Then enter next toY =. For a window, use the values 0 to 5 for\n1 2"
	},
	{
		"chunk_id" : "00001843",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 2\nand 10 to 10 for Press[GRAPH]. The graphs should intersect somewhere a little to right of\nFor a better approximation, press[2ND]then[CALC]. Select[5: intersect]and press[ENTER]three times. The\nx-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different\nwindow or use a different value forGuess?) So, to the nearest thousandth,\nTRY IT #9 Solve graphically. Round to the nearest thousandth.\nSummarizing Translations of the Logarithmic Function"
	},
	{
		"chunk_id" : "00001844",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now that we have worked with each type of translation for the logarithmic function, we can summarize each inTable 4to\narrive at the general equation for translating exponential functions.\nTranslations of the Parent Function\nTranslation Form\nShift\n Horizontally units to the left\n Vertically units up\nStretch and Compress\n Stretch if\n Compression if\nReflect about thex-axis\nReflect about they-axis\nGeneral equation for all translations\nTable4\n600 6  Exponential and Logarithmic Functions"
	},
	{
		"chunk_id" : "00001845",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "600 6  Exponential and Logarithmic Functions\nTranslations of Logarithmic Functions\nAll translations of the parent logarithmic function, have the form\nwhere the parent function, is\n shifted vertically up units.\n shifted horizontally to the left units.\n stretched vertically by a factor of if\n compressed vertically by a factor of if\n reflected about thex-axis when\nFor the graph of the parent function is reflected about they-axis.\nEXAMPLE10\nFinding the Vertical Asymptote of a Logarithm Graph"
	},
	{
		"chunk_id" : "00001846",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "What is the vertical asymptote of\nSolution\nThe vertical asymptote is at\nAnalysis\nThe coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left\nshifts the vertical asymptote to\nTRY IT #10 What is the vertical asymptote of\nEXAMPLE11\nFinding the Equation from a Graph\nFind a possible equation for the common logarithmic function graphed inFigure 15.\nFigure15\nSolution"
	},
	{
		"chunk_id" : "00001847",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure15\nSolution\nThis graph has a vertical asymptote at and has been vertically reflected. We do not know yet the vertical shift or\nthe vertical stretch. We know so far that the equation will have form:\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 601\nIt appears the graph passes through the points and Substituting\nNext, substituting in ,\nThis gives us the equation\nAnalysis\nWe can verify this answer by comparing the function values inTable 5with the points on the graph inFigure 15."
	},
	{
		"chunk_id" : "00001848",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 0 1 2 3\n1 0 0.58496 1 1.3219\n4 5 6 7 8\n1.5850 1.8074 2 2.1699 2.3219\nTable5\nTRY IT #11 Give the equation of the natural logarithm graphed inFigure 16.\nFigure16\nQ&A Is it possible to tell the domain and range and describe the end behavior of a function just by\nlooking at the graph?\nYes, if we know the function is a general logarithmic function. For example, look at the graph inFigure 16.\nThe graph approaches (or thereabouts) more and more closely, so is, or is very close to,"
	},
	{
		"chunk_id" : "00001849",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the vertical asymptote. It approaches from the right, so the domain is all points to the right,\n602 6  Exponential and Logarithmic Functions\nThe range, as with all general logarithmic functions, is all real numbers. And we can see\nthe end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is\nthat as and as  \nMEDIA\nAccess these online resources for additional instruction and practice with graphing logarithms."
	},
	{
		"chunk_id" : "00001850",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph an Exponential Function and Logarithmic Function(http://openstax.org/l/graphexplog)\nMatch Graphs with Exponential and Logarithmic Functions(http://openstax.org/l/matchexplog)\nFind the Domain of Logarithmic Functions(http://openstax.org/l/domainlog)\n6.4 SECTION EXERCISES\nVerbal\n1. The inverse of every 2. What type(s) of 3. What type(s) of\nlogarithmic function is an translation(s), if any, affect translation(s), if any, affect"
	},
	{
		"chunk_id" : "00001851",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential function and the range of a logarithmic the domain of a logarithmic\nvice-versa. What does this function? function?\ntell us about the relationship\nbetween the coordinates of\nthe points on the graphs of\neach?\n4. Consider the general 5. Does the graph of a general\nlogarithmic function logarithmic function have a\nWhy cant horizontal asymptote?\nbe zero? Explain.\nAlgebraic\nFor the following exercises, state the domain and range of the function.\n6. 7. 8.\n9. 10."
	},
	{
		"chunk_id" : "00001852",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7. 8.\n9. 10.\nFor the following exercises, state the domain and the vertical asymptote of the function.\n11. 12. 13.\n14. 15.\nFor the following exercises, state the domain, vertical asymptote, and end behavior of the function.\n16. 17. 18.\n19. 20.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 603\nFor the following exercises, state the domain, range, andx- andy-intercepts, if they exist. If they do not exist, write DNE.\n21. 22. 23.\n24. 25.\nGraphical"
	},
	{
		"chunk_id" : "00001853",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "21. 22. 23.\n24. 25.\nGraphical\nFor the following exercises, match each function inFigure 17with the letter corresponding to its graph.\nFigure17\n26. 27. 28.\n29. 30.\nFor the following exercises, match each function inFigure 18with the letter corresponding to its graph.\nFigure18\n604 6  Exponential and Logarithmic Functions\n31. 32. 33.\nFor the following exercises, sketch the graphs of each pair of functions on the same axis.\n34. and 35. and 36. and\n37. and"
	},
	{
		"chunk_id" : "00001854",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. and 35. and 36. and\n37. and\nFor the following exercises, match each function inFigure 19with the letter corresponding to its graph.\nFigure19\n38. 39. 40.\nFor the following exercises, sketch the graph of the indicated function.\n41. 42. 43.\n44. 45. 46.\nAccess for free at openstax.org\n6.4  Graphs of Logarithmic Functions 605\nFor the following exercises, write a logarithmic equation corresponding to the graph shown.\n47. Use as the parent 48. Use as the parent 49. Use as the parent"
	},
	{
		"chunk_id" : "00001855",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function. function. function.\n50. Use as the parent\nfunction.\nTechnology\nFor the following exercises, use a graphing calculator to find approximate solutions to each equation.\n51. 52.\n53. 54. 55.\nExtensions\n56. Let be any positive real 57. Explore and discuss the 58. Prove the conjecture made\nnumber such that graphs of in the previous exercise.\nWhat must be equal\nand\nto? Verify the result.\nMake a conjecture based\non the result.\n606 6  Exponential and Logarithmic Functions"
	},
	{
		"chunk_id" : "00001856",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "606 6  Exponential and Logarithmic Functions\n59. What is the domain of the 60. Use properties of\nfunction exponents to find the\nx-intercepts of the function\nDiscuss the result.\nalgebraically. Show the\nsteps for solving, and then\nverify the result by\ngraphing the function.\n6.5 Logarithmic Properties\nLearning Objectives\nIn this section, you will:\nUse the product rule for logarithms.\nUse the quotient rule for logarithms.\nUse the power rule for logarithms.\nExpand logarithmic expressions."
	},
	{
		"chunk_id" : "00001857",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Expand logarithmic expressions.\nCondense logarithmic expressions.\nUse the change-of-base formula for logarithms.\nFigure1 The pH of hydrochloric acid is tested with litmus paper. (credit: David Berardan)\nIn chemistry,pHis used as a measure of the acidity or alkalinity of a substance. The pH scale runs from 0 to 14.\nSubstances with a pH less than 7 are considered acidic, and substances with a pH greater than 7 are said to be basic. Our"
	},
	{
		"chunk_id" : "00001858",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bodies, for instance, must maintain a pH close to 7.35 in order for enzymes to work properly. To get a feel for what is\nacidic and what is basic, consider the following pH levels of some common substances:\n Battery acid: 0.8\n Stomach acid: 2.7\n Orange juice: 3.3\n Pure water: 7 (at 25 C)\n Human blood: 7.35\n Fresh coconut: 7.8\n Sodium hydroxide (lye): 14\nTo determine whether a solution is acidic or basic, we find its pH, which is a measure of the number of active positive"
	},
	{
		"chunk_id" : "00001859",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hydrogen ions in the solution. The pH is defined by the following formula, where is the concentration of hydrogen\nion in the solution\nThe equivalence of and is one of the logarithm properties we will examine in this section.\nAccess for free at openstax.org\n6.5  Logarithmic Properties 607\nUsing the Product Rule for Logarithms\nRecall that the logarithmic and exponential functions undo each other. This means that logarithms have similar"
	},
	{
		"chunk_id" : "00001860",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "properties to exponents. Some important properties of logarithms are given here. First, the following properties are easy\nto prove.\nFor example, since And since\nNext, we have the inverse property.\nFor example, to evaluate we can rewrite the logarithm as and then apply the inverse property\nto get\nTo evaluate we can rewrite the logarithm as and then apply the inverse property to get\nFinally, we have theone-to-oneproperty."
	},
	{
		"chunk_id" : "00001861",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finally, we have theone-to-oneproperty.\nWe can use the one-to-one property to solve the equation for Since the bases are the same,\nwe can apply the one-to-one property by setting the arguments equal and solving for\nBut what about the equation The one-to-one property does not help us in this instance.\nBefore we can solve an equation like this, we need a method for combining terms on the left side of the equation."
	},
	{
		"chunk_id" : "00001862",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Recall that we use theproduct rule of exponentsto combine the product of powers by adding exponents:\nWe have a similar property for logarithms, called theproduct rule for logarithms, which says that the logarithm of a\nproduct is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the\nexponents. We will use the inverse property to derive the product rule below.\nGiven any real number and positive real numbers and where we will show"
	},
	{
		"chunk_id" : "00001863",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Let and In exponential form, these equations are and It follows that\nNote that repeated applications of the product rule for logarithms allow us to simplify the logarithm of the product of\nany number of factors. For example, consider Using the product rule for logarithms, we can rewrite this\nlogarithm of a product as the sum of logarithms of its factors:\nThe Product Rule for Logarithms\nTheproduct rule for logarithmscan be used to simplify a logarithm of a product by rewriting it as a sum of"
	},
	{
		"chunk_id" : "00001864",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "individual logarithms.\n608 6  Exponential and Logarithmic Functions\n...\nHOW TO\nGiven the logarithm of a product, use the product rule of logarithms to write an equivalent sum of logarithms.\n1. Factor the argument completely, expressing each whole number factor as a product of primes.\n2. Write the equivalent expression by summing the logarithms of each factor.\nEXAMPLE1\nUsing the Product Rule for Logarithms\nExpand\nSolution\nWe begin by factoring the argument completely, expressing as a product of primes."
	},
	{
		"chunk_id" : "00001865",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Next we write the equivalent equation by summing the logarithms of each factor.\nTRY IT #1 Expand\nUsing the Quotient Rule for Logarithms\nFor quotients, we have a similar rule for logarithms. Recall that we use thequotient rule of exponentsto combine the\nquotient of exponents by subtracting: Thequotient rule for logarithmssays that the logarithm of a\nquotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive\nthe quotient rule."
	},
	{
		"chunk_id" : "00001866",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the quotient rule.\nGiven any real number and positive real numbers and where we will show\nLet and In exponential form, these equations are and It follows that\nFor example, to expand we must first express the quotient in lowest terms. Factoring and canceling we\nget,\nNext we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator.\nThen we apply the product rule.\nAccess for free at openstax.org\n6.5  Logarithmic Properties 609"
	},
	{
		"chunk_id" : "00001867",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.5  Logarithmic Properties 609\nThe Quotient Rule for Logarithms\nThequotient rule for logarithmscan be used to simplify a logarithm or a quotient by rewriting it as the difference of\nindividual logarithms.\n...\nHOW TO\nGiven the logarithm of a quotient, use the quotient rule of logarithms to write an equivalent difference of\nlogarithms.\n1. Express the argument in lowest terms by factoring the numerator and denominator and canceling common\nterms."
	},
	{
		"chunk_id" : "00001868",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms.\n2. Write the equivalent expression by subtracting the logarithm of the denominator from the logarithm of the\nnumerator.\n3. Check to see that each term is fully expanded. If not, apply the product rule for logarithms to expand completely.\nEXAMPLE2\nUsing the Quotient Rule for Logarithms\nExpand\nSolution\nFirst we note that the quotient is factored and in lowest terms, so we apply the quotient rule."
	},
	{
		"chunk_id" : "00001869",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule, noting that\nthe prime factors of the factor 15 are 3 and 5.\nAnalysis\nThere are exceptions to consider in this and later examples. First, because denominators must never be zero, this\nexpression is not defined for and Also, since the argument of a logarithm must be positive, we note as we\nobserve the expanded logarithm, that and Combining these conditions is beyond the"
	},
	{
		"chunk_id" : "00001870",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "scope of this section, and we will not consider them here or in subsequent exercises.\nTRY IT #2 Expand\nUsing the Power Rule for Logarithms\nWeve explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as One\nmethod is as follows:\nNotice that we used theproduct rule for logarithmsto find a solution for the example above. By doing so, we have\n610 6  Exponential and Logarithmic Functions"
	},
	{
		"chunk_id" : "00001871",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "610 6  Exponential and Logarithmic Functions\nderived thepower rule for logarithms, which says that the log of a power is equal to the exponent times the log of the\nbase. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to\na power. For example,\nThe Power Rule for Logarithms\nThepower rule for logarithmscan be used to simplify the logarithm of a power by rewriting it as the product of the\nexponent times the logarithm of the base.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001872",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor\nand a logarithm.\n1. Express the argument as a power, if needed.\n2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.\nEXAMPLE3\nExpanding a Logarithm with Powers\nExpand\nSolution\nThe argument is already written as a power, so we identify the exponent, 5, and the base, and rewrite the equivalent"
	},
	{
		"chunk_id" : "00001873",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expression by multiplying the exponent times the logarithm of the base.\nTRY IT #3 Expand\nEXAMPLE4\nRewriting an Expression as a Power before Using the Power Rule\nExpand using the power rule for logs.\nSolution\nExpressing the argument as a power, we get\nNext we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent\ntimes the logarithm of the base.\nTRY IT #4 Expand\nAccess for free at openstax.org\n6.5  Logarithmic Properties 611\nEXAMPLE5"
	},
	{
		"chunk_id" : "00001874",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6.5  Logarithmic Properties 611\nEXAMPLE5\nUsing the Power Rule in Reverse\nRewrite using the power rule for logs to a single logarithm with a leading coefficient of 1.\nSolution\nBecause the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the\nproduct of a number and a logarithm can be written as a power. For the expression we identify the factor, 4, as\nthe exponent and the argument, as the base, and rewrite the product as a logarithm of a power:"
	},
	{
		"chunk_id" : "00001875",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Rewrite using the power rule for logs to a single logarithm with a leading coefficient of 1.\nExpanding Logarithmic Expressions\nTaken together, the product rule, quotient rule, and power rule are often called laws of logs. Sometimes we apply\nmore than one rule in order to simplify an expression. For example:\nWe can use the power rule to expand logarithmic expressions involving negative and fractional exponents. Here is an"
	},
	{
		"chunk_id" : "00001876",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "alternate proof of the quotient rule for logarithms using the fact that a reciprocal is a negative power:\nWe can also apply the product rule to express a sum or difference of logarithms as the logarithm of a product.\nWith practice, we can look at a logarithmic expression and expand it mentally, writing the final answer. Remember,\nhowever, that we can only do this with products, quotients, powers, and rootsnever with addition or subtraction inside\nthe argument of the logarithm.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001877",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the argument of the logarithm.\nEXAMPLE6\nExpanding Logarithms Using Product, Quotient, and Power Rules\nRewrite as a sum or difference of logs.\nSolution\nFirst, because we have a quotient of two expressions, we can use the quotient rule:\nThen seeing the product in the first term, we use the product rule:\nFinally, we use the power rule on the first term:\nTRY IT #6 Expand\n612 6  Exponential and Logarithmic Functions\nEXAMPLE7\nUsing the Power Rule for Logarithms to Simplify the Logarithm of a Radical Expression"
	},
	{
		"chunk_id" : "00001878",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Expand\nSolution\nTRY IT #7 Expand\nQ&A Can we expand\nNo. There is no way to expand the logarithm of a sum or difference inside the argument of the logarithm.\nEXAMPLE8\nExpanding Complex Logarithmic Expressions\nExpand\nSolution\nWe can expand by applying the Product and Quotient Rules.\nTRY IT #8 Expand\nCondensing Logarithmic Expressions\nWe can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a"
	},
	{
		"chunk_id" : "00001879",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "single logarithm. It is important to remember that the logarithms must have the same base to be combined. We will\nlearn later how to change the base of any logarithm before condensing.\n...\nHOW TO\nGiven a sum, difference, or product of logarithms with the same base, write an equivalent expression as a\nsingle logarithm.\n1. Apply the power property first. Identify terms that are products of factors and a logarithm, and rewrite each as\nthe logarithm of a power."
	},
	{
		"chunk_id" : "00001880",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the logarithm of a power.\n2. Next apply the product property. Rewrite sums of logarithms as the logarithm of a product.\n3. Apply the quotient property last. Rewrite differences of logarithms as the logarithm of a quotient.\nAccess for free at openstax.org\n6.5  Logarithmic Properties 613\nEXAMPLE9\nUsing the Product and Quotient Rules to Combine Logarithms\nWrite as a single logarithm.\nSolution\nUsing the product and quotient rules\nThis reduces our original expression to\nThen, using the quotient rule"
	},
	{
		"chunk_id" : "00001881",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then, using the quotient rule\nTRY IT #9 Condense\nEXAMPLE10\nCondensing Complex Logarithmic Expressions\nCondense\nSolution\nWe apply the power rule first:\nNext we apply the product rule to the sum:\nFinally, we apply the quotient rule to the difference:\nTRY IT #10 Rewrite as a single logarithm.\nEXAMPLE11\nRewriting as a Single Logarithm\nRewrite as a single logarithm.\nSolution\nWe apply the power rule first:\nNext we rearrange and apply the product rule to the sum:\n614 6  Exponential and Logarithmic Functions"
	},
	{
		"chunk_id" : "00001882",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "614 6  Exponential and Logarithmic Functions\nFinally, we apply the quotient rule to the difference:\nTRY IT #11 Condense\nEXAMPLE12\nApplying of the Laws of Logs\nRecall that, in chemistry, If the concentration of hydrogen ions in a liquid is doubled, what is the effect\non pH?\nSolution\nSuppose is the original concentration of hydrogen ions, and is the original pH of the liquid. Then If the\nconcentration is doubled, the new concentration is Then the pH of the new liquid is\nUsing the product rule of logs"
	},
	{
		"chunk_id" : "00001883",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using the product rule of logs\nSince the new pH is\nWhen the concentration of hydrogen ions is doubled, the pH decreases by about 0.301.\nTRY IT #12 How does the pH change when the concentration of positive hydrogen ions is decreased by half?\nUsing the Change-of-Base Formula for Logarithms\nMost calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10"
	},
	{
		"chunk_id" : "00001884",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or we use thechange-of-base formulato rewrite the logarithm as the quotient of logarithms of any other base; when\nusing a calculator, we would change them to common or natural logs.\nTo derive the change-of-base formula, we use theone-to-oneproperty andpower rule for logarithms.\nGiven any positive real numbers and where and we show\nLet By exponentiating both sides with base , we arrive at an exponential form, namely It follows\nthat"
	},
	{
		"chunk_id" : "00001885",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that\nFor example, to evaluate using a calculator, we must first rewrite the expression as a quotient of common or\nnatural logs. We will use the common log.\nAccess for free at openstax.org\n6.5  Logarithmic Properties 615\nThe Change-of-Base Formula\nThechange-of-base formulacan be used to evaluate a logarithm with any base.\nFor any positive real numbers and where and\nIt follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of\ncommon or natural logs.\nand\n..."
	},
	{
		"chunk_id" : "00001886",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common or natural logs.\nand\n...\nHOW TO\nGiven a logarithm with the form use the change-of-base formula to rewrite it as a quotient of logs with\nany positive base where\n1. Determine the new base remembering that the common log, has base 10, and the natural log,\nhas base\n2. Rewrite the log as a quotient using the change-of-base formula\na. The numerator of the quotient will be a logarithm with base and argument\nb. The denominator of the quotient will be a logarithm with base and argument\nEXAMPLE13"
	},
	{
		"chunk_id" : "00001887",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE13\nChanging Logarithmic Expressions to Expressions Involving Only Natural Logs\nChange to a quotient of natural logarithms.\nSolution\nBecause we will be expressing as a quotient of natural logarithms, the new base,\nWe rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log\nwith argument 3. The denominator of the quotient will be the natural log with argument 5.\nTRY IT #13 Change to a quotient of natural logarithms."
	},
	{
		"chunk_id" : "00001888",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Can we change common logarithms to natural logarithms?\nYes. Remember that means So,\n616 6  Exponential and Logarithmic Functions\nEXAMPLE14\nUsing the Change-of-Base Formula with a Calculator\nEvaluate using the change-of-base formula with a calculator.\nSolution\nAccording to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our\ncalculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base"
	},
	{
		"chunk_id" : "00001889",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #14 Evaluate using the change-of-base formula.\nMEDIA\nAccess these online resources for additional instruction and practice with laws of logarithms.\nThe Properties of Logarithms(http://openstax.org/l/proplog)\nExpand Logarithmic Expressions(http://openstax.org/l/expandlog)\nEvaluate a Natural Logarithmic Expression(http://openstax.org/l/evaluatelog)\n6.5 SECTION EXERCISES\nVerbal\n1. How does the power rule for logarithms help when 2. What does the change-of-base formula do? Why is"
	},
	{
		"chunk_id" : "00001890",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solving logarithms with the form it useful when using a calculator?\nAlgebraic\nFor the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or\nproduct of logs.\n3. 4. 5.\n6. 7. 8.\nFor the following exercises, condense to a single logarithm if possible.\n9. 10.\n11. 12. 13.\n14.\nAccess for free at openstax.org\n6.5  Logarithmic Properties 617\nFor the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite"
	},
	{
		"chunk_id" : "00001891",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each expression as a sum, difference, or product of logs.\n15. 16. 17.\n18. 19.\nFor the following exercises, condense each expression to a single logarithm using the properties of logarithms.\n20. 21. 22.\n23. 24.\nFor the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.\n25. to base 26. to base\nFor the following exercises, suppose and Use the change-of-base formula along with"
	},
	{
		"chunk_id" : "00001892",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "properties of logarithms to rewrite each expression in terms of and Show the steps for solving.\n27. 28. 29.\nNumeric\nFor the following exercises, use properties of logarithms to evaluate without using a calculator.\n30. 31. 32.\nFor the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs.\nUse a calculator to approximate each to five decimal places.\n33. 34. 35.\n36. 37.\nExtensions"
	},
	{
		"chunk_id" : "00001893",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "33. 34. 35.\n36. 37.\nExtensions\n38. Use the product rule for logarithms to find all 39. Use the quotient rule for logarithms to find all\nvalues such that values such that\nShow the steps Show the steps for solving.\nfor solving.\n618 6  Exponential and Logarithmic Functions\n40. Can the power property of 41. Prove that 42. Does\nlogarithms be derived from for any\nthe power property of Verify the claim\npositive integers and\nexponents using the algebraically.\nequation If not,\nexplain why. If so, show the"
	},
	{
		"chunk_id" : "00001894",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation If not,\nexplain why. If so, show the\nderivation.\n6.6 Exponential and Logarithmic Equations\nLearning Objectives\nIn this section, you will:\nUse like bases to solve exponential equations.\nUse logarithms to solve exponential equations.\nUse the definition of a logarithm to solve logarithmic equations.\nUse the one-to-one property of logarithms to solve logarithmic equations.\nSolve applied problems involving exponential and logarithmic equations."
	},
	{
		"chunk_id" : "00001895",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1 Wild rabbits in Australia. The rabbit population grew so quickly in Australia that the event became known as\nthe rabbit plague. (credit: Richard Taylor, Flickr)\nIn 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia\nhad few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population\nnumbered in the millions."
	},
	{
		"chunk_id" : "00001896",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbered in the millions.\nUncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions.\nEquations resulting from those exponential functions can be solved to analyze and make predictions about exponential\ngrowth. In this section, we will learn techniques for solving exponential functions.\nUsing Like Bases to Solve Exponential Equations\nThe first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions"
	},
	{
		"chunk_id" : "00001897",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tells us that, for any real numbers and where if and only if\nIn other words, when anexponential equationhas the same base on each side, the exponents must be equal. This also\napplies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the\nrules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions\nare one-to-one to set the exponents equal to one another, and solve for the unknown."
	},
	{
		"chunk_id" : "00001898",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For example, consider the equation To solve for we use the division property of exponents to rewrite\nthe right side so that both sides have the common base, Then we apply the one-to-one property of exponents by\nsetting the exponents equal to one another and solving for :\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 619\nUsing the One-to-One Property of Exponential Functions to Solve Exponential Equations\nFor any algebraic expressions and any positive real number\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001899",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an exponential equation with the form where and are algebraic expressions with an\nunknown, solve for the unknown.\n1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form\n2. Use the one-to-one property to set the exponents equal.\n3. Solve the resulting equation, for the unknown.\nEXAMPLE1\nSolving an Exponential Equation with a Common Base\nSolve\nSolution\nTRY IT #1 Solve\nRewriting Equations So All Powers Have the Same Base"
	},
	{
		"chunk_id" : "00001900",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sometimes thecommon basefor an exponential equation is not explicitly shown. In these cases, we simply rewrite the\nterms in the equation as powers with a common base, and solve using the one-to-one property.\nFor example, consider the equation We can rewrite both sides of this equation as a power of Then we\napply the rules of exponents, along with the one-to-one property, to solve for\n620 6  Exponential and Logarithmic Functions\n...\nHOW TO"
	},
	{
		"chunk_id" : "00001901",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an exponential equation with unlike bases, use the one-to-one property to solve it.\n1. Rewrite each side in the equation as a power with a common base.\n2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form\n3. Use the one-to-one property to set the exponents equal.\n4. Solve the resulting equation, for the unknown.\nEXAMPLE2\nSolving Equations by Rewriting Them to Have a Common Base\nSolve\nSolution\nTRY IT #2 Solve\nEXAMPLE3"
	},
	{
		"chunk_id" : "00001902",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve\nSolution\nTRY IT #2 Solve\nEXAMPLE3\nSolving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base\nSolve\nSolution\nTRY IT #3 Solve\nQ&A Do all exponential equations have a solution? If not, how can we tell if there is a solution during the\nproblem-solving process?\nNo. Recall that the range of an exponential function is always positive. While solving the equation, we may\nobtain an expression that is undefined.\nEXAMPLE4\nSolving an Equation with Positive and Negative Powers\nSolve"
	},
	{
		"chunk_id" : "00001903",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve\nSolution\nThis equation has no solution. There is no real value of that will make the equation a true statement because any\npower of a positive number is positive.\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 621\nAnalysis\nFigure 2shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no\nsolution.\nFigure2\nTRY IT #4 Solve\nSolving Exponential Equations Using Logarithms"
	},
	{
		"chunk_id" : "00001904",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Exponential Equations Using Logarithms\nSometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by\ntaking the logarithm of each side. Recall, since is equivalent to we may apply logarithms with the\nsame base on both sides of an exponential equation.\n...\nHOW TO\nGiven an exponential equation in which a common base cannot be found, solve for the unknown.\n1. Apply the logarithm of both sides of the equation."
	},
	{
		"chunk_id" : "00001905",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If one of the terms in the equation has base 10, use the common logarithm.\nb. If none of the terms in the equation has base 10, use the natural logarithm.\n2. Use the rules of logarithms to solve for the unknown.\nEXAMPLE5\nSolving an Equation Containing Powers of Different Bases\nSolve\nSolution\n622 6  Exponential and Logarithmic Functions\nTRY IT #5 Solve\nQ&A Is there any way to solve\nYes. The solution is\nEquations Containinge"
	},
	{
		"chunk_id" : "00001906",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Yes. The solution is\nEquations Containinge\nOne common type of exponential equations are those with base This constant occurs again and again in nature, in\nmathematics, in science, in engineering, and in finance. When we have an equation with a base on either side, we can\nuse thenatural logarithmto solve it.\n...\nHOW TO\nGiven an equation of the form solve for\n1. Divide both sides of the equation by\n2. Apply the natural logarithm of both sides of the equation.\n3. Divide both sides of the equation by\nEXAMPLE6"
	},
	{
		"chunk_id" : "00001907",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Divide both sides of the equation by\nEXAMPLE6\nSolve an Equation of the Formy=Aekt\nSolve\nSolution\nAnalysis\nUsing laws of logs, we can also write this answer in the form If we want a decimal approximation of the\nanswer, we use a calculator.\nTRY IT #6 Solve\nQ&A Does every equation of the form have a solution?\nNo. There is a solution when and when and are either both 0 or neither 0, and they have the\nsame sign. An example of an equation with this form that has no solution is\nEXAMPLE7"
	},
	{
		"chunk_id" : "00001908",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nSolving an Equation That Can Be Simplified to the Formy=Aekt\nSolve\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 623\nSolution\nTRY IT #7 Solve\nExtraneous Solutions\nSometimes the methods used to solve an equation introduce anextraneous solution, which is a solution that is correct\nalgebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the"
	},
	{
		"chunk_id" : "00001909",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be\npositive. If the number we are evaluating in a logarithm function is negative, there is no output.\nEXAMPLE8\nSolving Exponential Functions in Quadratic Form\nSolve\nSolution\nAnalysis\nWhen we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the"
	},
	{
		"chunk_id" : "00001910",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unique property that when a product is zero, one or both of the factors must be zero. We reject the equation\nbecause a positive number never equals a negative number. The solution is not a real number, and in the real\nnumber system this solution is rejected as an extraneous solution.\nTRY IT #8 Solve\nQ&A Does every logarithmic equation have a solution?\nNo. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous\nsolutions."
	},
	{
		"chunk_id" : "00001911",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions.\nUsing the Definition of a Logarithm to Solve Logarithmic Equations\nWe have already seen that everylogarithmic equation is equivalent to the exponential equation We\ncan use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic\nexpression.\nFor example, consider the equation To solve this equation, we can use rules of logarithms\nto rewrite the left side in compact form and then apply the definition of logs to solve for"
	},
	{
		"chunk_id" : "00001912",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "624 6  Exponential and Logarithmic Functions\nUsing the Definition of a Logarithm to Solve Logarithmic Equations\nFor any algebraic expression and real numbers and where\nEXAMPLE9\nUsing Algebra to Solve a Logarithmic Equation\nSolve\nSolution\nTRY IT #9 Solve\nEXAMPLE10\nUsing Algebra Before and After Using the Definition of the Natural Logarithm\nSolve\nSolution\nTRY IT #10 Solve\nEXAMPLE11\nUsing a Graph to Understand the Solution to a Logarithmic Equation\nSolve\nSolution\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001913",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve\nSolution\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 625\nFigure 3represents the graph of the equation. On the graph, thex-coordinate of the point at which the two graphs\nintersect is close to 20. In other words A calculator gives a better approximation:\nFigure3 The graphs of and cross at the point which is approximately (20.0855, 3).\nTRY IT #11 Use a graphing calculator to estimate the approximate solution to the logarithmic equation\nto 2 decimal places."
	},
	{
		"chunk_id" : "00001914",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to 2 decimal places.\nUsing the One-to-One Property of Logarithms to Solve Logarithmic Equations\nAs with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one\nproperty of logarithmic functions tells us that, for any real numbers and any positive real number\nwhere\nFor example,\nSo, if then we can solve for and we get To check, we can substitute into the original equation:"
	},
	{
		"chunk_id" : "00001915",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In other words, when a logarithmic equation has the same base on each side, the arguments\nmust be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with\nlogs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we\nuse the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the\nunknown."
	},
	{
		"chunk_id" : "00001916",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unknown.\nFor example, consider the equation To solve this equation, we can use the rules of\nlogarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for\nTo check the result, substitute into\nUsing the One-to-One Property of Logarithms to Solve Logarithmic Equations\nFor any algebraic expressions and and any positive real number where\n626 6  Exponential and Logarithmic Functions"
	},
	{
		"chunk_id" : "00001917",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "626 6  Exponential and Logarithmic Functions\nNote, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an\nextraneous solution.\n...\nHOW TO\nGiven an equation containing logarithms, solve it using the one-to-one property.\n1. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form\n2. Use the one-to-one property to set the arguments equal.\n3. Solve the resulting equation, for the unknown.\nEXAMPLE12"
	},
	{
		"chunk_id" : "00001918",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE12\nSolving an Equation Using the One-to-One Property of Logarithms\nSolve\nSolution\nAnalysis\nThere are two solutions: or The solution is negative, but it checks when substituted into the original equation\nbecause the argument of the logarithm functions is still positive.\nTRY IT #12 Solve\nSolving Applied Problems Using Exponential and Logarithmic Equations\nIn previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen"
	},
	{
		"chunk_id" : "00001919",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve\nlogarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve\nequations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.\nOne such application is in science, in calculating the time it takes for half of the unstable material in a sample of a"
	},
	{
		"chunk_id" : "00001920",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radioactive substance to decay, called itshalf-life.Table 1lists the half-life for several of the more common radioactive\nsubstances.\nSubstance Use Half-life\ngallium-67 nuclear medicine 80 hours\ncobalt-60 manufacturing 5.3 years\ntechnetium-99m nuclear medicine 6 hours\nTable1\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 627\nSubstance Use Half-life\namericium-241 construction 432 years\ncarbon-14 archeological dating 5,715 years\nuranium-235 atomic power 703,800,000 years\nTable1"
	},
	{
		"chunk_id" : "00001921",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "uranium-235 atomic power 703,800,000 years\nTable1\nWe can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to\ncalculate the amount remaining after a specified time. We can use the formula for radioactive decay:\nwhere\n is the amount initially present\n is the half-life of the substance\n is the time period over which the substance is studied\n is the amount of the substance present after time\nEXAMPLE13"
	},
	{
		"chunk_id" : "00001922",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE13\nUsing the Formula for Radioactive Decay to Find the Quantity of a Substance\nHow long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?\nSolution\nAnalysis\nTen percent of 1000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams.\nTRY IT #13 How long will it take before twenty percent of our 1000-gram sample of uranium-235 has\ndecayed?\nMEDIA"
	},
	{
		"chunk_id" : "00001923",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decayed?\nMEDIA\nAccess these online resources for additional instruction and practice with exponential and logarithmic equations.\nSolving Logarithmic Equations(http://openstax.org/l/solvelogeq)\n628 6  Exponential and Logarithmic Functions\nSolving Exponential Equations with Logarithms(http://openstax.org/l/solveexplog)\n6.6 SECTION EXERCISES\nVerbal\n1. How can an exponential 2. When does an extraneous 3. When can the one-to-one\nequation be solved? solution occur? How can an property of logarithms be"
	},
	{
		"chunk_id" : "00001924",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "extraneous solution be used to solve an equation?\nrecognized? When can it not be used?\nAlgebraic\nFor the following exercises, use like bases to solve the exponential equation.\n4. 5. 6.\n7. 8. 9.\n10.\nFor the following exercises, use logarithms to solve.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23. 24. 25.\n26. 27. 28.\nFor the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.\n29. 30."
	},
	{
		"chunk_id" : "00001925",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "29. 30.\nFor the following exercises, use the definition of a logarithm to solve the equation.\n31. 32. 33.\n34. 35.\nAccess for free at openstax.org\n6.6  Exponential and Logarithmic Equations 629\nFor the following exercises, use the one-to-one property of logarithms to solve.\n36. 37. 38.\n39. 40. 41.\n42. 43.\nFor the following exercises, solve each equation for\n44. 45. 46.\n47. 48. 49.\n50.\nGraphical"
	},
	{
		"chunk_id" : "00001926",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "44. 45. 46.\n47. 48. 49.\n50.\nGraphical\nFor the following exercises, solve the equation for if there is a solution.Then graph both sides of the equation, and\nobserve the point of intersection (if it exists) to verify the solution.\n51. 52. 53.\n54. 55. 56.\n57. 58. 59.\n60. 61. 62.\n63. 64.\nFor the following exercises, solve for the indicated value, and graph the situation showing the solution point.\n65. An account with an initial 66. The formula for measuring 67. The population of a small"
	},
	{
		"chunk_id" : "00001927",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "deposit of earns sound intensity in decibels town is modeled by the\nannual interest, is defined by the equation\ncompounded continuously. equation where is measured in\nHow much will the account where years. In approximately\nbe worth after 20 years? how many years will the\nis the intensity of the\ntowns population reach\nsound in watts per square\nmeter and is\nthe lowest level of sound\nthat the average person\ncan hear. How many\ndecibels are emitted from a\njet plane with a sound\nintensity of watts"
	},
	{
		"chunk_id" : "00001928",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "jet plane with a sound\nintensity of watts\nper square meter?\n630 6  Exponential and Logarithmic Functions\nTechnology\nFor the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm.\nThen use a calculator to approximate the variable to 3 decimal places.\n68. using 69. using the natural 70. using the\nthe common log. log common log\n71. using the 72. using the\ncommon log natural log"
	},
	{
		"chunk_id" : "00001929",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common log natural log\nFor the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the\nnearest ten-thousandth.\n73. 74. 75.\n76. Atmospheric pressure in pounds per square 77. The magnitudeMof an earthquake is represented\ninch is represented by the formula by the equation where is the\nwhere is the number of miles\namount of energy released by the earthquake in\nabove sea level. To the nearest foot, how high is\njoules and is the assigned minimal"
	},
	{
		"chunk_id" : "00001930",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "joules and is the assigned minimal\nthe peak of a mountain with an atmospheric\nmeasure released by an earthquake. To the\npressure of pounds per square inch? (Hint:\nnearest hundredth, what would the magnitude be\nthere are 5280 feet in a mile)\nof an earthquake releasing joules of\nenergy?\nExtensions\n78. Use the definition of a 79. Recall the formula for 80. Recall the compound\nlogarithm along with the continually compounding interest formula\none-to-one property of interest, Use the Use the"
	},
	{
		"chunk_id" : "00001931",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one-to-one property of interest, Use the Use the\nlogarithms to prove that definition of a logarithm\ndefinition of a logarithm\nalong with properties of\nalong with properties of\nlogarithms to solve the\nlogarithms to solve the\nformula for time such\nformula for time\nthat is equal to a single\nlogarithm.\n81. Newtons Law of Cooling states that the\ntemperature of an object at any timetcan be\ndescribed by the equation\nwhere is the\ntemperature of the surrounding environment,"
	},
	{
		"chunk_id" : "00001932",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "temperature of the surrounding environment,\nis the initial temperature of the object, and is the\ncooling rate. Use the definition of a logarithm\nalong with properties of logarithms to solve the\nformula for time such that is equal to a single\nlogarithm.\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 631\n6.7 Exponential and Logarithmic Models\nLearning Objectives\nIn this section, you will:\nModel exponential growth and decay.\nUse Newtons Law of Cooling.\nUse logistic-growth models."
	},
	{
		"chunk_id" : "00001933",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use logistic-growth models.\nChoose an appropriate model for data.\nExpress an exponential model in base .\nFigure1 A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology\ncampus (credit: Georgia Tech Research Institute)\nWe have already explored some basic applications of exponential and logarithmic functions. In this section, we explore\nsome important applications in more depth, including radioactive isotopes and Newtons Law of Cooling."
	},
	{
		"chunk_id" : "00001934",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Modeling Exponential Growth and Decay\nIn real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar\ngeneral function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the\ncase of rapid growth, we may choose the exponential growth function:\nwhere is equal to the value at time zero, is Eulers constant, and is a positive constant that determines the rate"
	},
	{
		"chunk_id" : "00001935",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(percentage) of growth. We may use theexponential growthfunction in applications involvingdoubling time, the time it\ntakes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and\nnatural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we\ndiscuss the logistic equation, the logistic model sometimes fits the data better than the exponential model."
	},
	{
		"chunk_id" : "00001936",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably\nchoose theexponential decaymodel. Again, we have the form where is the starting value, and is\nEulers constant. Now is a negative constant that determines the rate of decay. We may use the exponential decay\nmodel when we are calculatinghalf-life, or the time it takes for a substance to exponentially decay to half of its original"
	},
	{
		"chunk_id" : "00001937",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quantity. We use half-life in applications involving radioactive isotopes.\nIn our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation\nand measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential\ngrowth and decay graphs have a distinctive shape, as we can see inFigure 2andFigure 3. It is important to remember"
	},
	{
		"chunk_id" : "00001938",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that, although parts of each of the two graphs seem to lie on thex-axis, they are really a tiny distance above thex-axis.\n632 6  Exponential and Logarithmic Functions\nFigure2 A graph showing exponential growth. The equation is\nFigure3 A graph showing exponential decay. The equation is\nExponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use"
	},
	{
		"chunk_id" : "00001939",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "orders of magnitude. Theorder of magnitudeis the power of ten, when the number is expressed in scientific notation,\nwith one digit to the left of the decimal. For example, the distance to the nearest star,Proxima Centauri, measured in\nkilometers, is 40,113,497,200,000 kilometers. Expressed in scientific notation, this is So, we could\ndescribe this number as having order of magnitude\nCharacteristics of the Exponential Function,\nAn exponential function with the form has the following characteristics:"
	},
	{
		"chunk_id" : "00001940",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " one-to-one function\n horizontal asymptote:\n domain:  \n range: \n x intercept: none\n y-intercept:\n increasing if (seeFigure 4)\n decreasing if (seeFigure 4)\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 633\nFigure4 An exponential function models exponential growth when and exponential decay when\nEXAMPLE1\nGraphing Exponential Growth\nA population of bacteria doubles every hour. If the culture started with 10 bacteria, graph the population as a function of\ntime.\nSolution"
	},
	{
		"chunk_id" : "00001941",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "time.\nSolution\nWhen an amount grows at a fixed percent per unit time, the growth is exponential. To find we use the fact that is\nthe amount at time zero, so To find use the fact that after one hour the population doubles from\nto The formula is derived as follows\nso Thus the equation we want to graph is The graph is shown inFigure 5.\nFigure5 The graph of\nAnalysis\nThe population of bacteria after ten hours is 10,240. We could describe this amount is being of the order of magnitude"
	},
	{
		"chunk_id" : "00001942",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The population of bacteria after twenty hours is 10,485,760 which is of the order of magnitude so we could say\n634 6  Exponential and Logarithmic Functions\nthat the population has increased by three orders of magnitude in ten hours.\nHalf-Life\nWe now turn toexponential decay. One of the common terms associated with exponential decay, as stated above, is\nhalf-life, the length of time it takes an exponentially decaying quantity to decrease to half its original amount. Every"
	},
	{
		"chunk_id" : "00001943",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radioactive isotope has a half-life, and the process describing the exponential decay of an isotope is called radioactive\ndecay.\nTo find the half-life of a function describing exponential decay, solve the following equation:\nWe find that the half-life depends only on the constant and not on the starting quantity\nThe formula is derived as follows\nSince the time, is positive, must, as expected, be negative. This gives us the half-life formula\n...\nHOW TO\nGiven the half-life, find the decay rate.\n1. Write"
	},
	{
		"chunk_id" : "00001944",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Write\n2. Replace by and replace by the given half-life.\n3. Solve to find Express as an exact value (do not round).\nNote:It is also possible to find the decay rate using\nEXAMPLE2\nFinding the Function that Describes Radioactive Decay\nThe half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time,\nSolution\nThis formula is derived as follows.\nThe function that describes this continuous decay is We observe that the coefficient of\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00001945",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n6.7  Exponential and Logarithmic Models 635\nis negative, as expected in the case of exponential decay.\nTRY IT #1 The half-life of plutonium-244 is 80,000,000 years. Find a function that gives the amount of\nplutonium-244 remaining as a function of time, measured in years.\nRadiocarbon Dating\nThe formula for radioactive decay is important inradiocarbon dating, which is used to calculate the approximate date a"
	},
	{
		"chunk_id" : "00001946",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his\ndiscovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the\nratio of those two isotopes in the air. It is believed to be accurate to within about 1% error for plants or animals that died\nwithin the last 60,000 years."
	},
	{
		"chunk_id" : "00001947",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "within the last 60,000 years.\nCarbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon\ndioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not\nradioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree\nrings and other organic samples of known datesalthough the ratio has changed slightly over the centuries."
	},
	{
		"chunk_id" : "00001948",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the\natmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to\ncarbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be\napproximated.\nSince the half-life of carbon-14 is 5,730 years, the formula for the amount of carbon-14 remaining after years is\nwhere"
	},
	{
		"chunk_id" : "00001949",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where\n is the amount of carbon-14 remaining\n is the amount of carbon-14 when the plant or animal began decaying.\nThis formula is derived as follows:\nTo find the age of an object, we solve this equation for\nOut of necessity, we neglect here the many details that a scientist takes into consideration when doing carbon-14 dating,\nand we only look at the basic formula. The ratio of carbon-14 to carbon-12 in the atmosphere is approximately"
	},
	{
		"chunk_id" : "00001950",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0.0000000001%. Let be the ratio of carbon-14 to carbon-12 in the organic artifact or fossil to be dated, determined by a\nmethod called liquid scintillation. From the equation we know the ratio of the percentage of\ncarbon-14 in the object we are dating to the initial amount of carbon-14 in the object when it was formed is\nWe solve this equation for to get\n636 6  Exponential and Logarithmic Functions\n...\nHOW TO\nGiven the percentage of carbon-14 in an object, determine its age."
	},
	{
		"chunk_id" : "00001951",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Express the given percentage of carbon-14 as an equivalent decimal,\n2. Substitute forkin the equation and solve for the age,\nEXAMPLE3\nFinding the Age of a Bone\nA bone fragment is found that contains 20% of its original carbon-14. To the nearest year, how old is the bone?\nSolution\nWe substitute for in the equation and solve for\nThe bone fragment is about 13,301 years old.\nAnalysis\nThe instruments that measure the percentage of carbon-14 are extremely sensitive and, as we mention above, a scientist"
	},
	{
		"chunk_id" : "00001952",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate to about 1%,\nso this age should be given as\nTRY IT #2 Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take\nmore or less than 230 years until only 1 milligram remains?\nCalculating Doubling Time\nFor decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we"
	},
	{
		"chunk_id" : "00001953",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity\nto double is called thedoubling time.\nGiven the basicexponential growthequation doubling time can be found by solving for when the original\nquantity has doubled, that is, by solving\nThe formula is derived as follows:\nThus the doubling time is\nEXAMPLE4\nFinding a Function That Describes Exponential Growth"
	},
	{
		"chunk_id" : "00001954",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "According to Moores Law, the doubling time for the number of transistors that can be put on a computer chip is\napproximately two years. Give a function that describes this behavior.\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 637\nSolution\nThe formula is derived as follows:\nThe function is\nTRY IT #3 Recent data suggests that, as of 2013, the rate of growth predicted by Moores Law no longer"
	},
	{
		"chunk_id" : "00001955",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "holds. Growth has slowed to a doubling time of approximately three years. Find the new function\nthat takes that longer doubling time into account.\nUsing Newtons Law of Cooling\nExponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower\ntemperature, the objects temperature will decrease exponentially, leveling off as it approaches the surrounding air"
	},
	{
		"chunk_id" : "00001956",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the\ntemperature of the surrounding air. Unless the room temperature is zero, this will correspond to avertical shiftof the\ngenericexponential decayfunction. This translation leads toNewtons Law of Cooling, the scientific formula for\ntemperature as a function of time as an objects temperature is equalized with the ambient temperature\nThis formula is derived as follows:\nNewtons Law of Cooling"
	},
	{
		"chunk_id" : "00001957",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Newtons Law of Cooling\nThe temperature of an object, in surrounding air with temperature will behave according to the formula\nwhere\n is time\n is the difference between the initial temperature of the object and the surroundings\n is a constant, the continuous rate of cooling of the object\n...\nHOW TO\nGiven a set of conditions, apply Newtons Law of Cooling.\n1. Set equal to they-coordinate of the horizontal asymptote (usually the ambient temperature)."
	},
	{
		"chunk_id" : "00001958",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Substitute the given values into the continuous growth formula to find the parameters and\n3. Substitute in the desired time to find the temperature or the desired temperature to find the time.\n638 6  Exponential and Logarithmic Functions\nEXAMPLE5\nUsing Newtons Law of Cooling\nA cheesecake is taken out of the oven with an ideal internal temperature of and is placed into a refrigerator.\nAfter 10 minutes, the cheesecake has cooled to If we must wait until the cheesecake has cooled to before we"
	},
	{
		"chunk_id" : "00001959",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "eat it, how long will we have to wait?\nSolution\nBecause the surrounding air temperature in the refrigerator is 35 degrees, the cheesecakes temperature will decay\nexponentially toward 35, following the equation\nWe know the initial temperature was 165, so\nWe were given another data point, which we can use to solve for\nThis gives us the equation for the cooling of the cheesecake:\nNow we can solve for the time it will take for the temperature to cool to 70 degrees."
	},
	{
		"chunk_id" : "00001960",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "It will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to\nTRY IT #4 A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the\ntemperature has risen to 45 degrees. How long will it take for the temperature to rise to 60\ndegrees?\nUsing Logistic Growth Models\nExponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall"
	},
	{
		"chunk_id" : "00001961",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the\nnumber of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-\nhalf-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time.\nEventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For"
	},
	{
		"chunk_id" : "00001962",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "this reason, it is often better to use a model with an upper bound instead of anexponential growthmodel, though the\nexponential growth model is still useful over a short term, before approaching the limiting value.\nThelogistic growth modelis approximately exponential at first, but it has a reduced rate of growth as the output\napproaches the models upper bound, called thecarrying capacity. For constants and the logistic growth of a\npopulation over time is represented by the model"
	},
	{
		"chunk_id" : "00001963",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "population over time is represented by the model\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 639\nThe graph inFigure 6shows how the growth rate changes over time. The graph increases from left to right, but the\ngrowth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase\ndecreases.\nFigure6\nLogistic Growth\nThe logistic growth model is\nwhere\n is the initial value\n is thecarrying capacity, orlimiting value"
	},
	{
		"chunk_id" : "00001964",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " is thecarrying capacity, orlimiting value\n is a constant determined by the rate of growth.\nEXAMPLE6\nUsing the Logistic-Growth Model\nAn influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people\nwho have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it\nspreads. These two factors make the logistic model a good one to study the spread of communicable diseases. And,"
	},
	{
		"chunk_id" : "00001965",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "clearly, there is a maximum value for the number of people infected: the entire population.\nFor example, at time there is one person in a community of 1,000 people who has the flu. So, in that community, at\nmost 1,000 people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant\nis Estimate the number of people in this community who will have had this flu after ten days. Predict how"
	},
	{
		"chunk_id" : "00001966",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "many people in this community will have had this flu after a long period of time has passed.\nSolution\nWe substitute the given data into the logistic growth model\nBecause at most 1,000 people, the entire population of the community, can get the flu, we know the limiting value is\nTo find we use the formula that the number of cases at time is from which it follows that\nThis model predicts that, after ten days, the number of people who have had the flu is"
	},
	{
		"chunk_id" : "00001967",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because the actual number must be a whole number (a person has either had the flu or\n640 6  Exponential and Logarithmic Functions\nnot) we round to 294. In the long term, the number of people who will contract the flu is the limiting value,\nAnalysis\nRemember that, because we are dealing with a virus, we cannot predict with certainty the number of people infected.\nThe model only approximates the number of people infected and will not give us exact or actual values."
	},
	{
		"chunk_id" : "00001968",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The graph inFigure 7gives a good picture of how this model fits the data.\nFigure7 The graph of\nTRY IT #5 Using the model inExample 6, estimate the number of cases of flu on day 15.\nChoosing an Appropriate Model for Data\nNow that we have discussed various mathematical models, we need to learn how to choose the appropriate model for\nthe raw data we have. Many factors influence the choice of a mathematical model, among which are experience,"
	},
	{
		"chunk_id" : "00001969",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "scientific laws, and patterns in the data itself. Not all data can be described by elementary functions. Sometimes, a\nfunction is chosen that approximates the data over a given interval. For instance, suppose data were gathered on the\nnumber of homes bought in the United States from the years 1960 to 2013. After plotting these data in a scatter plot, we\nnotice that the shape of the data from the years 2000 to 2013 follow a logarithmic curve. We could restrict the interval"
	},
	{
		"chunk_id" : "00001970",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from 2000 to 2010, apply regression analysis using a logarithmic model, and use it to predict the number of home buyers\nfor the year 2015.\nThree kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and\nlogarithmic functions. If the data lies on a straight line, or seems to lie approximately along a straight line, a linear model\nmay be best. If the data is non-linear, we often consider an exponential or logarithmic model, though other models, such"
	},
	{
		"chunk_id" : "00001971",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as quadratic models, may also be considered.\nIn choosing between an exponential model and a logarithmic model, we look at the way the data curves. This is called\nthe concavity. If we draw a line between two data points, and all (or most) of the data between those two points lies\nabove that line, we say the curve is concave down. We can think of it as a bowl that bends downward and therefore"
	},
	{
		"chunk_id" : "00001972",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cannot hold water. If all (or most) of the data between those two points lies below the line, we say the curve is concave\nup. In this case, we can think of a bowl that bends upward and can therefore hold water. An exponential curve, whether\nrising or falling, whether representing growth or decay, is always concave up away from its horizontal asymptote. A\nlogarithmic curve is always concave away from its vertical asymptote. In the case of positive data, which is the most"
	},
	{
		"chunk_id" : "00001973",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common case, an exponential curve is always concave up, and a logarithmic curve always concave down.\nA logistic curve changes concavity. It starts out concave up and then changes to concave down beyond a certain point,\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 641\ncalled a point of inflection.\nAfter using the graph to help us choose a type of function to use as a model, we substitute points, and solve to find the"
	},
	{
		"chunk_id" : "00001974",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parameters. We reduce round-off error by choosing points as far apart as possible.\nEXAMPLE7\nChoosing a Mathematical Model\nDoes a linear, exponential, logarithmic, or logistic model best fit the values listed inTable 1? Find the model, and use a\ngraph to check your choice.\n1 2 3 4 5 6 7 8 9\n0 1.386 2.197 2.773 3.219 3.584 3.892 4.159 4.394\nTable1\nSolution\nFirst, plot the data on a graph as inFigure 8. For the purpose of graphing, round the data to two decimal places.\nFigure8"
	},
	{
		"chunk_id" : "00001975",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nClearly, the points do not lie on a straight line, so we reject a linear model. If we draw a line between any two of the\npoints, most or all of the points between those two points lie above the line, so the graph is concave down, suggesting a\nlogarithmic model. We can try Plugging in the first point, gives We reject the case that\n(if it were, all outputs would be 0), so we know Thus and Next we can use the point\nto solve for\nBecause an appropriate model for the data is"
	},
	{
		"chunk_id" : "00001976",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because an appropriate model for the data is\nTo check the accuracy of the model, we graph the function together with the given points as inFigure 9.\n642 6  Exponential and Logarithmic Functions\nFigure9 The graph of\nWe can conclude that the model is a good fit to the data.\nCompareFigure 9to the graph of shown inFigure 10.\nFigure10 The graph of\nThe graphs appear to be identical when A quick check confirms this conclusion: for"
	},
	{
		"chunk_id" : "00001977",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "However, if the graph of includes a extra branch, as shown inFigure 11. This occurs because, while\ncannot have negative values in the domain (as such values would force the argument to be negative), the\nfunction can have negative domain values.\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 643\nFigure11\nTRY IT #6 Does a linear, exponential, or logarithmic model best fit the data inTable 2? Find the model.\n1 2 3 4 5 6 7 8 9"
	},
	{
		"chunk_id" : "00001978",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 2 3 4 5 6 7 8 9\n3.297 5.437 8.963 14.778 24.365 40.172 66.231 109.196 180.034\nTable2\nExpressing an Exponential Model in Base\nWhile powers and logarithms of any base can be used in modeling, the two most common bases are and In science\nand mathematics, the base is often preferred. We can use laws of exponents and laws of logarithms to change any\nbase to base\n...\nHOW TO\nGiven a model with the form change it to the form\n1. Rewrite as\n2. Use the power rule of logarithms to rewrite y as"
	},
	{
		"chunk_id" : "00001979",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Note that and in the equation\nEXAMPLE8\nChanging to basee\nChange the function so that this same function is written in the form\nSolution\nThe formula is derived as follows\nTRY IT #7 Change the function to one having as the base.\nMEDIA\nAccess these online resources for additional instruction and practice with exponential and logarithmic models.\n644 6  Exponential and Logarithmic Functions\nLogarithm Application  pH(http://openstax.org/l/logph)"
	},
	{
		"chunk_id" : "00001980",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Exponential Model  Age Using Half-Life(http://openstax.org/l/expmodelhalf)\nNewtons Law of Cooling(http://openstax.org/l/newtoncooling)\nExponential Growth Given Doubling Time(http://openstax.org/l/expgrowthdbl)\nExponential Growth  Find Initial Amount Given Doubling Time(http://openstax.org/l/initialdouble)\n6.7 SECTION EXERCISES\nVerbal\n1. With what kind of 2. What is carbon dating? Why 3. With what kind of\nexponential model would does it work? Give an exponential model would"
	},
	{
		"chunk_id" : "00001981",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "half-lifebe associated? What example in which carbon doubling timebe\nrole does half-life play in dating would be useful. associated? What role does\nthese models? doubling time play in these\nmodels?\n4. Define Newtons Law of 5. What is an order of\nCooling. Then name at least magnitude? Why are orders\nthree real-world situations of magnitude useful? Give\nwhere Newtons Law of an example to explain.\nCooling would be applied.\nNumeric\n6. The temperature of an\nobject in degrees Fahrenheit\naftertminutes is"
	},
	{
		"chunk_id" : "00001982",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "object in degrees Fahrenheit\naftertminutes is\nrepresented by the equation\nTo\nthe nearest degree, what is\nthe temperature of the\nobject after one and a half\nhours?\nFor the following exercises, use the logistic growth model\n7. Find and interpret 8. Find and interpret 9. Find the carrying capacity.\nRound to the nearest tenth. Round to the nearest tenth.\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 645\n10. Graph the model. 11. Determine whether the 12. Rewrite"
	},
	{
		"chunk_id" : "00001983",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "data from the table could as an exponential equation\nbest be represented as a with base to five decimal\nfunction that is linear, places.\nexponential, or\nlogarithmic. Then write a\nformula for a model that\nrepresents the data.\n2 0.694\n1 0.833\n0 1\n1 1.2\n2 1.44\n3 1.728\n4 2.074\n5 2.488\n646 6  Exponential and Logarithmic Functions\nTechnology\nFor the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter"
	},
	{
		"chunk_id" : "00001984",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.\n13. 14. 15.\n1 2 1 2.4 4 9.429\n2 4.079 2 2.88 5 9.972\n3 5.296 3 3.456 6 10.415\n4 6.159 4 4.147 7 10.79\n5 6.828 5 4.977 8 11.115\n6 7.375 6 5.972 9 11.401\n7 7.838 7 7.166 10 11.657\n8 8.238 8 8.6 11 11.889\n9 8.592 9 10.32 12 12.101\n10 8.908 10 12.383 13 12.295\n16.\n1.25 5.75\n2.25 8.75\n3.56 12.68\n4.2 14.6\n5.65 18.95\n6.75 22.25\n7.25 23.75\n8.6 27.8\n9.25 29.75\n10.5 33.5"
	},
	{
		"chunk_id" : "00001985",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.25 23.75\n8.6 27.8\n9.25 29.75\n10.5 33.5\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 647\nFor the following exercises, use a graphing calculator and this scenario: the population of a fish farm in years is\nmodeled by the equation\n17. Graph the function. 18. What is the initial 19. To the nearest tenth, what\npopulation of fish? is the doubling time for the\nfish population?\n20. To the nearest whole 21. To the nearest tenth, how 22. What is the carrying"
	},
	{
		"chunk_id" : "00001986",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number, what will the fish long will it take for the capacity for the fish\npopulation be after population to reach population? Justify your\nyears? answer using the graph of\nExtensions\n23. A substance has a half-life 24. The formula for an 25. Recall the formula for\nof 2.045 minutes. If the increasing population is calculating the magnitude\ninitial amount of the given by of an earthquake,\nsubstance was 132.8 where is the initial Show\ngrams, how many half-lives population and\neach step for solving this"
	},
	{
		"chunk_id" : "00001987",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each step for solving this\nwill have passed before the Derive a general formula\nequation algebraically for\nsubstance decays to 8.3 for the timetit takes for\nthe seismic moment\ngrams? What is the total the population to increase\ntime of decay? by a factor ofM.\n26. What is they-intercept of 27. Prove that for\nthe logistic growth model positive\nShow the\nsteps for calculation. What\ndoes this point tell us\nabout the population?\nReal-World Applications"
	},
	{
		"chunk_id" : "00001988",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "about the population?\nReal-World Applications\nFor the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by\nabout 30% each hour.\n28. To the nearest hour, what 29. Write an exponential 30. Using the model found in\nis the half-life of the drug? model representing the the previous exercise, find\namount of the drug and interpret the\nremaining in the patients result. Round to the\nsystem after hours. Then nearest hundredth.\nuse the formula to find the"
	},
	{
		"chunk_id" : "00001989",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "use the formula to find the\namount of the drug that\nwould remain in the\npatients system after 3\nhours. Round to the\nnearest milligram.\n648 6  Exponential and Logarithmic Functions\nFor the following exercises, use this scenario: A tumor is injected with grams of Iodine-125, which has a decay rate of\nper day.\n31. To the nearest day, how 32. Write an exponential 33. A scientist begins with\nlong will it take for half of model representing the grams of a radioactive"
	},
	{
		"chunk_id" : "00001990",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the Iodine-125 to decay? amount of Iodine-125 substance. After\nremaining in the tumor minutes, the sample has\nafter days. Then use the decayed to grams.\nformula to find the amount Rounding to five decimal\nof Iodine-125 that would places, write an\nremain in the tumor after exponential equation\n60 days. Round to the representing this situation.\nnearest tenth of a gram. To the nearest minute,\nwhat is the half-life of this\nsubstance?"
	},
	{
		"chunk_id" : "00001991",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "what is the half-life of this\nsubstance?\n34. The half-life of Radium-226 35. The half-life of Erbium-165 36. A wooden artifact from an\nis years. What is the is hours. What is the archeological dig contains\nannual decay rate? Express hourly decay rate? Express 60 percent of the\nthe decimal result to four the decimal result to four carbon-14 that is present in\ndecimal places and the decimal places and the living trees. To the nearest\npercentage to two decimal percentage to two decimal year, about how many"
	},
	{
		"chunk_id" : "00001992",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "places. places. years old is the artifact?\n(The half-life of carbon-14\nis years.)\n37. A research student is\nworking with a culture of\nbacteria that doubles in\nsize every twenty minutes.\nThe initial population count\nwas bacteria.\nRounding to five decimal\nplaces, write an\nexponential equation\nrepresenting this situation.\nTo the nearest whole\nnumber, what is the\npopulation size after\nhours?\nFor the following exercises, use this scenario: A biologist recorded a count of bacteria present in a culture after 5"
	},
	{
		"chunk_id" : "00001993",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minutes and 1000 bacteria present after 20 minutes.\n38. To the nearest whole 39. Rounding to six decimal\nnumber, what was the places, write an\ninitial population in the exponential equation\nculture? representing this situation.\nTo the nearest minute, how\nlong did it take the\npopulation to double?\nAccess for free at openstax.org\n6.7  Exponential and Logarithmic Models 649\nFor the following exercises, use this scenario: A pot of warm soup with an internal temperature of Fahrenheit was"
	},
	{
		"chunk_id" : "00001994",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "taken off the stove to cool in a room. After fifteen minutes, the internal temperature of the soup was\n40. Use Newtons Law of 41. To the nearest minute, how 42. To the nearest degree,\nCooling to write a formula long will it take the soup to what will the temperature\nthat models this situation. cool to be after and a half hours?\nFor the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of"
	},
	{
		"chunk_id" : "00001995",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and is allowed to cool in a room. After half an hour, the internal temperature of the turkey is\n43. Write a formula that 44. To the nearest degree, 45. To the nearest minute, how\nmodels this situation. what will the temperature long will it take the turkey\nbe after 50 minutes? to cool to\nFor the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the\nnearest thousandth.\n46. 47."
	},
	{
		"chunk_id" : "00001996",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nearest thousandth.\n46. 47.\n48. Plot each set of approximate values of intensity of 49. Recall the formula for calculating the magnitude\nsounds on a logarithmic scale: Whisper: of an earthquake, One\nVacuum: Jet:\nearthquake has magnitude on the MMS scale.\nIf a second earthquake has times as much\nenergy as the first, find the magnitude of the\nsecond quake. Round to the nearest hundredth.\nFor the following exercises, use this scenario: The equation models the number of people in a town"
	},
	{
		"chunk_id" : "00001997",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "who have heard a rumor aftertdays.\n50. How many people started 51. To the nearest whole 52. As increases without\nthe rumor? number, how many people bound, what value does\nwill have heard the rumor approach? Interpret\nafter 3 days? your answer.\nFor the following exercise, choose the correct answer choice.\n53. A doctor injects a patient with 13 milligrams of\nradioactive dye that decays exponentially. After 12\nminutes, there are 4.75 milligrams of dye\nremaining in the patients system. Which is an"
	},
	{
		"chunk_id" : "00001998",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "remaining in the patients system. Which is an\nappropriate model for this situation?\n \n\n\n650 6  Exponential and Logarithmic Functions\n6.8 Fitting Exponential Models to Data\nLearning Objectives\nIn this section, you will:\nBuild an exponential model from data.\nBuild a logarithmic model from data.\nBuild a logistic model from data.\nIn previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a"
	},
	{
		"chunk_id" : "00001999",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points\nexactly. In this section, we use a modeling technique calledregression analysisto find a curve that models data collected\nfrom real-world observations. Withregression analysis, we dont expect all the points to lie perfectly on the curve. The\nidea is to find a model that best fits the data. Then we use the model to make predictions about future events."
	},
	{
		"chunk_id" : "00002000",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Do not be confused by the wordmodel. In mathematics, we often use the termsfunction,equation, andmodel\ninterchangeably, even though they each have their own formal definition. The termmodelis typically used to indicate\nthat the equation or function approximates a real-world situation.\nWe will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having"
	},
	{
		"chunk_id" : "00002001",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of\ntheir graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each\nregression model is presented, key features and definitions of its associated function are included for review. Take a\nmoment to rethink each of these functions, reflect on the work weve done so far, and then explore the ways regression\nis used to model real-world phenomena."
	},
	{
		"chunk_id" : "00002002",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is used to model real-world phenomena.\nBuilding an Exponential Model from Data\nAs weve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment\ngrowth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these\nphenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But"
	},
	{
		"chunk_id" : "00002003",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thats not the whole story. Its thewaydata increase or decrease that helps us determine whether it is best modeled by\nan exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use\nexponential regression, so lets review exponential growth and decay.\nRecall that exponential functions have the form or When performing regression analysis, we use the"
	},
	{
		"chunk_id" : "00002004",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form most commonly used on graphing utilities, Take a moment to reflect on the characteristics weve already\nlearned about the exponential function (assume\n must be greater than zero and not equal to one.\n The initial value of the model is\n If the function models exponential growth. As increases, the outputs of the model increase slowly at\nfirst, but then increase more and more rapidly, without bound.\n If the function modelsexponential decay. As increases, the outputs for the model decrease rapidly"
	},
	{
		"chunk_id" : "00002005",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "at first and then level off to become asymptotic to thex-axis. In other words, the outputs never become equal to\nor less than zero.\nAs part of the results, your calculator will display a number known as thecorrelation coefficient, labeled by the variable\nor (You may have to change the calculators settings for these to be shown.) The values are an indication of the\ngoodness of fit of the regression equation to the data. We more commonly use the value of instead of but the"
	},
	{
		"chunk_id" : "00002006",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "closer either value is to 1, the better the regression equation approximates the data.\nExponential Regression\nExponential regressionis used to model situations in which growth begins slowly and then accelerates rapidly\nwithout bound, or where decay begins rapidly and then slows down to get closer and closer to zero. We use the\ncommand ExpReg on a graphing utility to fit an exponential function to a set of data points. This returns an\nequation of the form,\nNote that:\n must be non-negative."
	},
	{
		"chunk_id" : "00002007",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Note that:\n must be non-negative.\n when we have an exponential growth model.\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 651\n when we have an exponential decay model.\n...\nHOW TO\nGiven a set of data, perform exponential regression using a graphing utility.\n1. Use theSTATthenEDITmenu to enter given data.\na. Clear any existing data from the lists.\nb. List the input values in the L1 column.\nc. List the output values in the L2 column."
	},
	{
		"chunk_id" : "00002008",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "c. List the output values in the L2 column.\n2. Graph and observe a scatter plot of the data using theSTATPLOTfeature.\na. UseZOOM[9] to adjust axes to fit the data.\nb. Verify the data follow an exponential pattern.\n3. Find the equation that models the data.\na. Select ExpReg from theSTATthenCALCmenu.\nb. Use the values returned foraandbto record the model,\n4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.\nEXAMPLE1"
	},
	{
		"chunk_id" : "00002009",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nUsing Exponential Regression to Fit a Model to Data\nIn 2007, a university study was published investigating the crash risk of alcohol impaired driving. Data from 2,871\ncrashes were used to measure the association of a persons blood alcohol level (BAC) with the risk of being in an\naccident.Table 1shows results from the study9. Therelative riskis a measure of how many times more likely a person is"
	},
	{
		"chunk_id" : "00002010",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to crash. So, for example, a person with a BAC of 0.09 is 3.54 times as likely to crash as a person who has not been\ndrinking alcohol.\nBAC 0 0.01 0.03 0.05 0.07 0.09\nRelative Risk of Crashing 1 1.03 1.06 1.38 2.09 3.54\nBAC 0.11 0.13 0.15 0.17 0.19 0.21\nRelative Risk of Crashing 6.41 12.6 22.1 39.05 65.32 99.78\nTable1\na. Let represent the BAC level, and let represent the corresponding relative risk. Use exponential regression to fit a\nmodel to these data."
	},
	{
		"chunk_id" : "00002011",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "model to these data.\nb. After 6 drinks, a person weighing 160 pounds will have a BAC of about How many times more likely is a person\nwith this weight to crash if they drive after having a 6-pack of beer? Round to the nearest hundredth.\nSolution\na. Using theSTATthenEDITmenu on a graphing utility, list theBACvalues in L1 and the relative risk values in L2.\nThen use theSTATPLOTfeature to verify that the scatterplot follows the exponential pattern shown inFigure 1:"
	},
	{
		"chunk_id" : "00002012",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9 Source:Indiana University Center for Studies of Law in Action, 2007\n652 6  Exponential and Logarithmic Functions\nFigure1\nUse the ExpReg command from theSTATthenCALCmenu to obtain the exponential model,\nConverting from scientific notation, we have:\nNotice that which indicates the model is a good fit to the data. To see this, graph the model in the same\nwindow as the scatterplot to verify it is a good fit as shown inFigure 2:\nFigure2"
	},
	{
		"chunk_id" : "00002013",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nb. Use the model to estimate the risk associated with a BAC of Substitute for in the model and solve for\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 653\nIf a 160-pound person drives after having 6 drinks, they are about 26.35 times more likely to crash than if driving\nwhile sober.\nTRY IT #1 Table 2shows a recent graduates credit card balance each month after graduation.\nMonth 1 2 3 4 5 6 7 8\nDebt ($) 620.00 761.88 899.80 1039.93 1270.63 1589.04 1851.31 2154.92\nTable2"
	},
	{
		"chunk_id" : "00002014",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table2\n Use exponential regression to fit a model to these data.\n If spending continues at this rate, what will the graduates credit card debt be one year after\ngraduating?\nQ&A Is it reasonable to assume that an exponential regression model will represent a situation\nindefinitely?\nNo. Remember that models are formed by real-world data gathered for regression. It is usually\nreasonable to make estimates within the interval of original observation (interpolation). However, when a"
	},
	{
		"chunk_id" : "00002015",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "model is used to make predictions, it is important to use reasoning skills to determine whether the model\nmakes sense for inputs far beyond the original observation interval (extrapolation).\nBuilding a Logarithmic Model from Data\nJust as with exponential functions, there are many real-world applications for logarithmic functions: intensity of sound,\npH levels of solutions, yields of chemical reactions, production of goods, and growth of infants. As with exponential"
	},
	{
		"chunk_id" : "00002016",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "models, data modeled by logarithmic functions are either always increasing or always decreasing as time moves\nforward. Again, it is thewaythey increase or decrease that helps us determine whether alogarithmic modelis best.\nRecall that logarithmic functions increase or decrease rapidly at first, but then steadily slow as time moves on. By\nreflecting on the characteristics weve already learned about this function, we can better analyze real world situations"
	},
	{
		"chunk_id" : "00002017",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that reflect this type of growth or decay. When performing logarithmicregression analysis, we use the form of the\nlogarithmic function most commonly used on graphing utilities, For this function\n All input values, must be greater than zero.\n The point is on the graph of the model.\n If the model is increasing. Growth increases rapidly at first and then steadily slows over time.\n If the model is decreasing. Decay occurs rapidly at first and then steadily slows over time.\nLogarithmic Regression"
	},
	{
		"chunk_id" : "00002018",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Logarithmic Regression\nLogarithmic regressionis used to model situations where growth or decay accelerates rapidly at first and then slows\nover time. We use the command LnReg on a graphing utility to fit a logarithmic function to a set of data points. This\nreturns an equation of the form,\nNote that\n all input values, must be non-negative.\n when the model is increasing.\n when the model is decreasing.\n654 6  Exponential and Logarithmic Functions\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002019",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a set of data, perform logarithmic regression using a graphing utility.\n1. Use theSTATthenEDITmenu to enter given data.\na. Clear any existing data from the lists.\nb. List the input values in the L1 column.\nc. List the output values in the L2 column.\n2. Graph and observe a scatter plot of the data using theSTATPLOTfeature.\na. UseZOOM[9] to adjust axes to fit the data.\nb. Verify the data follow a logarithmic pattern.\n3. Find the equation that models the data."
	},
	{
		"chunk_id" : "00002020",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Find the equation that models the data.\na. Select LnReg from theSTATthenCALCmenu.\nb. Use the values returned foraandbto record the model,\n4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.\nEXAMPLE2\nUsing Logarithmic Regression to Fit a Model to Data\nDue to advances in medicine and higher standards of living, life expectancy has been increasing in most developed\ncountries since the beginning of the 20th century."
	},
	{
		"chunk_id" : "00002021",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table 3shows the average life expectancies, in years, of Americans from 1900201010.\nYear 1900 1910 1920 1930 1940 1950\nLife Expectancy(Years) 47.3 50.0 54.1 59.7 62.9 68.2\nYear 1960 1970 1980 1990 2000 2010\nLife Expectancy(Years) 69.7 70.8 73.7 75.4 76.8 78.7\nTable3\n Let represent time in decades starting with for the year 1900, for the year 1910, and so on. Let\nrepresent the corresponding life expectancy. Use logarithmic regression to fit a model to these data."
	},
	{
		"chunk_id" : "00002022",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Use the model to predict the average American life expectancy for the year 2030.\n10 Source:Center for Disease Control and Prevention, 2013\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 655\nSolution\n Using theSTATthenEDITmenu on a graphing utility, list the years using values 112 in L1 and the\ncorresponding life expectancy in L2. Then use theSTATPLOTfeature to verify that the scatterplot follows a\nlogarithmic pattern as shown inFigure 3:\nFigure3"
	},
	{
		"chunk_id" : "00002023",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "logarithmic pattern as shown inFigure 3:\nFigure3\nUse the LnReg command from theSTATthenCALCmenu to obtain the logarithmic model,\nNext, graph the model in the same window as the scatterplot to verify it is a good fit as shown inFigure 4:\nFigure4\n To predict the life expectancy of an American in the year 2030, substitute for the in the model and solve\nfor\nIf life expectancy continues to increase at this pace, the average life expectancy of an American will be 79.1 by the\nyear 2030."
	},
	{
		"chunk_id" : "00002024",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "year 2030.\n656 6  Exponential and Logarithmic Functions\nTRY IT #2 Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time\nmoved on.Table 4shows the number of games sold, in thousands, from the years 20002010.\nYear 2000 2001 2002 2003 2004 2005\nNumber Sold (thousands) 142 149 154 155 159 161\nYear 2006 2007 2008 2009 2010 -\nNumber Sold (thousands) 163 164 164 166 167 -\nTable4\n Let represent time in years starting with for the year 2000. Let represent the"
	},
	{
		"chunk_id" : "00002025",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of games sold in thousands. Use logarithmic regression to fit a model to these data.\n If games continue to sell at this rate, how many games will sell in 2015? Round to the nearest\nthousand.\nBuilding a Logistic Model from Data\nLike exponential and logarithmic growth, logistic growth increases over time. One of the most notable differences with\nlogistic growth models is that, at a certain point, growth steadily slows and the function approaches an upper bound, or"
	},
	{
		"chunk_id" : "00002026",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "limiting value. Because of this, logistic regression is best for modeling phenomena where there are limits in expansion,\nsuch as availability of living space or nutrients.\nIt is worth pointing out that logistic functions actually model resource-limited exponential growth. There are many\nexamples of this type of growth in real-world situations, including population growth and spread of disease, rumors, and"
	},
	{
		"chunk_id" : "00002027",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "even stains in fabric. When performing logisticregression analysis, we use the form most commonly used on graphing\nutilities:\nRecall that:\n is the initial value of the model.\n when the model increases rapidly at first until it reaches its point of maximum growth rate, At\nthat point, growth steadily slows and the function becomes asymptotic to the upper bound\n is the limiting value, sometimes called thecarrying capacity, of the model.\nLogistic Regression"
	},
	{
		"chunk_id" : "00002028",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Logistic Regression\nLogistic regressionis used to model situations where growth accelerates rapidly at first and then steadily slows to an\nupper limit. We use the command Logistic on a graphing utility to fit a logistic function to a set of data points. This\nreturns an equation of the form\nNote that\n The initial value of the model is\n Output values for the model grow closer and closer to as time increases.\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 657\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002029",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a set of data, perform logistic regression using a graphing utility.\n1. Use theSTATthenEDITmenu to enter given data.\na. Clear any existing data from the lists.\nb. List the input values in the L1 column.\nc. List the output values in the L2 column.\n2. Graph and observe a scatter plot of the data using theSTATPLOTfeature.\na. UseZOOM[9] to adjust axes to fit the data.\nb. Verify the data follow a logistic pattern.\n3. Find the equation that models the data."
	},
	{
		"chunk_id" : "00002030",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Find the equation that models the data.\na. Select Logistic from theSTATthenCALCmenu.\nb. Use the values returned for and to record the model,\n4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.\nEXAMPLE3\nUsing Logistic Regression to Fit a Model to Data\nMobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular"
	},
	{
		"chunk_id" : "00002031",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "service.Table 5shows the percentage of Americans with cellular service between the years 1995 and 201211.\nYear Americans with Cellular Service (%) Year Americans with Cellular Service (%)\n1995 12.69 2004 62.852\n1996 16.35 2005 68.63\n1997 20.29 2006 76.64\n1998 25.08 2007 82.47\n1999 30.81 2008 85.68\n2000 38.75 2009 89.14\n2001 45.00 2010 91.86\n2002 49.16 2011 95.28\n2003 55.15 2012 98.17\nTable5\n Let represent time in years starting with for the year 1995. Let represent the corresponding percentage of"
	},
	{
		"chunk_id" : "00002032",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "residents with cellular service. Use logistic regression to fit a model to these data.\n Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest\ntenth of a percent.\n Discuss the value returned for the upper limit, What does this tell you about the model? What would the limiting\n11 Source:The World Bank, 2013\n658 6  Exponential and Logarithmic Functions\nvalue be if the model were exact?\nSolution"
	},
	{
		"chunk_id" : "00002033",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value be if the model were exact?\nSolution\n Using theSTATthenEDITmenu on a graphing utility, list the years using values 015 in L1 and the corresponding\npercentage in L2. Then use theSTATPLOTfeature to verify that the scatterplot follows a logistic pattern as shown in\nFigure 5:\nFigure5\nUse the Logistic command from theSTATthenCALCmenu to obtain the logistic model,\nNext, graph the model in the same window as shown inFigure 6the scatterplot to verify it is a good fit:\nFigure6\n"
	},
	{
		"chunk_id" : "00002034",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\n\nTo approximate the percentage of Americans with cellular service in the year 2013, substitute for the in the model\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 659\nand solve for\nAccording to the model, about 99.3% of Americans had cellular service in 2013.\n\nThe model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with"
	},
	{
		"chunk_id" : "00002035",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the\nmodel were exact, the limiting value would be and the models outputs would get very close to, but never\nactually reach 100%. After all, there will always be someone out there without cellular service!\nTRY IT #3 Table 6shows the population, in thousands, of harbor seals in the Wadden Sea over the years\n1997 to 2012.\nYear Seal Population (Thousands) Year Seal Population (Thousands)"
	},
	{
		"chunk_id" : "00002036",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1997 3.493 2005 19.590\n1998 5.282 2006 21.955\n1999 6.357 2007 22.862\n2000 9.201 2008 23.869\n2001 11.224 2009 24.243\n2002 12.964 2010 24.344\n2003 16.226 2011 24.919\n2004 18.137 2012 25.108\nTable6\n Let represent time in years starting with for the year 1997. Let represent the\nnumber of seals in thousands. Use logistic regression to fit a model to these data.\n Use the model to predict the seal population for the year 2020.\n To the nearest whole number, what is the limiting value of this model?\nMEDIA"
	},
	{
		"chunk_id" : "00002037",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess this online resource for additional instruction and practice with exponential function models.\nExponential Regression on a Calculator(https://openstax.org/l/pregresscalc)\n660 6  Exponential and Logarithmic Functions\n6.8 SECTION EXERCISES\nVerbal\n1. What situations are best 2. What is a carrying capacity? 3. What is regression analysis?\nmodeled by a logistic What kind of model has a Describe the process of\nequation? Give an example, carrying capacity built into performing regression"
	},
	{
		"chunk_id" : "00002038",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and state a case for why the its formula? Why does this analysis on a graphing\nexample is a good fit. make sense? utility.\n4. What might a scatterplot of 5. What does they-intercept\ndata points look like if it on the graph of a logistic\nwere best described by a equation correspond to for\nlogarithmic model? a population modeled by\nthat equation?\nGraphical\nFor the following exercises, match the given function of best fit with the appropriate scatterplot inFigure 7through"
	},
	{
		"chunk_id" : "00002039",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 11. Answer using the letter beneath the matching graph.\nFigure7\nFigure8\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 661\nFigure9\nFigure10\nFigure11\n662 6  Exponential and Logarithmic Functions\n6. 7. 8.\n9. 10.\nNumeric\n11. To the nearest whole 12. Rewrite the exponential 13. A logarithmic model is given\nnumber, what is the initial model by the equation\nvalue of a population as an\nmodeled by the logistic equivalent model with base To the nearest hundredth,"
	},
	{
		"chunk_id" : "00002040",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation Express the exponent to for what value of does\nfour significant digits.\nWhat is the carrying\ncapacity?\n14. A logistic model is given by 15. What is they-intercept on\nthe equation the graph of the logistic\nTo the model given in the\nprevious exercise?\nnearest hundredth, for\nwhat value oftdoes\nTechnology\nFor the following exercises, use this scenario: The population of a koi pond over months is modeled by the function\n16. Graph the population 17. What was the initial 18. How many koi will the"
	},
	{
		"chunk_id" : "00002041",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "model to show the population of koi? pond have after one and a\npopulation over a span of half years?\nyears.\n19. How many months will it 20. Use the intersect feature to\ntake before there are approximate the number\nkoi in the pond? of months it will take\nbefore the population of\nthe pond reaches half its\ncarrying capacity.\nFor the following exercises, use this scenario: The population of an endangered species habitat for wolves is modeled\nby the function where is given in years."
	},
	{
		"chunk_id" : "00002042",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by the function where is given in years.\n21. Graph the population 22. What was the initial 23. How many wolves will the\nmodel to show the population of wolves habitat have after years?\npopulation over a span of transported to the habitat?\nyears.\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 663\n24. How many years will it take 25. Use the intersect feature to\nbefore there are approximate the number\nwolves in the habitat? of years it will take before\nthe population of the"
	},
	{
		"chunk_id" : "00002043",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the population of the\nhabitat reaches half its\ncarrying capacity.\nFor the following exercises, refer toTable 7.\nx 1 2 3 4 5 6\nf(x) 1125 1495 2310 3294 4650 6361\nTable7\n26. Use a graphing calculator 27. Use the regression feature 28. Write the exponential\nto create a scatter diagram to find an exponential function as an exponential\nof the data. function that best fits the equation with base\ndata in the table.\n29. Graph the exponential 30. Use the intersect feature to"
	},
	{
		"chunk_id" : "00002044",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation on the scatter find the value of for\ndiagram. which\nFor the following exercises, refer toTable 8.\nx 1 2 3 4 5 6\nf(x) 555 383 307 210 158 122\nTable8\n31. Use a graphing calculator 32. Use the regression feature 33. Write the exponential\nto create a scatter diagram to find an exponential function as an exponential\nof the data. function that best fits the equation with base\ndata in the table.\n34. Graph the exponential 35. Use the intersect feature to\nequation on the scatter find the value of for"
	},
	{
		"chunk_id" : "00002045",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation on the scatter find the value of for\ndiagram. which\nFor the following exercises, refer toTable 9.\nx 1 2 3 4 5 6\nf(x) 5.1 6.3 7.3 7.7 8.1 8.6\nTable9\n664 6  Exponential and Logarithmic Functions\n36. Use a graphing calculator 37. Use the LOGarithm option 38. Use the logarithmic\nto create a scatter diagram of the REGression feature function to find the value\nof the data. to find a logarithmic of the function when\nfunction of the form\nthat best\nfits the data in the table."
	},
	{
		"chunk_id" : "00002046",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that best\nfits the data in the table.\n39. Graph the logarithmic 40. Use the intersect feature to\nequation on the scatter find the value of for\ndiagram. which\nFor the following exercises, refer toTable 10.\nx 1 2 3 4 5 6 7 8\nf(x) 7.5 6 5.2 4.3 3.9 3.4 3.1 2.9\nTable10\n41. Use a graphing calculator 42. Use theLOGarithm option 43. Use the logarithmic\nto create a scatter diagram of theREGression feature function to find the value\nof the data. to find a logarithmic of the function when\nfunction of the form"
	},
	{
		"chunk_id" : "00002047",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function of the form\nthat best\nfits the data in the table.\n44. Graph the logarithmic 45. Use the intersect feature to\nequation on the scatter find the value of for\ndiagram. which\nFor the following exercises, refer toTable 11.\nx 1 2 3 4 5 6 7 8 9 10\nf(x) 8.7 12.3 15.4 18.5 20.7 22.5 23.3 24 24.6 24.8\nTable11\n46. Use a graphing calculator 47. Use the LOGISTIC 48. Graph the logistic equation\nto create a scatter diagram regression option to find a on the scatter diagram.\nof the data. logistic growth model of"
	},
	{
		"chunk_id" : "00002048",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the data. logistic growth model of\nthe form that\nbest fits the data in the\ntable.\n49. To the nearest whole 50. Use the intersect feature to\nnumber, what is the find the value of for\npredicted carrying capacity which the model reaches\nof the model? half its carrying capacity.\nAccess for free at openstax.org\n6.8  Fitting Exponential Models to Data 665\nFor the following exercises, refer toTable 12.\n0 2 4 5 7 8 10 11 15 17\n12 28.6 52.8 70.3 99.9 112.5 125.8 127.9 135.1 135.9\nTable12"
	},
	{
		"chunk_id" : "00002049",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table12\n51. Use a graphing calculator 52. Use the LOGISTIC 53. Graph the logistic equation\nto create a scatter diagram regression option to find a on the scatter diagram.\nof the data. logistic growth model of\nthe form that\nbest fits the data in the\ntable.\n54. To the nearest whole 55. Use the intersect feature to\nnumber, what is the find the value of for\npredicted carrying capacity which the model reaches\nof the model? half its carrying capacity.\nExtensions"
	},
	{
		"chunk_id" : "00002050",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Extensions\n56. Recall that the general form of a logistic equation 57. Use a graphing utility to find an exponential\nfor a population is given by such regression formula and a logarithmic\nregression formula for the points\nthat the initial population at time is\nand Round all numbers to 6 decimal\nShow algebraically that\nplaces. Graph the points and both formulas along\nwith the line on the same axis. Make a\nconjecture about the relationship of the\nregression formulas."
	},
	{
		"chunk_id" : "00002051",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "regression formulas.\n58. Verify the conjecture made in the previous 59. Find the inverse function for the logistic\nexercise. Round all numbers to six decimal places function Show all steps.\nwhen necessary.\n60. Use the result from the previous exercise to graph\nthe logistic model along with its\ninverse on the same axis. What are the intercepts\nand asymptotes of each function?\n666 6  Chapter Review\nChapter Review\nKey Terms"
	},
	{
		"chunk_id" : "00002052",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "666 6  Chapter Review\nChapter Review\nKey Terms\nannual percentage rate (APR) the yearly interest rate earned by an investment account, also callednominal rate\ncarrying capacity in a logistic model, the limiting value of the output\nchange-of-base formula a formula for converting a logarithm with any base to a quotient of logarithms with any\nother base.\ncommon logarithm the exponent to which 10 must be raised to get is written simply as"
	},
	{
		"chunk_id" : "00002053",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compound interest interest earned on the total balance, not just the principal\ndoubling time the time it takes for a quantity to double\nexponential growth a model that grows by a rate proportional to the amount present\nextraneous solution a solution introduced while solving an equation that does not satisfy the conditions of the\noriginal equation\nhalf-life the length of time it takes for a substance to exponentially decay to half of its original quantity"
	},
	{
		"chunk_id" : "00002054",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "logarithm the exponent to which must be raised to get written\nlogistic growth model a function of the form where is the initial value, is the carrying capacity,\nor limiting value, and is a constant determined by the rate of growth\nnatural logarithm the exponent to which the number must be raised to get is written as\nNewtons Law of Cooling the scientific formula for temperature as a function of time as an objects temperature is\nequalized with the ambient temperature"
	},
	{
		"chunk_id" : "00002055",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equalized with the ambient temperature\nnominal rate the yearly interest rate earned by an investment account, also calledannual percentage rate\norder of magnitude the power of ten, when a number is expressed in scientific notation, with one non-zero digit to\nthe left of the decimal\npower rule for logarithms a rule of logarithms that states that the log of a power is equal to the product of the\nexponent and the log of its base"
	},
	{
		"chunk_id" : "00002056",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponent and the log of its base\nproduct rule for logarithms a rule of logarithms that states that the log of a product is equal to a sum of logarithms\nquotient rule for logarithms a rule of logarithms that states that the log of a quotient is equal to a difference of\nlogarithms\nKey Equations\ndefinition of the exponential\nfunction\ndefinition of exponential\ngrowth\ncompound interest formula\nis the number of unit time periods of growth"
	},
	{
		"chunk_id" : "00002057",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the number of unit time periods of growth\ncontinuous growth formula is the starting amount (in the continuous compounding formula a is replaced\nwith P, the principal)\nis the mathematical constant,\nGeneral Form for the Translation of the Parent Function\nAccess for free at openstax.org\n6  Chapter Review 667\nFor\nDefinition of the logarithmic function\nif and only if\nDefinition of the common logarithm For if and only if\nDefinition of the natural logarithm For if and only if"
	},
	{
		"chunk_id" : "00002058",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "General Form for the Translation of the Parent Logarithmic Function\nThe Product Rule for Logarithms\nThe Quotient Rule for Logarithms\nThe Power Rule for Logarithms\nThe Change-of-Base Formula\nFor any algebraic expressions and and any positive real number\nOne-to-one property for exponential\nwhere\nfunctions\nif and only if\nFor any algebraic expressionSand positive real numbers and where\nDefinition of a logarithm\nif and only if\nFor any algebraic expressionsSandTand any positive real number"
	},
	{
		"chunk_id" : "00002059",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "One-to-one property for logarithmic\nwhere\nfunctions\nif and only if\nHalf-life formula If the half-life is\nCarbon-14 dating is the amount of carbon-14 when the plant or animal died\nis the amount of carbon-14 remaining today\nis the age of the fossil in years\nDoubling time\nIf the doubling time is\nformula\nNewtons Law of where is the ambient temperature, and is the\nCooling continuous rate of cooling.\nKey Concepts\n6.1Exponential Functions"
	},
	{
		"chunk_id" : "00002060",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Key Concepts\n6.1Exponential Functions\n An exponential function is defined as a function with a positive constant other than raised to a variable exponent.\nSeeExample 1.\n668 6  Chapter Review\n A function is evaluated by solving at a specific value. SeeExample 2andExample 3.\n An exponential model can be found when the growth rate and initial value are known. SeeExample 4.\n An exponential model can be found when the two data points from the model are known. SeeExample 5."
	},
	{
		"chunk_id" : "00002061",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " An exponential model can be found using two data points from the graph of the model. SeeExample 6.\n An exponential model can be found using two data points from the graph and a calculator. SeeExample 7.\n The value of an account at any time can be calculated using the compound interest formula when the principal,\nannual interest rate, and compounding periods are known. SeeExample 8.\n The initial investment of an account can be found using the compound interest formula when the value of the"
	},
	{
		"chunk_id" : "00002062",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "account, annual interest rate, compounding periods, and life span of the account are known. SeeExample 9.\n The number is a mathematical constant often used as the base of real world exponential growth and decay\nmodels. Its decimal approximation is\n Scientific and graphing calculators have the key or for calculating powers of SeeExample 10.\n Continuous growth or decay models are exponential models that use as the base. Continuous growth and decay"
	},
	{
		"chunk_id" : "00002063",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "models can be found when the initial value and growth or decay rate are known. SeeExample 11andExample 12.\n6.2Graphs of Exponential Functions\n The graph of the function has ay-intercept at domain   range  and horizontal\nasymptote SeeExample 1.\n If the function is increasing. The left tail of the graph will approach the asymptote and the right tail will\nincrease without bound.\n If the function is decreasing. The left tail of the graph will increase without bound, and the right tail will"
	},
	{
		"chunk_id" : "00002064",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "approach the asymptote\n The equation represents a vertical shift of the parent function\n The equation represents a horizontal shift of the parent function SeeExample 2.\n Approximate solutions of the equation can be found using a graphing calculator. SeeExample 3.\n The equation where represents a vertical stretch if or compression if of the\nparent function SeeExample 4.\n When the parent function is multiplied by the result, is a reflection about thex-axis."
	},
	{
		"chunk_id" : "00002065",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "When the input is multiplied by the result, is a reflection about they-axis. SeeExample 5.\n All translations of the exponential function can be summarized by the general equation SeeTable\n3.\n Using the general equation we can write the equation of a function given its description. See\nExample 6.\n6.3Logarithmic Functions\n The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an\nexponential function."
	},
	{
		"chunk_id" : "00002066",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential function.\n Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See\nExample 1.\n Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See\nExample 2.\n Logarithmic functions with base can be evaluated mentally using previous knowledge of powers of SeeExample\n3andExample 4.\n Common logarithms can be evaluated mentally using previous knowledge of powers of SeeExample 5."
	},
	{
		"chunk_id" : "00002067",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " When common logarithms cannot be evaluated mentally, a calculator can be used. SeeExample 6.\n Real-world exponential problems with base can be rewritten as a common logarithm and then evaluated using a\ncalculator. SeeExample 7.\n Natural logarithms can be evaluated using a calculatorExample 8.\n6.4Graphs of Logarithmic Functions\n To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and\nsolve for SeeExample 1andExample 2"
	},
	{
		"chunk_id" : "00002068",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solve for SeeExample 1andExample 2\n The graph of the parent function has anx-intercept at domain  range  \nvertical asymptote and\n if the function is increasing.\n if the function is decreasing.\nAccess for free at openstax.org\n6  Chapter Review 669\nSeeExample 3.\n The equation shifts the parent function horizontally\n left units if\n right units if\nSeeExample 4.\n The equation shifts the parent function vertically\n up units if\n down units if\nSeeExample 5.\n For any constant the equation"
	},
	{
		"chunk_id" : "00002069",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 5.\n For any constant the equation\n stretches the parent function vertically by a factor of if\n compresses the parent function vertically by a factor of if\nSeeExample 6andExample 7.\n When the parent function is multiplied by the result is a reflection about thex-axis. When the input\nis multiplied by the result is a reflection about they-axis.\n The equation represents a reflection of the parent function about thex-axis."
	},
	{
		"chunk_id" : "00002070",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The equation represents a reflection of the parent function about they-axis.\nSeeExample 8.\n A graphing calculator may be used to approximate solutions to some logarithmic equations SeeExample 9.\n All translations of the logarithmic function can be summarized by the general equation\nSeeTable 4.\n Given an equation with the general form we can identify the vertical asymptote for\nthe transformation. SeeExample 10.\n Using the general equation we can write the equation of a logarithmic function given its"
	},
	{
		"chunk_id" : "00002071",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph. SeeExample 11.\n6.5Logarithmic Properties\n We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. SeeExample 1.\n We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See\nExample 2.\n We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log\nof its base. SeeExample 3,Example 4, andExample 5."
	},
	{
		"chunk_id" : "00002072",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with\na complex input. SeeExample 6,Example 7,andExample 8.\n The rules of logarithms can also be used to condense sums, differences, and products with the same base as a\nsingle logarithm. SeeExample 9,Example 10,Example 11, andExample 12.\n We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base\nformula. SeeExample 13."
	},
	{
		"chunk_id" : "00002073",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "formula. SeeExample 13.\n The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and as the quotient of\nnatural or common logs. That way a calculator can be used to evaluate. SeeExample 14.\n6.6Exponential and Logarithmic Equations\n We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the\nsame base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one"
	},
	{
		"chunk_id" : "00002074",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "another and solve for the unknown.\n When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents\nequal to one another and solve for the unknown. SeeExample 1.\n When we are given an exponential equation where the bases arenotexplicitly shown as being equal, rewrite each\nside of the equation as powers of the same base, then set the exponents equal to one another and solve for the\nunknown. SeeExample 2,Example 3, andExample 4."
	},
	{
		"chunk_id" : "00002075",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unknown. SeeExample 2,Example 3, andExample 4.\n When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side.\nSeeExample 5.\n We can solve exponential equations with base by applying the natural logarithm of both sides because\nexponential and logarithmic functions are inverses of each other. SeeExample 6andExample 7.\n After solving an exponential equation, check each solution in the original equation to find and eliminate any\n670 6  Chapter Review"
	},
	{
		"chunk_id" : "00002076",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "670 6  Chapter Review\nextraneous solutions. SeeExample 8.\n When given an equation of the form where is an algebraic expression, we can use the definition of a\nlogarithm to rewrite the equation as the equivalent exponential equation and solve for the unknown. See\nExample 9andExample 10.\n We can also use graphing to solve equations with the form We graph both equations and\non the same coordinate plane and identify the solution as thex-value of the intersecting point. SeeExample\n11."
	},
	{
		"chunk_id" : "00002077",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.\n When given an equation of the form where and are algebraic expressions, we can use the one-\nto-one property of logarithms to solve the equation for the unknown. SeeExample 12.\n Combining the skills learned in this and previous sections, we can solve equations that model real world situations,\nwhether the unknown is in an exponent or in the argument of a logarithm. SeeExample 13.\n6.7Exponential and Logarithmic Models\n The basic exponential function is If we have exponential growth; if we have"
	},
	{
		"chunk_id" : "00002078",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential decay.\n We can also write this formula in terms of continuous growth as where is the starting value. If is\npositive, then we have exponential growth when and exponential decay when SeeExample 1.\n In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use\nthe model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See\nExample 2."
	},
	{
		"chunk_id" : "00002079",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 2.\n We can find the age, of an organic artifact by measuring the amount, of carbon-14 remaining in the artifact and\nusing the formula to solve for SeeExample 3.\n Given a substances doubling time or half-time, we can find a function that represents its exponential growth or\ndecay. SeeExample 4.\n We can use Newtons Law of Cooling to find how long it will take for a cooling object to reach a desired\ntemperature, or to find what temperature an object will be after a given time. SeeExample 5."
	},
	{
		"chunk_id" : "00002080",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " We can use logistic growth functions to model real-world situations where the rate of growth changes over time,\nsuch as population growth, spread of disease, and spread of rumors. SeeExample 6.\n We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic,\nand logistic graphs help us to develop models that best fit our data. SeeExample 7.\n Any exponential function with the form can be rewritten as an equivalent exponential function with the"
	},
	{
		"chunk_id" : "00002081",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form where SeeExample 8.\n6.8Fitting Exponential Models to Data\n Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without\nbound, or where decay begins rapidly and then slows down to get closer and closer to zero.\n We use the command ExpReg on a graphing utility to fit function of the form to a set of data points. See\nExample 1.\n Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows"
	},
	{
		"chunk_id" : "00002082",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "over time.\n We use the command LnReg on a graphing utility to fit a function of the form to a set of data\npoints. SeeExample 2.\n Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as\nthe function approaches an upper limit.\n We use the command Logistic on a graphing utility to fit a function of the form to a set of data\npoints. SeeExample 3.\nAccess for free at openstax.org\n6  Exercises 671\nExercises\nReview Exercises\nExponential Functions"
	},
	{
		"chunk_id" : "00002083",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Exercises\nReview Exercises\nExponential Functions\n1. Determine whether the 2. The population of a herd of 3. Find an exponential\nfunction deer is represented by the equation that passes\nrepresents exponential function through the points\ngrowth, exponential decay, where is given in years. To and\nor neither. Explain the nearest whole number,\nwhat will the herd\npopulation be after years?\n4. Determine whetherTable 1 5. A retirement account is 6. Hsu-Mei wants to save"
	},
	{
		"chunk_id" : "00002084",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "could represent a function that opened with an initial $5,000 for a down payment\nis linear, exponential, or neither. deposit of $8,500 and earns on a car. To the nearest\nIf it appears to be exponential, interest compounded dollar, how much will she\nfind a function that passes monthly. What will the need to invest in an account\nthrough the points. account be worth in now with APR,\nyears? compounded daily, in order\nx 1 2 3 4 to reach her goal in years?\nf(x) 3 0.9 0.27 0.081\nTable1"
	},
	{
		"chunk_id" : "00002085",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "f(x) 3 0.9 0.27 0.081\nTable1\n7. Does the equation 8. Suppose an investment\nrepresent account is opened with an\ncontinuous growth, initial deposit of\ncontinuous decay, or earning interest,\nneither? Explain. compounded continuously.\nHow much will the account\nbe worth after years?\nGraphs of Exponential Functions\n9. Graph the function State the 10. Graph the function and its\ndomain and range and give they-intercept. reflection about they-axis on the same axes, and\ngive they-intercept.\n672 6  Exercises"
	},
	{
		"chunk_id" : "00002086",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "give they-intercept.\n672 6  Exercises\n11. The graph of is reflected about the 12. The graph below shows transformations of the\ny-axis and stretched vertically by a factor of graph of What is the equation for the\nWhat is the equation of the new function, transformation?\nState itsy-intercept, domain, and range.\nFigure1\nLogarithmic Functions\n13. Rewrite 14. Rewrite as an 15. Rewrite as an\nas an equivalent equivalent exponential equivalent logarithmic\nexponential equation. equation. equation."
	},
	{
		"chunk_id" : "00002087",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exponential equation. equation. equation.\n16. Rewrite as an 17. Solve forxif by converting the\nequivalent logarithmic logarithmic equation to exponential\nequation.\nform.\n18. Evaluate 19. Evaluate without using a calculator.\nwithout using a calculator.\n20. Evaluate using a 21. Evaluate 22. Evaluate using a\ncalculator. Round to the without using a calculator. calculator. Round to the\nnearest thousandth.\nnearest thousandth.\nGraphs of Logarithmic Functions\n23. Graph the function 24. Graph the function"
	},
	{
		"chunk_id" : "00002088",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. Graph the function 24. Graph the function\n25. State the domain, vertical asymptote, and end\nbehavior of the function\nLogarithmic Properties\n26. Rewrite in expanded form. 27. Rewrite\nin compact form.\nAccess for free at openstax.org\n6  Exercises 673\n28. Rewrite in expanded form. 29. Rewrite in compact form.\n30. Rewrite as a product. 31. Rewrite as a single logarithm.\n32. Use properties of logarithms to expand 33. Use properties of logarithms to expand"
	},
	{
		"chunk_id" : "00002089",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. Condense the expression 35. Condense the expression\nto a single logarithm. to a single logarithm.\n36. Rewrite to base 37. Rewrite as a logarithm. Then apply\nthe change of base formula to solve for using\nthe common log. Round to the nearest\nthousandth.\nExponential and Logarithmic Equations\n38. Solve 39. Solve by 40. Use logarithms to find the\nby exact solution for\nrewriting each side with a rewriting each side with a If there\ncommon base. common base. is no solution, writeno\nsolution."
	},
	{
		"chunk_id" : "00002090",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution.\n41. Use logarithms to find the 42. Find the exact solution for 43. Find the exact solution for\nexact solution for . If there is no If there\nIf there solution, writeno solution. is no solution, writeno\nis no solution, writeno solution.\nsolution.\n44. Find the exact solution for 45. Find the exact solution for 46. Use the definition of a\nIf there is If logarithm to solve.\nno solution, writeno there is no solution, write\nsolution. no solution."
	},
	{
		"chunk_id" : "00002091",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution. no solution.\n47. Use the definition of a 48. Use the one-to-one property of 49. Use the one-to-one property\nlogarithm to find the exact logarithms to find an exact of logarithms to find an exact\nsolution for solution for solution for\nIf there is no solution, writeno If there is no solution, writeno\nsolution. solution.\n50. The formula for measuring sound intensity in 51. The population of a city is modeled by the\ndecibels is defined by the equation equation where is"
	},
	{
		"chunk_id" : "00002092",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where is the intensity of the measured in years. If the city continues to grow at\nthis rate, how many years will it take for the\nsound in watts per square meter and\npopulation to reach one million?\nis the lowest level of sound that the average\nperson can hear. How many decibels are emitted\nfrom a large orchestra with a sound intensity of\nwatts per square meter?\n674 6  Exercises\n52. Find the inverse function for the exponential 53. Find the inverse function for the logarithmic\nfunction function"
	},
	{
		"chunk_id" : "00002093",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function function\nExponential and Logarithmic Models\nFor the following exercises, use this scenario: A doctor prescribes milligrams of a therapeutic drug that decays by\nabout each hour.\n54. To the nearest minute, what is the half-life of the 55. Write an exponential model representing the\ndrug? amount of the drug remaining in the patients\nsystem after hours. Then use the formula to find\nthe amount of the drug that would remain in the\npatients system after hours. Round to the\nnearest hundredth of a gram."
	},
	{
		"chunk_id" : "00002094",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nearest hundredth of a gram.\nFor the following exercises, use this scenario: A soup with an internal temperature of Fahrenheit was taken off the\nstove to cool in a room. After fifteen minutes, the internal temperature of the soup was\n56. Use Newtons Law of 57. How many minutes will it\nCooling to write a formula take the soup to cool to\nthat models this situation.\nFor the following exercises, use this scenario: The equation models the number of people in a\nschool who have heard a rumor after days."
	},
	{
		"chunk_id" : "00002095",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "school who have heard a rumor after days.\n58. How many people started 59. To the nearest tenth, how 60. What is the carrying\nthe rumor? many days will it be before capacity?\nthe rumor spreads to half\nthe carrying capacity?\nAccess for free at openstax.org\n6  Exercises 675\nFor the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter\nplots. Determine whether the data from the table would likely represent a function that is linear, exponential, or"
	},
	{
		"chunk_id" : "00002096",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "logarithmic.\n61. 62. 63. Find a formula for an\nx f(x) x f(x) exponential equation that\ngoes through the points\n1 3.05 0.5 18.05 and Then\nexpress the formula as an\nequivalent equation with\n2 4.42 1 17\nbasee.\n3 6.4 3 15.33\n4 9.28 5 14.55\n5 13.46 7 14.04\n6 19.52 10 13.5\n7 28.3 12 13.22\n8 41.04 13 13.1\n9 59.5 15 12.88\n10 86.28 17 12.69\n20 12.45\nFitting Exponential Models to Data\n64. What is the carrying capacity for a population 65. The population of a culture of bacteria is modeled"
	},
	{
		"chunk_id" : "00002097",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "modeled by the logistic equation by the logistic equation\nWhat is the initial population\nwhere is in days. To the nearest tenth, how many\nfor the model? days will it take the culture to reach of its\ncarrying capacity?\n676 6  Exercises\nFor the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the\nshape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic"
	},
	{
		"chunk_id" : "00002098",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round\nvalues to five decimal places.\n66. 67. 68.\nx f(x) x f(x) x f(x)\n1 409.4 0.15 36.21 0 9\n2 260.7 0.25 28.88 2 22.6\n3 170.4 0.5 24.39 4 44.2\n4 110.6 0.75 18.28 5 62.1\n5 74 1 16.5 7 96.9\n6 44.7 1.5 12.99 8 113.4\n7 32.4 2 9.91 10 133.4\n8 19.5 2.25 8.57 11 137.6\n9 12.7 2.75 7.23 15 148.4\n10 8.1 3 5.99 17 149.3\n3.5 4.81\nPractice Test"
	},
	{
		"chunk_id" : "00002099",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10 8.1 3 5.99 17 149.3\n3.5 4.81\nPractice Test\n1. The population of a pod of 2. Find an exponential 3. Drew wants to save $2,500\nbottlenose dolphins is equation that passes to go to the next World Cup.\nmodeled by the function through the points To the nearest dollar, how\nwhere is and much will he need to invest\ngiven in years. To the in an account now with\nnearest whole number, what APR, compounding\nwill the pod population be daily, in order to reach his\nafter years? goal in years?"
	},
	{
		"chunk_id" : "00002100",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "after years? goal in years?\nAccess for free at openstax.org\n6  Exercises 677\n4. An investment account was 5. Graph the function 6. The graph shows transformations\nopened with an initial and its of the graph of What\ndeposit of $9,600 and earns reflection across they-axis is the equation for the\ninterest, compounded on the same axes, and give transformation?\ncontinuously. How much will they-intercept.\nthe account be worth after\nyears?\n7. Rewrite 8. Rewrite as an 9. Solve for by converting the"
	},
	{
		"chunk_id" : "00002101",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as an equivalent exponential equivalent logarithmic logarithmic equation\nequation. equation. to exponential\nform.\n10. Evaluate 11. Evaluate using a 12. Graph the function\nwithout using a calculator. calculator. Round to the\nnearest thousandth.\n13. State the domain, vertical 14. Rewrite as a 15. Rewrite\nasymptote, and end sum. in compact form.\nbehavior of the function\n16. Rewrite as a 17. Use properties of\nlogarithm to expand\nproduct.\n18. Condense the expression 19. Rewrite as"
	},
	{
		"chunk_id" : "00002102",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "18. Condense the expression 19. Rewrite as\nto a single a logarithm. Then apply the\nchange of base formula to\nlogarithm.\nsolve for using the\nnatural log. Round to the\nnearest thousandth.\n20. Solve by rewriting each 21. Use logarithms to find the\nside with a common base. exact solution for\n. If\nthere is no solution, write\nno solution.\n678 6  Exercises\n22. Find the exact solution for 23. Find the exact solution for 24. Find the exact solution for\nIf there If If there is"
	},
	{
		"chunk_id" : "00002103",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If there If If there is\nis no solution, writeno there is no solution, write no solution, writeno\nsolution. no solution. solution.\n25. Find the exact solution for 26. Use the definition of a\nIf there logarithm to find the exact\nis no solution, writeno solution for\nsolution.\n27. Use the one-to-one property of logarithms to find 28. The formula for measuring\nan exact solution for sound intensity in decibels\nIf there is no is defined by the\nsolution, writeno solution. equation\nwhere\nis the intensity of the"
	},
	{
		"chunk_id" : "00002104",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where\nis the intensity of the\nsound in watts per square\nmeter and is\nthe lowest level of sound\nthat the average person\ncan hear. How many\ndecibels are emitted from a\nrock concert with a sound\nintensity of\nwatts per square meter?\n29. A radiation safety officer is 30. Write the formula found in 31. A bottle of soda with a\nworking with grams of the previous exercise as an temperature of\na radioactive substance. equivalent equation with Fahrenheit was taken off a"
	},
	{
		"chunk_id" : "00002105",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "After days, the sample base Express the shelf and placed in a\nhas decayed to grams. exponent to five significant refrigerator with an\nRounding to five significant digits. internal temperature of\ndigits, write an exponential After ten minutes,\nequation representing this the internal temperature of\nsituation. To the nearest the soda was Use\nday, what is the half-life of Newtons Law of Cooling to\nthis substance? write a formula that\nmodels this situation. To\nthe nearest degree, what"
	},
	{
		"chunk_id" : "00002106",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the nearest degree, what\nwill the temperature of the\nsoda be after one hour?\nAccess for free at openstax.org\n6  Exercises 679\n32. The population of a wildlife 33. Enter the data fromTable 1 34. The population of a lake of\nhabitat is modeled by the into a graphing calculator fish is modeled by the\nequation and graph the resulting logistic equation\nwhere scatter plot. Determine where\nwhether the data from the\nis given in years. How is time in years. To the\ntable would likely"
	},
	{
		"chunk_id" : "00002107",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "table would likely\nmany animals were nearest hundredth, how\nrepresent a function that is\noriginally transported to many years will it take the\nlinear, exponential, or\nthe habitat? How many lake to reach of its\nlogarithmic.\nyears will it take before the carrying capacity?\nhabitat reaches half its\nx f(x)\ncapacity?\n1 3\n2 8.55\n3 11.79\n4 14.09\n5 15.88\n6 17.33\n7 18.57\n8 19.64\n9 20.58\n10 21.42\nTable1\n680 6  Exercises"
	},
	{
		"chunk_id" : "00002108",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8 19.64\n9 20.58\n10 21.42\nTable1\n680 6  Exercises\nFor the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the\nshape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic\nmodel. Then use the appropriate regression feature to find an equation that models the data. When necessary, round\nvalues to five decimal places.\n35. 36. 37.\nx f(x) x f(x) x f(x)\n1 20 3 13.98 0 2.2"
	},
	{
		"chunk_id" : "00002109",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x f(x) x f(x) x f(x)\n1 20 3 13.98 0 2.2\n2 21.6 4 17.84 0.5 2.9\n3 29.2 5 20.01 1 3.9\n4 36.4 6 22.7 1.5 4.8\n5 46.6 7 24.1 2 6.4\n6 55.7 8 26.15 3 9.3\n7 72.6 9 27.37 4 12.3\n8 87.1 10 28.38 5 15\n9 107.2 11 29.97 6 16.2\n10 138.1 12 31.07 7 17.3\n13 31.43 8 17.9\nAccess for free at openstax.org\n7  Introduction 681\n7 THE UNIT CIRCLE: SINE AND COSINE FUNCTIONS\nThe tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr)\nChapter Outline\n7.1Angles\n7.2Right Triangle Trigonometry"
	},
	{
		"chunk_id" : "00002110",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.1Angles\n7.2Right Triangle Trigonometry\n7.3Unit Circle\n7.4The Other Trigonometric Functions\nIntroduction to The Unit Circle: Sine and Cosine Functions\nLife is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to\nthe gravitational pull of the moon. And as a result of the motion of the moon itself, the tides occur with different"
	},
	{
		"chunk_id" : "00002111",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "strengths. Throughout history, many Indigenous peoples have used this regularity to build cultural narratives and direct\nkey activities, such as agriculture, hunting, and fishing. Aboriginal people in the Torres Strait area (the northern tip) of\nAustralia used the tidal peaks to determine the best times to fish. Their elders explain that the stronger spring tides\nstirred up sediment and obscured fish vision, leaving them more likely to take in lures and resulting in a larger catch.1"
	},
	{
		"chunk_id" : "00002112",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In mathematics, a function that repeats its values in regular intervals is known as a periodic function. The graphs of such\nfunctions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has\nthe same output at exactly the same place in every cycle. And this translates to all the cycles of the function\nhavingexactlythe same length. So, if we know all the details ofone full cycleof a true periodic function, then we know"
	},
	{
		"chunk_id" : "00002113",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the state of the functions outputs atalltimes, future and past. In this chapter, we will investigate various examples of\nperiodic functions.\n1 Hamacher, D.W., Tapim, A., Passi, S., and Barsa, J. (2018). Dancing with the stars  astronomy and music in the Torres Strait. In Imagining\nOther Worlds: Explorations in Astronomy and Culture.\n682 7  The Unit Circle: Sine and Cosine Functions\n7.1 Angles\nLearning Objectives\nIn this section you will:\nDraw angles in standard position."
	},
	{
		"chunk_id" : "00002114",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Draw angles in standard position.\nConvert between degrees and radians.\nFind coterminal angles.\nFind the length of a circular arc.\nUse linear and angular speed to describe motion on a circular path.\nA golfer swings to hit a ball over a sand trap and onto the green. An airline pilot maneuvers a plane toward a narrow\nrunway. A dress designer creates the latest fashion. What do they all have in common? They all work with angles, and so"
	},
	{
		"chunk_id" : "00002115",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "do all of us at one time or another. Sometimes we need to measure angles exactly with instruments. Other times we\nestimate them or judge them by eye. Either way, the proper angle can make the difference between success and failure\nin many undertakings. In this section, we will examine properties of angles.\nDrawing Angles in Standard Position\nProperly defining an angle first requires that we define a ray. Arayis a directed line segment. It consists of one point on"
	},
	{
		"chunk_id" : "00002116",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a line and all points extending in one direction from that point. The first point is called theendpointof the ray. We can\nrefer to a specific ray by stating its endpoint and any other point on it. The ray inFigure 1can be named as ray EF, or in\nsymbol form\nFigure1\nAnangleis the union of two rays having a common endpoint. The endpoint is called thevertexof the angle, and the\ntwo rays are the sides of the angle. The angle inFigure 2is formed from and . Angles can be named using a"
	},
	{
		"chunk_id" : "00002117",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "point on each ray and the vertex, such as angleDEF, or in symbol form\nFigure2\nGreek letters are often used as variables for the measure of an angle.Table 1is a list of Greek letters commonly used to\nrepresent angles, and a sample angle is shown inFigure 3.\nor\ntheta phi alpha beta gamma\nTable1\nAccess for free at openstax.org\n7.1  Angles 683\nFigure3 Angle theta, shown as\nAngle creation is a dynamic process. We start with two rays lying on top of one another. We leave one fixed in place, and"
	},
	{
		"chunk_id" : "00002118",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rotate the other. The fixed ray is theinitial side,and the rotated ray is theterminal side. In order to identify the\ndifferent sides, we indicate the rotation with a small arrow close to the vertex as inFigure 4.\nFigure4\nAs we discussed at the beginning of the section, there are many applications for angles, but in order to use them\ncorrectly, we must be able to measure them. Themeasure of an angleis the amount of rotation from the initial side to"
	},
	{
		"chunk_id" : "00002119",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the terminal side. Probably the most familiar unit of angle measurement is the degree. Onedegreeis of a circular\nrotation, so a complete circular rotation contains degrees. An angle measured in degrees should always include the\nunit degrees after the number, or include the degree symbol For example,\nTo formalize our work, we will begin by drawing angles on anx-ycoordinate plane. Angles can occur in any position on"
	},
	{
		"chunk_id" : "00002120",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the coordinate plane, but for the purpose of comparison, the convention is to illustrate them in the same position\nwhenever possible. An angle is instandard positionif its vertex is located at the origin, and its initial side extends along\nthe positivex-axis. SeeFigure 5.\nFigure5\nIf the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a"
	},
	{
		"chunk_id" : "00002121",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "positive angle. If the angle is measured in a clockwise direction, the angle is said to be anegative angle.\n684 7  The Unit Circle: Sine and Cosine Functions\nDrawing an angle in standard position always starts the same waydraw the initial side along the positivex-axis. To\nplace the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by\ndividing the angle measure in degrees by For example, to draw a angle, we calculate that So, the"
	},
	{
		"chunk_id" : "00002122",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positivex-axis. To draw\na angle, we calculate that So the terminal side will be 1 complete rotation around the circle, moving\ncounterclockwise from the positivex-axis. In this case, the initial side and the terminal side overlap. SeeFigure 6.\nFigure6\nSince we define an angle instandard positionby its terminal side, we have a special type of angle whose terminal side"
	},
	{
		"chunk_id" : "00002123",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "lies on an axis, aquadrantal angle. This type of angle can have a measure of 0, 90, 180, 270, or 360. SeeFigure 7.\nFigure7 Quadrantal angles have a terminal side that lies along an axis. Examples are shown.\nQuadrantal Angles\nAn angle is aquadrantal angleif its terminal side lies on an axis, including 0, 90, 180, 270, or 360.\n...\nHOW TO\nGiven an angle measure in degrees, draw the angle in standard position.\n1. Express the angle measure as a fraction of\n2. Reduce the fraction to simplest form."
	},
	{
		"chunk_id" : "00002124",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Reduce the fraction to simplest form.\n3. Draw an angle that contains that same fraction of the circle, beginning on the positivex-axis and moving\ncounterclockwise for positive angles and clockwise for negative angles.\nAccess for free at openstax.org\n7.1  Angles 685\nEXAMPLE1\nDrawing an Angle in Standard Position Measured in Degrees\n Sketch an angle of in standard position.  Sketch an angle of in standard position.\nSolution\n\nDivide the angle measure by"
	},
	{
		"chunk_id" : "00002125",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\n\nDivide the angle measure by\nTo rewrite the fraction in a more familiar fraction, we can recognize that\nOne-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at as\ninFigure 8.\nFigure8\n686 7  The Unit Circle: Sine and Cosine Functions\n\nDivide the angle measure by\nIn this case, we can recognize that\nThree-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-"
	},
	{
		"chunk_id" : "00002126",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "half of another quarter, as inFigure 9.\nFigure9\nTRY IT #1 Show an angle of on a circle in standard position.\nConverting Between Degrees and Radians\nDividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may\nchoose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop"
	},
	{
		"chunk_id" : "00002127",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "before the circle is completed. The portion that you drew is referred to as an arc. Anarcmay be a portion of a full circle, a\nfull circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire\ncircle is called thecircumferenceof that circle.\nThe circumference of a circle is If we divide both sides of this equation by we create the ratio of the"
	},
	{
		"chunk_id" : "00002128",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "circumference, which is always to the radius, regardless of the length of the radius. So the circumference of any circle\nis times the length of the radius. That means that if we took a string as long as the radius and used it to\nmeasure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more\nthan a quarter of a seventh, as shown inFigure 10.\nAccess for free at openstax.org\n7.1  Angles 687\nFigure10"
	},
	{
		"chunk_id" : "00002129",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.1  Angles 687\nFigure10\nThis brings us to our new angle measure. Oneradianis the measure of a central angle of a circle that intercepts an arc\nequal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii.\nBecause the total circumference equals times the radius, a full circular rotation is radians.\nSeeFigure 11. Note that when an angle is described without a specific unit, it refers to radian measure. For example, an"
	},
	{
		"chunk_id" : "00002130",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length\n(circumference) divided by a length (radius) and the length units cancel.\nFigure11 The angle sweeps out a measure of one radian. Note that the length of the intercepted arc is the same as\nthe length of the radius of the circle.\nRelating Arc Lengths to Radius\nAnarc length is the length of the curve along the arc. Just as the full circumference of a circle always has a constant"
	},
	{
		"chunk_id" : "00002131",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of\nthe length of the radius.\nThis ratio, called the radian measure, is the same regardless of the radius of the circleit depends only on the angle.\nThis property allows us to define a measure of any angle as the ratio of the arc length to the radiusr. SeeFigure 12.\nIf then\n688 7  The Unit Circle: Sine and Cosine Functions"
	},
	{
		"chunk_id" : "00002132",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12 (a) In an angle of 1 radian, the arc length equals the radius (b) An angle of 2 radians has an arc length\n(c) A full revolution is or about 6.28 radians.\nTo elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of\na circle is where is the radius. The smaller circle then has circumference and the larger has\ncircumference Now we draw a angle on the two circles, as inFigure 13."
	},
	{
		"chunk_id" : "00002133",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure13 A angle contains one-eighth of the circumference of a circle, regardless of the radius.\nNotice what happens if we find the ratio of the arc length divided by the radius of the circle.\nSince both ratios are the angle measures of both circles are the same, even though the arc length and radius differ.\nRadians\nOneradianis the measure of the central angle of a circle such that the length of the arc between the initial side and"
	},
	{
		"chunk_id" : "00002134",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the terminal side is equal to the radius of the circle. A full revolution equals radians. A half revolution\nis equivalent to radians.\nTheradian measureof an angle is the ratio of the length of the arc subtended by the angle to the radius of the circle.\nIn other words, if is the length of an arc of a circle, and is the radius of the circle, then the central angle containing\nthat arc measures radians. In a circle of radius 1, the radian measure corresponds to the length of the arc."
	},
	{
		"chunk_id" : "00002135",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.1  Angles 689\nQ&A A measure of 1 radian looks to be about Is that correct?\nYes. It is approximately Because radians equals radian equals\nUsing Radians\nBecauseradianmeasure is the ratio of two lengths, it is a unitless measure. For example, inFigure 12, suppose the\nradius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the"
	},
	{
		"chunk_id" : "00002136",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle, the inches cancel, and we have a result without units. Therefore, it is not necessary to write the label radians\nafter a radian measure, and if we see an angle that is not labeled with degrees or the degree symbol, we can assume\nthat it is a radian measure.\nConsidering the most basic case, theunit circle(a circle with radius 1), we know that 1 rotation equals 360 degrees,\nWe can also track one rotation around a circle by finding the circumference, and for the unit circle"
	},
	{
		"chunk_id" : "00002137",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "These two different ways to rotate around a circle give us a way to convert from degrees to radians.\nIdentifying Special Angles Measured in Radians\nIn addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full\nrevolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar.\nIt is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown inFigure 14. Memorizing"
	},
	{
		"chunk_id" : "00002138",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "these angles will be very useful as we study the properties associated with angles.\nFigure14 Commonly encountered angles measured in degrees\nNow, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in\nFigure 14, which are shown inFigure 15. Be sure you can verify each of these measures.\n690 7  The Unit Circle: Sine and Cosine Functions\nFigure15 Commonly encountered angles measured in radians\nEXAMPLE2\nFinding a Radian Measure"
	},
	{
		"chunk_id" : "00002139",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nFinding a Radian Measure\nFind the radian measure of one-third of a full rotation.\nSolution\nFor any circle, the arc length along such a rotation would be one-third of the circumference. We know that\nSo,\nThe radian measure would be the arc length divided by the radius.\nTRY IT #2 Find the radian measure of three-fourths of a full rotation.\nConverting Between Radians and Degrees\nBecause degrees and radians both measure angles, we need to be able to convert between them. We can easily do so"
	},
	{
		"chunk_id" : "00002140",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "using a proportion where is the measure of the angle in degrees and is the measure of the angle in radians.\nThis proportion shows that the measure of angle in degrees divided by 180 equals the measure of angle in radians\ndivided by Or, phrased another way, degrees is to 180 as radians is to\nAccess for free at openstax.org\n7.1  Angles 691\nConverting between Radians and Degrees\nTo convert between degrees and radians, use the proportion\nEXAMPLE3\nConverting Radians to Degrees"
	},
	{
		"chunk_id" : "00002141",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nConverting Radians to Degrees\nConvert each radian measure to degrees.\n  3\nSolution\nBecause we are given radians and we want degrees, we should set up a proportion and solve it.\n We use the proportion, substituting the given information.\n We use the proportion, substituting the given information.\nTRY IT #3 Convert radians to degrees.\nEXAMPLE4\nConverting Degrees to Radians\nConvert degrees to radians.\nSolution"
	},
	{
		"chunk_id" : "00002142",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Convert degrees to radians.\nSolution\nIn this example, we start with degrees and want radians, so we again set up a proportion, but we substitute the given\ninformation into a different part of the proportion.\nAnalysis\nAnother way to think about this problem is by remembering that Because we can find that\nis\nTRY IT #4 Convert to radians.\n692 7  The Unit Circle: Sine and Cosine Functions\nFinding Coterminal Angles"
	},
	{
		"chunk_id" : "00002143",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Coterminal Angles\nConverting between degrees and radians can make working with angles easier in some applications. For other\napplications, we may need another type of conversion. Negative angles and angles greater than a full revolution are\nmore awkward to work with than those in the range of to or to It would be convenient to replace those\nout-of-range angles with a corresponding angle within the range of a single revolution."
	},
	{
		"chunk_id" : "00002144",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "It is possible for more than one angle to have the same terminal side. Look atFigure 16. The angle of is a positive\nangle, measured counterclockwise. The angle of is a negative angle, measured clockwise. But both angles have\nthe same terminal side. If two angles in standard position have the same terminal side, they arecoterminal angles. Every\nangle greater than or less than is coterminal with an angle between and and it is often more convenient"
	},
	{
		"chunk_id" : "00002145",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to find the coterminal angle within the range of to than to work with an angle that is outside that range.\nFigure16 An angle of and an angle of are coterminal angles.\nAny angle has infinitely many coterminal angles because each time we add to that angleor subtract from\nitthe resulting value has a terminal side in the same location. For example, and are coterminal for this\nreason, as is\nAn angles reference angle is the measure of the smallest, positive, acute angle formed by the terminal side of the"
	},
	{
		"chunk_id" : "00002146",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can\nbe used as models for angles in other quadrants. SeeFigure 17for examples of reference angles for angles in different\nquadrants.\nFigure17\nCoterminal and Reference Angles\nCoterminal anglesare two angles in standard position that have the same terminal side.\nAccess for free at openstax.org\n7.1  Angles 693"
	},
	{
		"chunk_id" : "00002147",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.1  Angles 693\nAn anglesreference angleis the size of the smallest acute angle, formed by the terminal side of the angle and\nthe horizontal axis.\n...\nHOW TO\nGiven an angle greater than find a coterminal angle between and\n1. Subtract from the given angle.\n2. If the result is still greater than subtract again till the result is between and\n3. The resulting angle is coterminal with the original angle.\nEXAMPLE5\nFinding an Angle Coterminal with an Angle of Measure Greater Than"
	},
	{
		"chunk_id" : "00002148",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the least positive angle that is coterminal with an angle measuring where\nSolution\nAn angle with measure is coterminal with an angle with measure but is still greater than\nso we subtract again to find another coterminal angle:\nThe angle is coterminal with To put it another way, equals plus two full rotations, as shown in\nFigure 18.\nFigure18\nTRY IT #5 Find an angle that is coterminal with an angle measuring where\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002149",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an angle with measure less than find a coterminal angle having a measure between and\n1. Add to the given angle.\n2. If the result is still less than add again until the result is between and\n3. The resulting angle is coterminal with the original angle.\n694 7  The Unit Circle: Sine and Cosine Functions\nEXAMPLE6\nFinding an Angle Coterminal with an Angle Measuring Less Than\nShow the angle with measure on a circle and find a positive coterminal angle such that\nSolution"
	},
	{
		"chunk_id" : "00002150",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSince is half of we can start at the positive horizontal axis and measure clockwise half of a angle.\nBecause we can find coterminal angles by adding or subtracting a full rotation of we can find a positive coterminal\nangle here by adding\nWe can then show the angle on a circle, as inFigure 19.\nFigure19\nTRY IT #6 Find an angle that is coterminal with an angle measuring such that\nFinding Coterminal Angles Measured in Radians"
	},
	{
		"chunk_id" : "00002151",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Coterminal Angles Measured in Radians\nWe can findcoterminal anglesmeasured in radians in much the same way as we have found them using degrees. In\nboth cases, we find coterminal angles by adding or subtracting one or more full rotations.\n...\nHOW TO\nGiven an angle greater than find a coterminal angle between 0 and\n1. Subtract from the given angle.\n2. If the result is still greater than subtract again until the result is between and\n3. The resulting angle is coterminal with the original angle."
	},
	{
		"chunk_id" : "00002152",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nFinding Coterminal Angles Using Radians\nFind an angle that is coterminal with where\nSolution\nWhen working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in\nradians, we can find coterminal angles by adding or subtracting full rotations of radians:\nThe angle is coterminal, but not less than so we subtract another rotation.\nAccess for free at openstax.org\n7.1  Angles 695\nThe angle is coterminal with as shown inFigure 20.\nFigure20"
	},
	{
		"chunk_id" : "00002153",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure20\nTRY IT #7 Find an angle of measure that is coterminal with an angle of measure where\nDetermining the Length of an Arc\nRecall that the radian measure of an angle was defined as the ratio of thearc length of a circular arc to the radius of\nthe circle, From this relationship, we can find arc length along a circle, given an angle.\nArc Length on a Circle\nIn a circle of radiusr, the length of an arc subtended by an angle with measure in radians, shown inFigure 21, is\nFigure21\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002154",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure21\n...\nHOW TO\nGiven a circle of radius calculate the length of the arc subtended by a given angle of measure\n1. If necessary, convert to radians.\n696 7  The Unit Circle: Sine and Cosine Functions\n2. Multiply the radius\nEXAMPLE8\nFinding the Length of an Arc\nAssume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.\n In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?"
	},
	{
		"chunk_id" : "00002155",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Use your answer from part (a) to determine the radian measure for Mercurys movement in one Earth day.\nSolution\n Lets begin by finding the circumference of Mercurys orbit.\nSince Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled.\n Now, we convert to radians.\nTRY IT #8 Find the arc length along a circle of radius 10 units subtended by an angle of\nFinding the Area of a Sector of a Circle"
	},
	{
		"chunk_id" : "00002156",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Area of a Sector of a Circle\nIn addition to arc length, we can also use angles to find the area of asector of a circle. A sector is a region of a circle\nbounded by two radii and the intercepted arc, like a slice of pizza or pie. Recall that the area of a circle with radius can\nbe found using the formula If the two radii form an angle of measured in radians, then is the ratio of the"
	},
	{
		"chunk_id" : "00002157",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of\nthe circle. Thus, thearea of a sectoris the fraction multiplied by the entire area. (Always remember that this formula\nonly applies if is in radians.)\nArea of a Sector\nThearea of a sectorof a circle with radius subtended by an angle measured in radians, is\nSeeFigure 22.\nAccess for free at openstax.org\n7.1  Angles 697"
	},
	{
		"chunk_id" : "00002158",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.1  Angles 697\nFigure22 The area of the sector equals half the square of the radius times the central angle measured in radians.\n...\nHOW TO\nGiven a circle of radius find the area of a sector defined by a given angle\n1. If necessary, convert to radians.\n2. Multiply half the radian measure of by the square of the radius\nEXAMPLE9\nFinding the Area of a Sector\nAn automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown inFigure 23. What is the"
	},
	{
		"chunk_id" : "00002159",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "area of the sector of grass the sprinkler waters?\nFigure23 The sprinkler sprays 20 ft within an arc of\nSolution\nFirst, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already\nknow the equivalent radian measure, but we can also convert:\nThe area of the sector is then\nSo the area is about\n698 7  The Unit Circle: Sine and Cosine Functions\nTRY IT #9 In central pivot irrigation, which creates the field shapes similar to the image at the beginning of"
	},
	{
		"chunk_id" : "00002160",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Equations and Inequalities, a large irrigation pipe on wheels rotates around a center point. A\nfarmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her\nto water 150 thousand square meters a day, what angle should she set the system to cover? Write\nthe answer in radian measure to two decimal places.\nUse Linear and Angular Speed to Describe Motion on a Circular Path"
	},
	{
		"chunk_id" : "00002161",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object\ntraveling in a circular path has two types of speed.Linear speedis speed along a straight path and can be determined\nby the distance it moves along (itsdisplacement) in a given time interval. For instance, if a wheel with radius 5 inches\nrotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or inches,"
	},
	{
		"chunk_id" : "00002162",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "every second. So the linear speed of the point is in./s. The equation for linear speed is as follows where is linear\nspeed, is displacement, and is time.\nAngular speed results from circular motion and can be determined by the angle through which a point rotates in a given\ntime interval. In other words,angular speedis angular rotation per unit time. So, for instance, if a gear makes a full\nrotation every 4 seconds, we can calculate its angular speed as 90 degrees per second. Angular speed can"
	},
	{
		"chunk_id" : "00002163",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed\nis as follows, where (read as omega) is angular speed, is the angle traversed, and is time.\nCombining the definition of angular speed with the arc length equation, we can find a relationship between\nangular and linear speeds. The angular speed equation can be solved for giving Substituting this into the arc\nlength equation gives:\nSubstituting this into the linear speed equation gives:"
	},
	{
		"chunk_id" : "00002164",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Angular and Linear Speed\nAs a point moves along a circle of radius itsangular speed, is the angular rotation per unit time,\nThelinear speed, of the point can be found as the distance traveled, arc length per unit time,\nWhen the angular speed is measured in radians per unit time, linear speed and angular speed are related by the\nequation\nThis equation states that the angular speed in radians, representing the amount of rotation occurring in a unit of"
	},
	{
		"chunk_id" : "00002165",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "time, can be multiplied by the radius to calculate the total arc length traveled in a unit of time, which is the definition\nof linear speed.\nAccess for free at openstax.org\n7.1  Angles 699\n...\nHOW TO\nGiven the amount of angle rotation and the time elapsed, calculate the angular speed.\n1. If necessary, convert the angle measure to radians.\n2. Divide the angle in radians by the number of time units elapsed:\n3. The resulting speed will be in radians per time unit."
	},
	{
		"chunk_id" : "00002166",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Water wheels have been used for thousands of years to transfer the power of flowing water to other devices. The image\nbelow depicts the design of the the 3rd century Roman water wheel in Hierapolis, a city in what is now Turkey. Water\nturned the wheel, which in turn rotated a crank connected to two saws used to cut blocks. These design elements were\nused in water wheel applications throughout the world, and even provided the underlying principle for the steam\nengine, invented about 1500 years later."
	},
	{
		"chunk_id" : "00002167",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "engine, invented about 1500 years later.\nEXAMPLE10\nFinding Angular Speed\nA water wheel, shown inFigure 24, completes 1 rotation every 5 seconds. Find the angular speed in radians per second.\nFigure24\nSolution\nThe wheel completes 1 rotation, or passes through an angle of radians in 5 seconds, so the angular speed would be\nradians per second.\nTRY IT #10 A vintage vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per\nminute. Find the angular speed in radians per second.\n..."
	},
	{
		"chunk_id" : "00002168",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.\n1. Convert the total rotation to radians if necessary.\n2. Divide the total rotation in radians by the elapsed time to find the angular speed: apply\n700 7  The Unit Circle: Sine and Cosine Functions\n3. Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length"
	},
	{
		"chunk_id" : "00002169",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unit used for the radius and the time unit used for the elapsed time: apply\nEXAMPLE11\nFinding a Linear Speed\nA bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per\nminute). Find the speed the bicycle is traveling down the road.\nSolution\nHere, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside\nof the tires is the speed at which the bicycle travels down the road."
	},
	{
		"chunk_id" : "00002170",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make\nthis conversion:\nUsing the formula from above along with the radius of the wheels, we can find the linear speed:\nRemember that radians are a unitless measure, so it is not necessary to include them.\nFinally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour."
	},
	{
		"chunk_id" : "00002171",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #11 A satellite is rotating around Earth at 0.25 radian per hour at an altitude of 242 km above Earth. If\nthe radius of Earth is 6378 kilometers, find the linear speed of the satellite in kilometers per hour.\nMEDIA\nAccess these online resources for additional instruction and practice with angles, arc length, and areas of sectors.\nAngles in Standard Position(http://openstax.org/l/standardpos)\nAngle of Rotation(http://openstax.org/l/angleofrotation)\nCoterminal Angles(http://openstax.org/l/coterminal)"
	},
	{
		"chunk_id" : "00002172",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determining Coterminal Angles(http://openstax.org/l/detcoterm)\nPositive and Negative Coterminal Angles(http://openstax.org/l/posnegcoterm)\nRadian Measure(http://openstax.org/l/radianmeas)\nCoterminal Angles in Radians(http://openstax.org/l/cotermrad)\nArc Length and Area of a Sector(http://openstax.org/l/arclength)\n7.1 SECTION EXERCISES\nVerbal\n1. Draw an angle in standard 2. Explain why there are an 3. State what a positive or"
	},
	{
		"chunk_id" : "00002173",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "position. Label the vertex, infinite number of angles negative angle signifies, and\ninitial side, and terminal that are coterminal to a explain how to draw each.\nside. certain angle.\nAccess for free at openstax.org\n7.1  Angles 701\n4. How does radian measure 5. Explain the differences\nof an angle compare to the between linear speed and\ndegree measure? Include an angular speed when\nexplanation of 1 radian in describing motion along a\nyour paragraph. circular path.\nGraphical"
	},
	{
		"chunk_id" : "00002174",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "your paragraph. circular path.\nGraphical\nFor the following exercises, draw an angle in standard position with the given measure.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n18. 19. 20.\n21.\nFor the following exercises, refer toFigure 25. Round to two decimal places.\nFigure25\n22. Find the arc length. 23. Find the area of the sector.\n702 7  The Unit Circle: Sine and Cosine Functions\nFor the following exercises, refer toFigure 26. Round to two decimal places.\nFigure26"
	},
	{
		"chunk_id" : "00002175",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure26\n24. Find the arc length. 25. Find the area of the sector.\nAlgebraic\nFor the following exercises, convert angles in radians to degrees.\n26. radians 27. radians 28. radians\n29. radians 30. radians 31. radians\n32. radians\nFor the following exercises, convert angles in degrees to radians.\n33. 34. 35.\n36. 37. 38.\n39.\nFor the following exercises, use the given information to find the length of a circular arc. Round to two decimal places."
	},
	{
		"chunk_id" : "00002176",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "40. Find the length of the arc 41. Find the length of the arc 42. Find the length of the arc\nof a circle of radius 12 of a circle of radius 5.02 of a circle of diameter 14\ninches subtended by a miles subtended by the meters subtended by the\ncentral angle of radians. central angle of central angle of\n43. Find the length of the arc 44. Find the length of the arc 45. Find the length of the arc\nof a circle of radius 10 of a circle of radius 5 of a circle of diameter 12"
	},
	{
		"chunk_id" : "00002177",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "centimeters subtended by inches subtended by the meters subtended by the\nthe central angle of central angle of central angle is\nAccess for free at openstax.org\n7.1  Angles 703\nFor the following exercises, use the given information to find the area of the sector. Round to four decimal places.\n46. A sector of a circle has a 47. A sector of a circle has a 48. A sector of a circle with\ncentral angle of and a central angle of and a diameter 10 feet and an\nradius 6 cm. radius of 20 cm. angle of radians."
	},
	{
		"chunk_id" : "00002178",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radius 6 cm. radius of 20 cm. angle of radians.\n49. A sector of a circle with\nradius of 0.7 inches and an\nangle of radians.\nFor the following exercises, find the angle between and that is coterminal to the given angle.\n50. 51. 52.\n53.\nFor the following exercises, find the angle between 0 and in radians that is coterminal to the given angle.\n54. 55. 56.\n57.\nReal-World Applications\n58. A truck with 32-inch 59. A bicycle with 24-inch 60. A wheel of radius 8 inches"
	},
	{
		"chunk_id" : "00002179",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "diameter wheels is diameter wheels is is rotating 15/s. What is\ntraveling at 60 mi/h. Find traveling at 15 mi/h. Find the linear speed the\nthe angular speed of the the angular speed of the angular speed in RPM, and\nwheels in rad/min. How wheels in rad/min. How the angular speed in rad/s?\nmany revolutions per many revolutions per\nminute do the wheels minute do the wheels\nmake? make?\n61. A wheel of radius inches 62. A computer hard drive disc 63. When being burned in a"
	},
	{
		"chunk_id" : "00002180",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is rotating rad/s. What has diameter of 120 writable CD-R drive, the\nis the linear speed the millimeters. When playing angular speed of a CD is\nangular speed in RPM, and audio, the angular speed often much faster than\nthe angular speed in deg/ varies to keep the linear when playing audio, but\ns? speed constant where the the angular speed still\ndisc is being read. When varies to keep the linear\nreading along the outer speed constant where the\nedge of the disc, the disc is being written. When"
	},
	{
		"chunk_id" : "00002181",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "edge of the disc, the disc is being written. When\nangular speed is about 200 writing along the outer\nRPM (revolutions per edge of the disc, the\nminute). Find the linear angular speed of one drive\nspeed. is about 4800 RPM\n(revolutions per minute).\nFind the linear speed if the\nCD has diameter of 120\nmillimeters.\n704 7  The Unit Circle: Sine and Cosine Functions\n64. A person is standing on the 65. Find the distance along an 66. Find the distance along an"
	},
	{
		"chunk_id" : "00002182",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equator of Earth (radius arc on the surface of Earth arc on the surface of Earth\n3960 miles). What are their that subtends a central that subtends a central\nlinear and angular speeds? angle of 5 minutes angle of 7 minutes\n. .\nThe radius of Earth is 3960 The radius of Earth is\nmiles. miles.\n67. Consider a clock with an\nhour hand and minute\nhand. What is the measure\nof the angle the minute\nhand traces in minutes?\nExtensions\n68. Two cities have the same 69. A city is located at 40 70. A city is located at 75"
	},
	{
		"chunk_id" : "00002183",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "longitude. The latitude of degrees north latitude. degrees north latitude.\ncity A is 9.00 degrees north Assume the radius of the Assume the radius of the\nand the latitude of city B is earth is 3960 miles and the earth is 3960 miles and the\n30.00 degree north. earth rotates once every 24 earth rotates once every 24\nAssume the radius of the hours. Find the linear hours. Find the linear\nearth is 3960 miles. Find speed of a person who speed of a person who"
	},
	{
		"chunk_id" : "00002184",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the distance between the resides in this city. resides in this city.\ntwo cities.\n71. Find the linear speed of the 72. A bicycle has wheels 28 73. A car travels 3 miles. Its\nmoon if the average inches in diameter. A tires make 2640\ndistance between the earth tachometer determines revolutions. What is the\nand moon is 239,000 miles, that the wheels are radius of a tire in inches?\nassuming the orbit of the rotating at 180 RPM\nmoon is circular and (revolutions per minute)."
	},
	{
		"chunk_id" : "00002185",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "moon is circular and (revolutions per minute).\nrequires about 28 days. Find the speed the bicycle\nExpress answer in miles is travelling down the road.\nper hour.\n74. A wheel on a tractor has a\n24-inch diameter. How\nmany revolutions does the\nwheel make if the tractor\ntravels 4 miles?\n7.2 Right Triangle Trigonometry\nLearning Objectives\nIn this section you will:\nUse right triangles to evaluate trigonometric functions.\nFind function values for and\nUse equal cofunctions of complementary angles."
	},
	{
		"chunk_id" : "00002186",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use equal cofunctions of complementary angles.\nUse the definitions of trigonometric functions of any angle.\nUse right-triangle trigonometry to solve applied problems.\nMt. Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its\nheight is no easy task. In fact, the actual measurement has been a source of controversy for hundreds of years. The\nAccess for free at openstax.org\n7.2  Right Triangle Trigonometry 705"
	},
	{
		"chunk_id" : "00002187",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.2  Right Triangle Trigonometry 705\nmeasurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section,\nwe will define a new group of functions known as trigonometric functions, and find out how they can be used to\nmeasure heights, such as those of the tallest mountains.\nUsing Right Triangles to Evaluate Trigonometric Functions\nFigure 1shows aright trianglewith a vertical side of length and a horizontal side has length Notice that the triangle"
	},
	{
		"chunk_id" : "00002188",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is inscribed in a circle of radius 1. Such a circle, with a center at the origin and a radius of 1, is known as aunit circle.\nFigure1\nWe can define the trigonometric functions in terms an angletand the lengths of the sides of the triangle. Theadjacent\nsideis the side closest to the angle,x. (Adjacent means next to.) Theopposite sideis the side across from the angle,y.\nThehypotenuseis the side of the triangle opposite the right angle, 1. These sides are labeled inFigure 2."
	},
	{
		"chunk_id" : "00002189",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2 The sides of a right triangle in relation to angle\nGiven a right triangle with an acute angle of the first three trigonometric functions are listed.\nA common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of Sine is\nopposite overhypotenuse,Cosine isadjacent overhypotenuse,Tangent isopposite overadjacent.\nFor the triangle shown inFigure 1, we have the following.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002190",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the side lengths of a right triangle and one of the acute angles, find the sine, cosine, and tangent of that\nangle.\n1. Find the sine as the ratio of the opposite side to the hypotenuse.\n2. Find the cosine as the ratio of the adjacent side to the hypotenuse.\n3. Find the tangent as the ratio of the opposite side to the adjacent side.\n706 7  The Unit Circle: Sine and Cosine Functions\nEXAMPLE1\nEvaluating a Trigonometric Function of a Right Triangle"
	},
	{
		"chunk_id" : "00002191",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given the triangle shown inFigure 3, find the value of\nFigure3\nSolution\nThe side adjacent to the angle is 15, and the hypotenuse of the triangle is 17.\nTRY IT #1 Given the triangle shown inFigure 4, find the value of\nFigure4\nReciprocal Functions\nIn addition to sine, cosine, and tangent, there are three more functions. These too are defined in terms of the sides of\nthe triangle.\nTake another look at these definitions. These functions are the reciprocals of the first three functions."
	},
	{
		"chunk_id" : "00002192",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle.\nIn fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle inFigure 5. The\nside opposite one acute angle is the side adjacent to the other acute angle, and vice versa.\nAccess for free at openstax.org\n7.2  Right Triangle Trigonometry 707\nFigure5 The side adjacent to one angle is opposite the other angle."
	},
	{
		"chunk_id" : "00002193",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find\nthe sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.\n...\nHOW TO\nGiven the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.\n1. If needed, draw the right triangle and label the angle provided."
	},
	{
		"chunk_id" : "00002194",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.\n3. Find the required function:\n sine as the ratio of the opposite side to the hypotenuse\n cosine as the ratio of the adjacent side to the hypotenuse\n tangent as the ratio of the opposite side to the adjacent side\n secant as the ratio of the hypotenuse to the adjacent side\n cosecant as the ratio of the hypotenuse to the opposite side"
	},
	{
		"chunk_id" : "00002195",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " cotangent as the ratio of the adjacent side to the opposite side\nEXAMPLE2\nEvaluating Trigonometric Functions of Angles Not in Standard Position\nUsing the triangle shown inFigure 6, evaluate\nFigure6\nSolution\nAnalysis\nAnother approach would have been to find sine, cosine, and tangent first. Then find their reciprocals to determine the\nother functions.\n708 7  The Unit Circle: Sine and Cosine Functions\nTRY IT #2 Using the triangle shown inFigure 7,evaluate\nFigure7"
	},
	{
		"chunk_id" : "00002196",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure7\nFinding Trigonometric Functions of Special Angles Using Side Lengths\nIt is helpful to evaluate the trigonometric functions as they relate to the special anglesmultiples of and\nRemember, however, that when dealing with right triangles, we are limited to angles between\nSuppose we have a triangle, which can also be described as a triangle. The sides have lengths in\nthe relation The sides of a triangle, which can also be described as a triangle, have"
	},
	{
		"chunk_id" : "00002197",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "lengths in the relation These relations are shown inFigure 8.\nFigure8 Side lengths of special triangles\nWe can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.\n...\nHOW TO\nGiven trigonometric functions of a special angle, evaluate using side lengths.\n1. Use the side lengths shown inFigure 8for the special angle you wish to evaluate.\n2. Use the ratio of side lengths appropriate to the function you wish to evaluate.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002198",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.2  Right Triangle Trigonometry 709\nEXAMPLE3\nEvaluating Trigonometric Functions of Special Angles Using Side Lengths\nFind the exact value of the trigonometric functions of using side lengths.\nSolution\nTRY IT #3 Find the exact value of the trigonometric functions of using side lengths.\nUsing Equal Cofunction of Complements\nIf we look more closely at the relationship between the sine and cosine of the special angles, we notice a pattern. In a"
	},
	{
		"chunk_id" : "00002199",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "right triangle with angles of and we see that the sine of namely is also the cosine of while the sine of\nnamely is also the cosine of\nSeeFigure 9.\nFigure9 The sine of equals the cosine of and vice versa.\nThis result should not be surprising because, as we see fromFigure 9, the side opposite the angle of is also the side\nadjacent to so and are exactly the same ratio of the same two sides, and Similarly,\nand are also the same ratio using the same two sides, and"
	},
	{
		"chunk_id" : "00002200",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The interrelationship between the sines and cosines of and also holds for the two acute angles in any right triangle,\nsince in every case, the ratio of the same two sides would constitute the sine of one angle and the cosine of the other.\n710 7  The Unit Circle: Sine and Cosine Functions\nSince the three angles of a triangle add to and the right angle is the remaining two angles must also add up to\nThat means that a right triangle can be formed with any two angles that add to in other words, any two"
	},
	{
		"chunk_id" : "00002201",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "complementary angles. So we may state acofunction identity: If any two angles are complementary, the sine of one is\nthe cosine of the other, and vice versa. This identity is illustrated inFigure 10.\nFigure10 Cofunction identity of sine and cosine of complementary angles\nUsing this identity, we can state without calculating, for instance, that the sine of equals the cosine of and that\nthe sine of equals the cosine of We can also state that if, for a given angle then as\nwell.\nCofunction Identities"
	},
	{
		"chunk_id" : "00002202",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "well.\nCofunction Identities\nThecofunction identitiesin radians are listed inTable 1.\nTable1\n...\nHOW TO\nGiven the sine and cosine of an angle, find the sine or cosine of its complement.\n1. To find the sine of the complementary angle, find the cosine of the original angle.\n2. To find the cosine of the complementary angle, find the sine of the original angle.\nEXAMPLE4\nUsing Cofunction Identities\nIf find\nSolution\nAccording to the cofunction identities for sine and cosine, we have the following."
	},
	{
		"chunk_id" : "00002203",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.2  Right Triangle Trigonometry 711\nSo\nTRY IT #4 If find\nUsing Trigonometric Functions\nIn previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of\nright-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.\n...\nHOW TO\nGiven a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides."
	},
	{
		"chunk_id" : "00002204",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. For each side, select the trigonometric function that has the unknown side as either the numerator or the\ndenominator. The known side will in turn be the denominator or the numerator.\n2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.\n3. Using the value of the trigonometric function and the known side length, solve for the missing side length.\nEXAMPLE5\nFinding Missing Side Lengths Using Trigonometric Ratios"
	},
	{
		"chunk_id" : "00002205",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the unknown sides of the triangle inFigure 11.\nFigure11\nSolution\nWe know the angle and the opposite side, so we can use the tangent to find the adjacent side.\nWe rearrange to solve for\nWe can use the sine to find the hypotenuse.\nAgain, we rearrange to solve for\n712 7  The Unit Circle: Sine and Cosine Functions\nTRY IT #5 A right triangle has one angle of and a hypotenuse of 20. Find the unknown sides and angle of\nthe triangle.\nUsing Right Triangle Trigonometry to Solve Applied Problems"
	},
	{
		"chunk_id" : "00002206",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a\ntriangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape\nmeasure along its height. We do so by measuring a distance from the base of the object to a point on the ground some\ndistance away, where we can look up to the top of the tall object at an angle. Theangle of elevationof an object above"
	},
	{
		"chunk_id" : "00002207",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "an observer relative to the observer is the angle between the horizontal and the line from the object to the observer's\neye. The right triangle this position creates has sides that represent the unknown height, the measured distance from\nthe base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the\nbase of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height."
	},
	{
		"chunk_id" : "00002208",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, we can form a triangle from the top of a tall object by looking downward. Theangle of depressionof an object\nbelow an observer relative to the observer is the angle between the horizontal and the line from the object to the\nobserver's eye. SeeFigure 12.\nFigure12\n...\nHOW TO\nGiven a tall object, measure its height indirectly.\n1. Make a sketch of the problem situation to keep track of known and unknown information."
	},
	{
		"chunk_id" : "00002209",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.\n3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight\nmakes with the horizontal.\n4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line\nof sight.\n5. Solve the equation for the unknown height.\nEXAMPLE6\nMeasuring a Distance Indirectly"
	},
	{
		"chunk_id" : "00002210",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nMeasuring a Distance Indirectly\nTo find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of\nbetween a line of sight to the top of the tree and the ground, as shown inFigure 13. Find the height of the tree.\nAccess for free at openstax.org\n7.2  Right Triangle Trigonometry 713\nFigure13\nSolution\nWe know that the angle of elevation is and the adjacent side is 30 ft long. The opposite side is the unknown height."
	},
	{
		"chunk_id" : "00002211",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we\nwill state our information in terms of the tangent of letting be the unknown height.\nThe tree is approximately 46 feet tall.\nTRY IT #6 How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests\nagainst the building making an angle of with the ground? Round to the nearest foot.\nMEDIA"
	},
	{
		"chunk_id" : "00002212",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with right triangle trigonometry.\nFinding Trig Functions on Calculator(http://openstax.org/l/findtrigcal)\nFinding Trig Functions Using a Right Triangle(http://openstax.org/l/trigrttri)\nRelate Trig Functions to Sides of a Right Triangle(http://openstax.org/l/reltrigtri)\nDetermine Six Trig Functions from a Triangle(http://openstax.org/l/sixtrigfunc)\nDetermine Length of Right Triangle Side(http://openstax.org/l/rttriside)"
	},
	{
		"chunk_id" : "00002213",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.2 SECTION EXERCISES\nVerbal\n1. For the given right triangle, label the adjacent side, 2. When a right triangle with a hypotenuse of 1 is\nopposite side, and hypotenuse for the indicated placed in a circle of radius 1, which sides of the\nangle. triangle correspond to thex- andy-coordinates?\n714 7  The Unit Circle: Sine and Cosine Functions\n3. The tangent of an angle 4. What is the relationship 5. Explain the cofunction\ncompares which sides of the between the two acute identity."
	},
	{
		"chunk_id" : "00002214",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "right triangle? angles in a right triangle?\nAlgebraic\nFor the following exercises, use cofunctions of complementary angles.\n6. 7. 8.\n9.\nFor the following exercises, find the lengths of the missing sides if side is opposite angle side is opposite angle\nand side is the hypotenuse.\n10. 11. 12.\n13. 14. 15.\n16.\nGraphical\nFor the following exercises, useFigure 14to evaluate each trigonometric function of angle\nFigure14\n17. 18. 19.\n20. 21. 22.\nAccess for free at openstax.org\n7.2  Right Triangle Trigonometry 715"
	},
	{
		"chunk_id" : "00002215",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.2  Right Triangle Trigonometry 715\nFor the following exercises, useFigure 15to evaluate each trigonometric function of angle\nFigure15\n23. 24. 25.\n26. 27. 28.\nFor the following exercises, solve for the unknown sides of the given triangle.\n29. 30. 31.\nTechnology\nFor the following exercises, use a calculator to find the length of each side to four decimal places.\n32. 33. 34.\n35. 36.\n37. 38. 39.\n40. 41.\n716 7  The Unit Circle: Sine and Cosine Functions\nExtensions\n42. Find 43. Find 44. Find"
	},
	{
		"chunk_id" : "00002216",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Extensions\n42. Find 43. Find 44. Find\n45. Find 46. A radio tower is located 400 47. A radio tower is located 325\nfeet from a building. From feet from a building. From\na window in the building, a a window in the building, a\nperson determines that the person determines that the\nangle of elevation to the angle of elevation to the\ntop of the tower is and top of the tower is and\nthat the angle of that the angle of\ndepression to the bottom depression to the bottom\nof the tower is How of the tower is How"
	},
	{
		"chunk_id" : "00002217",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the tower is How of the tower is How\ntall is the tower? tall is the tower?\n48. A 200-foot tall monument 49. A 400-foot tall monument 50. There is an antenna on the\nis located in the distance. is located in the distance. top of a building. From a\nFrom a window in a From a window in a location 300 feet from the\nbuilding, a person building, a person base of the building, the\ndetermines that the angle determines that the angle angle of elevation to the"
	},
	{
		"chunk_id" : "00002218",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of elevation to the top of of elevation to the top of top of the building is\nthe monument is and the monument is and measured to be From\nthat the angle of that the angle of the same location, the\ndepression to the bottom depression to the bottom angle of elevation to the\nof the monument is of the monument is top of the antenna is\nHow far is the person from How far is the person from measured to be Find\nthe monument? the monument? the height of the antenna.\n51. There is lightning rod on"
	},
	{
		"chunk_id" : "00002219",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "51. There is lightning rod on\nthe top of a building. From\na location 500 feet from the\nbase of the building, the\nangle of elevation to the\ntop of the building is\nmeasured to be From\nthe same location, the\nangle of elevation to the\ntop of the lightning rod is\nmeasured to be Find\nthe height of the lightning\nrod.\nAccess for free at openstax.org\n7.3  Unit Circle 717\nReal-World Applications\n52. A 33-ft ladder leans against 53. A 23-ft ladder leans against 54. The angle of elevation to"
	},
	{
		"chunk_id" : "00002220",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a building so that the angle a building so that the angle the top of a building in\nbetween the ground and between the ground and Charlotte is found to be 9\nthe ladder is How high the ladder is How high degrees from the ground\ndoes the ladder reach up does the ladder reach up at a distance of 1 mile from\nthe side of the building? the side of the building? the base of the building.\nUsing this information, find\nthe height of the building.\n55. The angle of elevation to 56. Assuming that a 370-foot"
	},
	{
		"chunk_id" : "00002221",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the top of a building in tall giant redwood grows\nSeattle is found to be 2 vertically, if I walk a certain\ndegrees from the ground distance from the tree and\nat a distance of 2 miles measure the angle of\nfrom the base of the elevation to the top of the\nbuilding. Using this tree to be how far\ninformation, find the from the base of the tree\nheight of the building. am I?\n7.3 Unit Circle\nLearning Objectives\nIn this section you will:\nFind function values for the sine and cosine of and"
	},
	{
		"chunk_id" : "00002222",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identify the domain and range of sine and cosine functions.\nFind reference angles.\nUse reference angles to evaluate trigonometric functions.\nFigure1 The Singapore Flyer was the worlds tallest Ferris wheel until being overtaken by the High Roller in Las Vegas\nand the Ain Dubai in Dubai. (credit: Vibin JK/Flickr)\nLooking for a thrill? Then consider a ride on the Ain Dubai, the world's tallest Ferris wheel. Located in Dubai, the most"
	},
	{
		"chunk_id" : "00002223",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "populous city and the financial and tourism hub of the United Arab Emirates, the wheel soars to 820 feet, about 1.5\ntenths of a mile. Described as an observation wheel, riders enjoy spectacular views of the Burj Khalifa (the world's tallest\nbuilding) and the Palm Jumeirah (a human-made archipelago home to over 10,000 people and 20 resorts) as they travel\nfrom the ground to the peak and down again in a repeating pattern. In this section, we will examine this type of"
	},
	{
		"chunk_id" : "00002224",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "revolving motion around a circle. To do so, we need to define the type of circle first, and then place that circle on a\ncoordinate system. Then we can discuss circular motion in terms of the coordinate pairs.\nFinding Trigonometric Functions Using the Unit Circle\nWe have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in\n718 7  The Unit Circle: Sine and Cosine Functions"
	},
	{
		"chunk_id" : "00002225",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms of the unit circle. Recall that a unit circle is a circle centered at the origin with radius 1, as shown inFigure 2. The\nangle (in radians) that intercepts forms an arc of length Using the formula and knowing that we see\nthat for a unit circle,\nThex-andy-axes divide the coordinate plane into four quarters called quadrants. We label these quadrants to mimic the\ndirection a positive angle would sweep. The four quadrants are labeled I, II, III, and IV."
	},
	{
		"chunk_id" : "00002226",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For any angle we can label the intersection of the terminal side and the unit circle as by its coordinates, The\ncoordinates and will be the outputs of the trigonometric functions and respectively. This\nmeans and\nFigure2 Unit circle where the central angle is radians\nUnit Circle\nAunit circlehas a center at and radius In a unit circle, the length of the intercepted arc is equal to the radian\nmeasure of the central angle"
	},
	{
		"chunk_id" : "00002227",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "measure of the central angle\nLet be the endpoint on the unit circle of an arc ofarc length The coordinates of this point can be\ndescribed as functions of the angle.\nDefining Sine and Cosine Functions from the Unit Circle\nThe sine function relates a real number to they-coordinate of the point where the corresponding angle intercepts the\nunit circle. More precisely, the sine of an angle equals they-value of the endpoint on the unit circle of an arc of length"
	},
	{
		"chunk_id" : "00002228",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "InFigure 2, the sine is equal to Like all functions, thesine functionhas an input and an output. Its input is the\nmeasure of the angle; its output is they-coordinate of the corresponding point on the unit circle.\nThecosine functionof an angle equals thex-value of the endpoint on the unit circle of an arc of length InFigure 3,\nthe cosine is equal to\nFigure3\nBecause it is understood that sine and cosine are functions, we do not always need to write them with parentheses:"
	},
	{
		"chunk_id" : "00002229",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the same as and is the same as Likewise, is a commonly used shorthand notation for\nAccess for free at openstax.org\n7.3  Unit Circle 719\nBe aware that many calculators and computers do not recognize the shorthand notation. When in doubt, use the extra\nparentheses when entering calculations into a calculator or computer.\nSine and Cosine Functions\nIf is a real number and a point on the unit circle corresponds to a central angle then\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002230",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a pointP on the unit circle corresponding to an angle of find the sine and cosine.\n1. The sine of is equal to they-coordinate of point\n2. The cosine of is equal to thex-coordinate of point\nEXAMPLE1\nFinding Function Values for Sine and Cosine\nPoint is a point on the unit circle corresponding to an angle of as shown inFigure 4. Find and\nFigure4\nSolution\nWe know that is thex-coordinate of the corresponding point on the unit circle and is they-coordinate of the"
	},
	{
		"chunk_id" : "00002231",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "corresponding point on the unit circle. So:\nTRY IT #1 A certain angle corresponds to a point on the unit circle at as shown inFigure 5.\nFind and\n720 7  The Unit Circle: Sine and Cosine Functions\nFigure5\nFinding Sines and Cosines of Angles on an Axis\nFor quadrantral angles, the corresponding point on the unit circle falls on thex-ory-axis. In that case, we can easily\ncalculate cosine and sine from the values of and\nEXAMPLE2\nCalculating Sines and Cosines along an Axis\nFind and\nSolution"
	},
	{
		"chunk_id" : "00002232",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find and\nSolution\nMoving counterclockwise around the unit circle from the positivex-axis brings us to the top of the circle, where the\ncoordinates are as shown inFigure 6.\nFigure6\nWe can then use our definitions of cosine and sine.\nThe cosine of is 0; the sine of is 1.\nTRY IT #2 Find cosine and sine of the angle\nAccess for free at openstax.org\n7.3  Unit Circle 721\nThe Pythagorean Identity"
	},
	{
		"chunk_id" : "00002233",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3  Unit Circle 721\nThe Pythagorean Identity\nNow that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the\nequation for the unit circle is Because and we can substitute for and to get\nThis equation, is known as thePythagorean Identity. SeeFigure 7.\nFigure7\nWe can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or vice versa. However, because"
	},
	{
		"chunk_id" : "00002234",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct\nsign. If we know the quadrant where the angle is, we can easily choose the correct solution.\nPythagorean Identity\nThePythagorean Identitystates that, for any real number\n...\nHOW TO\nGiven the sine of some angle and its quadrant location, find the cosine of\n1. Substitute the known value of into the Pythagorean Identity.\n2. Solve for"
	},
	{
		"chunk_id" : "00002235",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Solve for\n3. Choose the solution with the appropriate sign for thex-values in the quadrant where is located.\nEXAMPLE3\nFinding a Cosine from a Sine or a Sine from a Cosine\nIf and is in the second quadrant, find\nSolution\nIf we drop a vertical line from the point on the unit circle corresponding to we create a right triangle, from which we\ncan see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. SeeFigure 8.\n722 7  The Unit Circle: Sine and Cosine Functions\nFigure8"
	},
	{
		"chunk_id" : "00002236",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nSubstituting the known value for sine into the Pythagorean Identity,\nBecause the angle is in the second quadrant, we know thex-value is a negative real number, so the cosine is also\nnegative.\nTRY IT #3 If and is in the fourth quadrant, find\nFinding Sines and Cosines of Special Angles\nWe have already learned some properties of the special angles, such as the conversion from radians to degrees, and we"
	},
	{
		"chunk_id" : "00002237",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "found their sines and cosines using right triangles. We can also calculate sines and cosines of the special angles using\nthe Pythagorean Identity.\nFinding Sines and Cosines of Angles\nFirst, we will look at angles of or as shown inFigure 9. A triangle is an isosceles triangle, so thex-\nandy-coordinates of the corresponding point on the circle are the same. Because thex-andy-values are the same, the\nsine and cosine values will also be equal.\nFigure9"
	},
	{
		"chunk_id" : "00002238",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nAt which is 45 degrees, the radius of the unit circle bisects the first quadrantal angle. This means the radius lies\nalong the line A unit circle has a radius equal to 1 so the right triangle formed below the line has sides\nAccess for free at openstax.org\n7.3  Unit Circle 723\nand and radius = 1. SeeFigure 10.\nFigure10\nFrom the Pythagorean Theorem we get\nWe can then substitute\nNext we combine like terms.\nAnd solving for we get\nIn quadrant I,\nAt or 45 degrees,"
	},
	{
		"chunk_id" : "00002239",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In quadrant I,\nAt or 45 degrees,\nIf we then rationalize the denominators, we get\nTherefore, the coordinates of a point on a circle of radius at an angle of are\nFinding Sines and Cosines of and Angles\nNext, we will find the cosine and sine at an angle of or First, we will draw a triangle inside a circle with one side at\n724 7  The Unit Circle: Sine and Cosine Functions\nan angle of and another at an angle of as shown inFigure 11. If the resulting two right triangles are combined"
	},
	{
		"chunk_id" : "00002240",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "into one large triangle, notice that all three angles of this larger triangle will be as shown inFigure 12.\nFigure11\nFigure12\nBecause all the angles are equal, the sides are also equal. The vertical line has length and since the sides are all\nequal, we can also conclude that or Since\nAnd since in our unit circle,\nUsing the Pythagorean Identity, we can find the cosine value.\nThe coordinates for the point on a circle of radius at an angle of are At the radius of"
	},
	{
		"chunk_id" : "00002241",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the unit circle, 1, serves as the hypotenuse of a 30-60-90 degree right triangle, as shown inFigure 13. Angle has\nmeasure At point we draw an angle with measure of We know the angles in a triangle sum to so\nthe measure of angle is also Now we have an equilateral triangle. Because each side of the equilateral triangle\nis the same length, and we know one side is the radius of the unit circle, all sides must be of length 1.\nAccess for free at openstax.org\n7.3  Unit Circle 725\nFigure13"
	},
	{
		"chunk_id" : "00002242",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3  Unit Circle 725\nFigure13\nThe measure of angle is 30. Angle is double angle so its measure is 60. is the perpendicular\nbisector of so it cuts in half. This means that is the radius, or Notice that is thex-coordinate of\npoint which is at the intersection of the 60 angle and the unit circle. This gives us a triangle with hypotenuse of\n1 and side of length\nFrom the Pythagorean Theorem, we get\nSubstituting we get\nSolving for we get"
	},
	{
		"chunk_id" : "00002243",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substituting we get\nSolving for we get\nSince has the terminal side in quadrant I where they-coordinate is positive, we choose the positive value.\nAt (60), the coordinates for the point on a circle of radius at an angle of are so we can find\nthe sine and cosine.\nWe have now found the cosine and sine values for all of the most commonly encountered angles in the first quadrant of\nthe unit circle.Table 1summarizes these values.\nAngle or or or or\nTable1\n726 7  The Unit Circle: Sine and Cosine Functions"
	},
	{
		"chunk_id" : "00002244",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Cosine 1 0\nSine 0 1\nTable1\nFigure 14shows the common angles in the first quadrant of the unit circle.\nFigure14\nUsing a Calculator to Find Sine and Cosine\nTo find the cosine and sine of angles other than the special angles, we turn to a computer or calculator.Be aware: Most\ncalculators can be set into degree or radian mode, which tells the calculator the units for the input value. When we"
	},
	{
		"chunk_id" : "00002245",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "evaluate on our calculator, it will evaluate it as the cosine of 30 degrees if the calculator is in degree mode, or the\ncosine of 30 radians if the calculator is in radian mode.\n...\nHOW TO\nGiven an angle in radians, use a graphing calculator to find the cosine.\n1. If the calculator has degree mode and radian mode, set it to radian mode.\n2. Press the COS key.\n3. Enter the radian value of the angle and press the close-parentheses key \")\"\"."
	},
	{
		"chunk_id" : "00002246",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Press ENTER.\nEXAMPLE4\nUsing a Graphing Calculator to Find Sine and Cosine\nEvaluate using a graphing calculator or computer.\nSolution\nEnter the following keystrokes:\nAccess for free at openstax.org\n7.3  Unit Circle 727\nAnalysis\nWe can find the cosine or sine of an angle in degrees directly on a calculator with degree mode. For calculators or\nsoftware that use only radian mode, we can find the sine of for example, by including the conversion factor to\nradians as part of the input:\nTRY IT #4 Evaluate"
	},
	{
		"chunk_id" : "00002247",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radians as part of the input:\nTRY IT #4 Evaluate\nIdentifying the Domain and Range of Sine and Cosine Functions\nNow that we can find the sine and cosine of an angle, we need to discuss their domains and ranges. What are the\ndomains of the sine and cosine functions? That is, what are the smallest and largest numbers that can be inputs of the\nfunctions? Because angles smaller than and angles larger than can still be graphed on the unit circle and have real"
	},
	{
		"chunk_id" : "00002248",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values of and there is no lower or upper limit to the angles that can be inputs to the sine and cosine functions.\nThe input to the sine and cosine functions is the rotation from the positivex-axis, and that may be any real number.\nWhat are the ranges of the sine and cosine functions? What are the least and greatest possible values for their output?\nWe can see the answers by examining the unit circle, as shown inFigure 15. The bounds of thex-coordinate are"
	},
	{
		"chunk_id" : "00002249",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The bounds of they-coordinate are also Therefore, the range of both the sine and cosine functions is\nFigure15\nFinding Reference Angles\nWe have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another\nquadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value.\nBecause the sine value is they-coordinate on the unit circle, the other angle with the same sine will share the same"
	},
	{
		"chunk_id" : "00002250",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "y-value, but have the oppositex-value. Therefore, its cosine value will be the opposite of the first angles cosine value.\nLikewise, there will be an angle in the fourth quadrant with the same cosine as the original angle. The angle with the\nsame cosine will share the samex-value but will have the oppositey-value. Therefore, its sine value will be the opposite\nof the original angles sine value.\nAs shown inFigure 16, angle has the same sine value as angle the cosine values are opposites. Angle has the same"
	},
	{
		"chunk_id" : "00002251",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cosine value as angle the sine values are opposites.\n728 7  The Unit Circle: Sine and Cosine Functions\nFigure16\nRecall that an angles reference angle is the acute angle, formed by the terminal side of the angle and the horizontal\naxis. A reference angle is always an angle between and or and radians. As we can see fromFigure 17, for any\nangle in quadrants II, III, or IV, there is a reference angle in quadrant I.\nFigure17\n...\nHOW TO\nGiven an angle between and find its reference angle."
	},
	{
		"chunk_id" : "00002252",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. An angle in the first quadrant is its own reference angle.\n2. For an angle in the second or third quadrant, the reference angle is or\n3. For an angle in the fourth quadrant, the reference angle is or\n4. If an angle is less than or greater than add or subtract as many times as needed to find an equivalent\nangle between and\nEXAMPLE5\nFinding a Reference Angle\nFind the reference angle of as shown inFigure 18.\nAccess for free at openstax.org\n7.3  Unit Circle 729\nFigure18\nSolution"
	},
	{
		"chunk_id" : "00002253",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3  Unit Circle 729\nFigure18\nSolution\nBecause is in the third quadrant, the reference angle is\nTRY IT #5 Find the reference angle of\nUsing Reference Angles\nNow lets take a moment to reconsider the Ferris wheel introduced at the beginning of this section. Suppose a rider\nsnaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way\naround the circle. What is the riders new elevation? To answer questions such as this one, we need to evaluate the sine"
	},
	{
		"chunk_id" : "00002254",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or cosine functions at angles that are greater than 90 degrees or at a negative angle. Reference angles make it possible\nto evaluate trigonometric functions for angles outside the first quadrant. They can also be used to find coordinates\nfor those angles. We will use thereference angleof the angle of rotation combined with the quadrant in which the\nterminal side of the angle lies.\nUsing Reference Angles to Evaluate Trigonometric Functions"
	},
	{
		"chunk_id" : "00002255",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The\nabsolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the\nquadrant of the original angle. The cosine will be positive or negative depending on the sign of thex-values in that\nquadrant. The sine will be positive or negative depending on the sign of they-values in that quadrant.\nUsing Reference Angles to Find Cosine and Sine"
	},
	{
		"chunk_id" : "00002256",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Reference Angles to Find Cosine and Sine\nAngles have cosines and sines with the same absolute value as their reference angles. The sign (positive or negative)\ncan be determined from the quadrant of the angle.\n...\nHOW TO\nGiven an angle in standard position, find the reference angle, and the cosine and sine of the original angle.\n1. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference\nangle."
	},
	{
		"chunk_id" : "00002257",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle.\n2. Determine the values of the cosine and sine of the reference angle.\n3. Give the cosine the same sign as thex-values in the quadrant of the original angle.\n730 7  The Unit Circle: Sine and Cosine Functions\n4. Give the sine the same sign as they-values in the quadrant of the original angle.\nEXAMPLE6\nUsing Reference Angles to Find Sine and Cosine\n Using a reference angle, find the exact value of and\n Using the reference angle, find and\nSolution"
	},
	{
		"chunk_id" : "00002258",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Using the reference angle, find and\nSolution\n is located in the second quadrant. The angle it makes with thex-axis is so the reference\nangle is\nThis tells us that has the same sine and cosine values as except for the sign.\nSince is in the second quadrant, thex-coordinate of the point on the circle is negative, so the cosine value is\nnegative. They-coordinate is positive, so the sine value is positive.\n is in the third quadrant. Its reference angle is The cosine and sine of are both In the third"
	},
	{
		"chunk_id" : "00002259",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadrant, both and are negative, so:\nTRY IT #6  Use the reference angle of to find and\n Use the reference angle of to find and\nUsing Reference Angles to Find Coordinates\nNow that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use\nsymmetry and reference angles to fill in cosine and sine values for the rest of the special angles on theunit circle. They"
	},
	{
		"chunk_id" : "00002260",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "are shown inFigure 19. Take time to learn the coordinates of all of the major angles in the first quadrant.\nFigure19 Special angles and coordinates of corresponding points on the unit circle\nIn addition to learning the values for special angles, we can use reference angles to find coordinates of any point\non the unit circle, using what we know of reference angles along with the identities\nAccess for free at openstax.org\n7.3  Unit Circle 731"
	},
	{
		"chunk_id" : "00002261",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3  Unit Circle 731\nFirst we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the\nreference angle, and give them the signs corresponding to they- andx-values of the quadrant.\n...\nHOW TO\nGiven the angle of a point on a circle and the radius of the circle, find the coordinates of the point.\n1. Find the reference angle by measuring the smallest angle to thex-axis.\n2. Find the cosine and sine of the reference angle."
	},
	{
		"chunk_id" : "00002262",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Determine the appropriate signs for and in the given quadrant.\nEXAMPLE7\nUsing the Unit Circle to Find Coordinates\nFind the coordinates of the point on the unit circle at an angle of\nSolution\nWe know that the angle is in the third quadrant.\nFirst, lets find the reference angle by measuring the angle to thex-axis. To find the reference angle of an angle whose\nterminal side is in quadrant III, we find the difference of the angle and\nNext, we will find the cosine and sine of the reference angle."
	},
	{
		"chunk_id" : "00002263",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We must determine the appropriate signs forxandyin the given quadrant. Because our original angle is in the third\nquadrant, where both and are negative, both cosine and sine are negative.\nNow we can calculate the coordinates using the identities and\nThe coordinates of the point are on the unit circle.\nTRY IT #7 Find the coordinates of the point on the unit circle at an angle of\nMEDIA\nAccess these online resources for additional instruction and practice with sine and cosine functions."
	},
	{
		"chunk_id" : "00002264",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Trigonometric Functions Using the Unit Circle(http://openstax.org/l/trigunitcir)\nSine and Cosine from the Unit(http://openstax.org/l/sincosuc)\nSine and Cosine from the Unit Circle and Multiples of Pi Divided by Six(http://openstax.org/l/sincosmult)\nSine and Cosine from the Unit Circle and Multiples of Pi Divided by Four(http://openstax.org/l/sincosmult4)\nTrigonometric Functions Using Reference Angles(http://openstax.org/l/trigrefang)\n732 7  The Unit Circle: Sine and Cosine Functions\n7.3 SECTION EXERCISES"
	},
	{
		"chunk_id" : "00002265",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3 SECTION EXERCISES\nVerbal\n1. Describe the unit circle. 2. What do thex-and 3. Discuss the difference\ny-coordinates of the points between a coterminal angle\non the unit circle represent? and a reference angle.\n4. Explain how the cosine of an 5. Explain how the sine of an\nangle in the second angle in the second\nquadrant differs from the quadrant differs from the\ncosine of its reference angle sine of its reference angle in\nin the unit circle. the unit circle.\nAlgebraic"
	},
	{
		"chunk_id" : "00002266",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "in the unit circle. the unit circle.\nAlgebraic\nFor the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal\npoint determined by lies.\n6. and 7. and 8. and\n9. and\nFor the following exercises, find the exact value of each trigonometric function.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22.\nNumeric\nFor the following exercises, state the reference angle for the given angle.\n23. 24. 25.\n26. 27. 28.\n29. 30. 31.\n32. 33."
	},
	{
		"chunk_id" : "00002267",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. 24. 25.\n26. 27. 28.\n29. 30. 31.\n32. 33.\nAccess for free at openstax.org\n7.3  Unit Circle 733\nFor the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each\nangle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n43. 44. 45.\n46. 47. 48.\n49.\nFor the following exercises, find the requested value."
	},
	{
		"chunk_id" : "00002268",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "50. If and is in the 51. If and is in the 52. If and is in the\nfourth quadrant, find first quadrant, find second quadrant, find\n53. If and is in 54. Find the coordinates of the 55. Find the coordinates of the\nthe third quadrant, find point on a circle with radius point on a circle with radius\n15 corresponding to an 20 corresponding to an\nangle of angle of\n56. Find the coordinates of the 57. Find the coordinates of the 58. State the domain of the"
	},
	{
		"chunk_id" : "00002269",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "point on a circle with radius point on a circle with radius sine and cosine functions.\n8 corresponding to an 16 corresponding to an\nangle of angle of\n59. State the range of the sine\nand cosine functions.\nGraphical\nFor the following exercises, use the given point on the unit circle to find the value of the sine and cosine of\n60. 61. 62.\n734 7  The Unit Circle: Sine and Cosine Functions\n63. 64. 65.\n66. 67. 68.\n69. 70. 71.\n72. 73. 74.\n75. 76. 77.\nAccess for free at openstax.org\n7.3  Unit Circle 735\n78. 79."
	},
	{
		"chunk_id" : "00002270",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3  Unit Circle 735\n78. 79.\nTechnology\nFor the following exercises, use a graphing calculator to evaluate.\n80. 81. 82.\n83. 84. 85.\n86. 87. 88.\n89.\nExtensions\nFor the following exercises, evaluate.\n90. 91. 92.\n93. 94. 95.\n96. 97. 98.\n99.\nReal-World Applications\nFor the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around.\nThe child enters at the point that is, on the due north position. Assume the carousel revolves counter clockwise."
	},
	{
		"chunk_id" : "00002271",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "100. What are the coordinates 101. What are the coordinates 102. What are the coordinates\nof the child after 45 of the child after 90 of the child after 125\nseconds? seconds? seconds?\n736 7  The Unit Circle: Sine and Cosine Functions\n103. When will the child have 104. When will the child have\ncoordinates coordinates\nif the ride if the ride lasts 6 minutes?\nlasts 6 minutes? (There\nare multiple answers.)\n7.4 The Other Trigonometric Functions\nLearning Objectives\nIn this section you will:"
	},
	{
		"chunk_id" : "00002272",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section you will:\nFind exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of and\nUse reference angles to evaluate the trigonometric functions secant, tangent, and cotangent.\nUse properties of even and odd trigonometric functions.\nRecognize and use fundamental identities.\nEvaluate trigonometric functions with a calculator.\nA wheelchair ramp that meets the standards of the Americans with Disabilities Act must make an angle with the ground"
	},
	{
		"chunk_id" : "00002273",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "whose tangent is or less, regardless of its length. A tangent represents a ratio, so this means that for every 1 inch of\nrise, the ramp must have 12 inches of run. Trigonometric functions allow us to specify the shapes and proportions of\nobjects independent of exact dimensions. We have already defined the sine and cosine functions of an angle. Though\nsine and cosine are the trigonometric functions most often used, there are four others. Together they make up the set of"
	},
	{
		"chunk_id" : "00002274",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "six trigonometric functions. In this section, we will investigate the remaining functions.\nFinding Exact Values of the Trigonometric Functions Secant, Cosecant, Tangent,\nand Cotangent\nWe can also define the remaining functions in terms of the unit circle with a point corresponding to an angle of\nas shown inFigure 1. As with the sine and cosine, we can use the coordinates to find the other functions.\nFigure1"
	},
	{
		"chunk_id" : "00002275",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nThe first function we will define is the tangent. Thetangentof an angle is the ratio of they-value to thex-value of the\ncorresponding point on the unit circle. InFigure 1, the tangent of angle is equal to Because they-value is\nequal to the sine of and thex-value is equal to the cosine of the tangent of angle can also be defined as\nThe tangent function is abbreviated as The remaining three functions can all be expressed as\nreciprocals of functions we have already defined."
	},
	{
		"chunk_id" : "00002276",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reciprocals of functions we have already defined.\n Thesecantfunction is the reciprocal of the cosine function. InFigure 1, the secant of angle is equal to\nThe secant function is abbreviated as\n Thecotangentfunction is the reciprocal of the tangent function. InFigure 1, the cotangent of angle is equal to\nThe cotangent function is abbreviated as\n Thecosecantfunction is the reciprocal of the sine function. InFigure 1, the cosecant of angle is equal to\nThe cosecant function is abbreviated as"
	},
	{
		"chunk_id" : "00002277",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The cosecant function is abbreviated as\nTangent, Secant, Cosecant, and Cotangent Functions\nIf is a real number and is a point where the terminal side of an angle of radians intercepts the unit circle, then\nAccess for free at openstax.org\n7.4  The Other Trigonometric Functions 737\nEXAMPLE1\nFinding Trigonometric Functions from a Point on the Unit Circle\nThe point is on the unit circle, as shown inFigure 2. Find and\nFigure2\nSolution"
	},
	{
		"chunk_id" : "00002278",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nSolution\nBecause we know the coordinates of the point on the unit circle indicated by angle we can use those coordinates\nto find the six functions:\nTRY IT #1 The point is on the unit circle, as shown inFigure 3. Find\nand\n738 7  The Unit Circle: Sine and Cosine Functions\nFigure3\nEXAMPLE2\nFinding the Trigonometric Functions of an Angle\nFind and when\nSolution\nWe have previously used the properties of equilateral triangles to demonstrate that and We"
	},
	{
		"chunk_id" : "00002279",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to\nfind the remaining function values.\nTRY IT #2 Find and when\nBecause we know the sine and cosine values for the common first-quadrant angles, we can find the other function\nvalues for those angles as well by setting equal to the cosine and equal to the sine and then using the definitions of\ntangent, secant, cosecant, and cotangent. The results are shown inTable 1."
	},
	{
		"chunk_id" : "00002280",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.4  The Other Trigonometric Functions 739\nAngle\nCosine 1 0\nSine 0 1\nTangent 0 1 Undefined\nSecant 1 2 Undefined\nCosecant Undefined 2 1\nCotangent Undefined 1 0\nTable1\nUsing Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent\nWe can evaluatetrigonometric functionsof angles outside the first quadrant using reference angles as we have already\ndone with the sine and cosine functions. The procedure is the same: Find thereference angleformed by the terminal"
	},
	{
		"chunk_id" : "00002281",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same\nas those for the reference angle, except for the positive or negative sign, which is determined byx- andy-values in the\noriginal quadrant.Figure 4shows which functions are positive in which quadrant.\nTo help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic"
	},
	{
		"chunk_id" : "00002282",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "phrase A Smart Trig Class. Each of the four words in the phrase corresponds to one of the four quadrants, starting\nwith quadrant I and rotating counterclockwise. In quadrant I, which is A,all of the six trigonometric functions are\npositive. In quadrant II, Smart, onlysine and its reciprocal function, cosecant, are positive. In quadrant III, Trig, only\ntangent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, Class, onlycosine and its reciprocal"
	},
	{
		"chunk_id" : "00002283",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function, secant, are positive.\nFigure4 The trigonometric functions are each listed in the quadrants in which they are positive.\n...\nHOW TO\nGiven an angle not in the first quadrant, use reference angles to find all six trigonometric functions.\n1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference\n740 7  The Unit Circle: Sine and Cosine Functions\nangle.\n2. Evaluate the function at the reference angle."
	},
	{
		"chunk_id" : "00002284",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Evaluate the function at the reference angle.\n3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine\nwhether the output is positive or negative.\nEXAMPLE3\nUsing Reference Angles to Find Trigonometric Functions\nUse reference angles to find all six trigonometric functions of\nSolution\nThe angle between this angles terminal side and thex-axis is so that is the reference angle. Since is in the third"
	},
	{
		"chunk_id" : "00002285",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadrant, where both and are negative, cosine, sine, secant, and cosecant will be negative, while tangent and\ncotangent will be positive.\nTRY IT #3 Use reference angles to find all six trigonometric functions of\nUsing Even and Odd Trigonometric Functions\nTo be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine\nhow each function treats a negative input. As it turns out, there is an important difference among the functions in this\nregard."
	},
	{
		"chunk_id" : "00002286",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "regard.\nConsider the function shown inFigure 5. The graph of the function is symmetrical about they-axis. All along\nthe curve, any two points with oppositex-values have the same function value. This matches the result of calculation:\nand so on. So is an even function, a function such that two inputs that are\nopposites have the same output. That means\nFigure5 The function is an even function.\nNow consider the function shown inFigure 6. The graph is not symmetrical about they-axis. All along the"
	},
	{
		"chunk_id" : "00002287",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph, any two points with oppositex-values also have oppositey-values. So is an odd function, one such that\ntwo inputs that are opposites have outputs that are also opposites. That means\nAccess for free at openstax.org\n7.4  The Other Trigonometric Functions 741\nFigure6 The function is an odd function.\nWe can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle,"
	},
	{
		"chunk_id" : "00002288",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as inFigure 7. The sine of the positive angle is The sine of the negative angle is The sine function, then, is an odd\nfunction. We can test each of the six trigonometric functions in this fashion. The results are shown inTable 2.\nFigure7\nTable2\nEven and Odd Trigonometric Functions\nAn even function is one in which\nAn odd function is one in which\nCosine and secant are even:\nSine, tangent, cosecant, and cotangent are odd:\n742 7  The Unit Circle: Sine and Cosine Functions\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002289",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nUsing Even and Odd Properties of Trigonometric Functions\nIf the secant of angle is 2, what is the secant of\nSolution\nSecant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle is\n2, the secant of is also 2.\nTRY IT #4 If the cotangent of angle is what is the cotangent of\nRecognizing and Using Fundamental Identities\nWe have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further,"
	},
	{
		"chunk_id" : "00002290",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and derive some fundamental identities. Identities are statements that are true for all values of the input on which they\nare defined. Usually, identities can be derived from definitions and relationships we already know. For example, the\nPythagorean Identitywe learned earlier was derived from the Pythagorean Theorem and the definitions of sine and\ncosine.\nFundamental Identities\nWe can derive some usefulidentitiesfrom the six trigonometric functions. The other four trigonometric functions"
	},
	{
		"chunk_id" : "00002291",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can be related back to the sine and cosine functions using these basic relationships:\nEXAMPLE5\nUsing Identities to Evaluate Trigonometric Functions\n Given evaluate\n Given evaluate\nSolution\nBecause we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.\nAccess for free at openstax.org\n7.4  The Other Trigonometric Functions 743\n\n\nTRY IT #5 Evaluate\nEXAMPLE6\nUsing Identities to Simplify Trigonometric Expressions\nSimplify\nSolution"
	},
	{
		"chunk_id" : "00002292",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Simplify\nSolution\nWe can simplify this by rewriting both functions in terms of sine and cosine.\nBy showing that can be simplified to we have, in fact, established a new identity.\nTRY IT #6 Simplify\nAlternate Forms of the Pythagorean Identity\nWe can use these fundamental identities to derive alternate forms of the Pythagorean Identity, One\nform is obtained by dividing both sides by\nThe other form is obtained by dividing both sides by\nAlternate Forms of the Pythagorean Identity"
	},
	{
		"chunk_id" : "00002293",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Alternate Forms of the Pythagorean Identity\n744 7  The Unit Circle: Sine and Cosine Functions\nEXAMPLE7\nUsing Identities to Relate Trigonometric Functions\nIf and is in quadrant IV, as shown inFigure 8, find the values of the other five trigonometric functions.\nFigure8\nSolution\nWe can find the sine using the Pythagorean Identity, and the remaining functions by relating them to\nsine and cosine."
	},
	{
		"chunk_id" : "00002294",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sine and cosine.\nThe sign of the sine depends on they-values in the quadrant where the angle is located. Since the angle is in quadrant\nIV, where they-values are negative, its sine is negative,\nThe remaining functions can be calculated using identities relating them to sine and cosine.\nTRY IT #7 If and find the values of the other five functions.\nAs we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002295",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.4  The Other Trigonometric Functions 745\nperiodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant\nand secant, a revolution of one circle, or will result in the same outputs for these functions. And for tangent and\ncotangent, only a half a revolution will result in the same outputs.\nOther functions can also be periodic. For example, the lengths of months repeat every four years. If represents the"
	},
	{
		"chunk_id" : "00002296",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "length time, measured in years, and represents the number of days in February, then This pattern\nrepeats over and over through time. In other words, every four years, February is guaranteed to have the same number\nof days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is\ncalled the period. Aperiodis the shortest interval over which a function completes one full cyclein this example, the"
	},
	{
		"chunk_id" : "00002297",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "period is 4 and represents the time it takes for us to be certain February has the same number of days.\nPeriod of a Function\nTheperiod of a repeating function is the number representing the interval such that for any\nvalue of\nThe period of the cosine, sine, secant, and cosecant functions is\nThe period of the tangent and cotangent functions is\nEXAMPLE8\nFinding the Values of Trigonometric Functions\nFind the values of the six trigonometric functions of angle based onFigure 9.\nFigure9\nSolution"
	},
	{
		"chunk_id" : "00002298",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nSolution\n746 7  The Unit Circle: Sine and Cosine Functions\nTRY IT #8 Find the values of the six trigonometric functions of angle based onFigure 10.\nFigure10\nEXAMPLE9\nFinding the Value of Trigonometric Functions\nIf\nSolution\nTRY IT #9\nEvaluating Trigonometric Functions with a Calculator\nWe have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them"
	},
	{
		"chunk_id" : "00002299",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a\nscientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm\nthe correct mode is chosen before making a calculation.\nEvaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra"
	},
	{
		"chunk_id" : "00002300",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may\nnot be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a\nsine, cosine, or tangent.\nIf we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees\nmultiplied by the conversion factor to convert the degrees to radians. To find the secant of we could press"
	},
	{
		"chunk_id" : "00002301",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n7.4  The Other Trigonometric Functions 747\n...\nHOW TO\nGiven an angle measure in radians, use a scientific calculator to find the cosecant.\n1. If the calculator has degree mode and radian mode, set it to radian mode.\n2. Enter:\n3. Enter the value of the angle inside parentheses.\n4. Press the SIN key.\n5. Press the = key.\n...\nHOW TO\nGiven an angle measure in radians, use a graphing utility/calculator to find the cosecant."
	},
	{
		"chunk_id" : "00002302",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If the graphing utility has degree mode and radian mode, set it to radian mode.\n Enter:\n Press the SIN key.\n Enter the value of the angle inside parentheses.\n Press the ENTER key.\nEXAMPLE10\nEvaluating the Cosecant Using Technology\nEvaluate the cosecant of\nSolution\nFor a scientific calculator, enter information as follows:\nTRY IT #10 Evaluate the cotangent of\nMEDIA\nAccess these online resources for additional instruction and practice with other trigonometric functions."
	},
	{
		"chunk_id" : "00002303",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determing Trig Function Values(http://openstax.org/l/trigfuncval)\nMore Examples of Determining Trig Functions(http://openstax.org/l/moretrigfun)\nPythagorean Identities(http://openstax.org/l/pythagiden)\nTrig Functions on a Calculator(http://openstax.org/l/trigcalc)\n748 7  The Unit Circle: Sine and Cosine Functions\n7.4 SECTION EXERCISES\nVerbal\n1. On an interval of 2. What would you estimate 3. For any angle in quadrant II,\ncan the sine and cosine the cosine of degrees to if you knew the sine of the"
	},
	{
		"chunk_id" : "00002304",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values of a radian measure be? Explain your reasoning. angle, how could you\never be equal? If so, where? determine the cosine of the\nangle?\n4. Describe the secant 5. Tangent and cotangent have\nfunction. a period of What does\nthis tell us about the output\nof these functions?\nAlgebraic\nFor the following exercises, find the exact value of each expression.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\nFor the following exercises, use reference angles to evaluate the expression.\n18. 19. 20.\n21. 22. 23."
	},
	{
		"chunk_id" : "00002305",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "18. 19. 20.\n21. 22. 23.\n24. 25. 26.\n27. 28. 29.\n30. 31. 32.\n33. 34. 35.\n36. 37. 38. If and is in\nquadrant II, find\nand\nAccess for free at openstax.org\n7.4  The Other Trigonometric Functions 749\n39. If and is in 40. If , and , find 41. If and\nquadrant III, find find\nand\nand\n42. If and 43. If what is the 44. If what is the\nfind\nand\n45. If what is the 46. If what is the 47. If what is the\n48. If what is the\nGraphical"
	},
	{
		"chunk_id" : "00002306",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "48. If what is the\nGraphical\nFor the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.\n49. 50. 51.\nTechnology\nFor the following exercises, use a graphing calculator to evaluate to three decimal places.\n52. 53. 54.\n55. 56. 57.\n58. 59. 60.\n61.\n750 7  The Unit Circle: Sine and Cosine Functions\nExtensions\nFor the following exercises, use identities to evaluate the expression.\n62. If and 63. If and 64. If and\nfind find find"
	},
	{
		"chunk_id" : "00002307",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "62. If and 63. If and 64. If and\nfind find find\n65. If and 66. Determine whether the function\nfind is even, odd, or neither.\n67. Determine whether the function 68. Determine whether the function\nis even, odd, or is even, odd, or neither.\nneither.\n69. Determine whether the function\nis even, odd, or neither.\nFor the following exercises, use identities to simplify the expression.\n70. 71.\nReal-World Applications\n72. The amount of sunlight in a 73. The amount of sunlight in a 74. The equation"
	},
	{
		"chunk_id" : "00002308",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "certain city can be modeled certain city can be modeled\nby the function by the function models the blood pressure,\nwhere where where represents time\nrepresents the hours of represents the hours of in seconds. (a) Find the\nsunlight, and is the day sunlight, and is the day blood pressure after 15\nof the year. Use the of the year. Use the seconds. (b) What are the\nequation to find how many equation to find how many maximum and minimum\nhours of sunlight there are hours of sunlight there are blood pressures?"
	},
	{
		"chunk_id" : "00002309",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on February 11, the 42nd on September 24, the\nday of the year. State the 267th day of the year. State\nperiod of the function. the period of the function.\n75. The height of a piston, in 76. The height of a piston, in\ninches, can be modeled by inches, can be modeled by\nthe equation the equation\nwhere where\nrepresents the crank angle. represents the crank angle.\nFind the height of the Find the height of the\npiston when the crank piston when the crank\nangle is angle is\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002310",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle is angle is\nAccess for free at openstax.org\n7  Chapter Review 751\nChapter Review\nKey Terms\nadjacent side in a right triangle, the side between a given angle and the right angle\nangle the union of two rays having a common endpoint\nangle of depression the angle between the horizontal and the line from the object to the observers eye, assuming the\nobject is positioned lower than the observer"
	},
	{
		"chunk_id" : "00002311",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "object is positioned lower than the observer\nangle of elevation the angle between the horizontal and the line from the object to the observers eye, assuming the\nobject is positioned higher than the observer\nangular speed the angle through which a rotating object travels in a unit of time\narc length the length of the curve formed by an arc\narea of a sector area of a portion of a circle bordered by two radii and the intercepted arc; the fraction multiplied\nby the area of the entire circle"
	},
	{
		"chunk_id" : "00002312",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by the area of the entire circle\ncosecant the reciprocal of the sine function: on the unit circle,\ncosine function thex-value of the point on a unit circle corresponding to a given angle\ncotangent the reciprocal of the tangent function: on the unit circle,\ncoterminal angles description of positive and negative angles in standard position sharing the same terminal side\ndegree a unit of measure describing the size of an angle as one-360th of a full revolution of a circle"
	},
	{
		"chunk_id" : "00002313",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hypotenuse the side of a right triangle opposite the right angle\nidentities statements that are true for all values of the input on which they are defined\ninitial side the side of an angle from which rotation begins\nlinear speed the distance along a straight path a rotating object travels in a unit of time; determined by the arc length\nmeasure of an angle the amount of rotation from the initial side to the terminal side\nnegative angle description of an angle measured clockwise from the positivex-axis"
	},
	{
		"chunk_id" : "00002314",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opposite side in a right triangle, the side most distant from a given angle\nperiod the smallest interval of a repeating function such that\npositive angle description of an angle measured counterclockwise from the positivex-axis\nPythagorean Identity a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle\nplus the square of the sine of that angle equals 1\nquadrantal angle an angle whose terminal side lies on an axis"
	},
	{
		"chunk_id" : "00002315",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radian the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle\nradian measure the ratio of the arc length formed by an angle divided by the radius of the circle\nray one point on a line and all points extending in one direction from that point; one side of an angle\nreference angle the measure of the acute angle formed by the terminal side of the angle and the horizontal axis\nsecant the reciprocal of the cosine function: on the unit circle,"
	},
	{
		"chunk_id" : "00002316",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sine function they-value of the point on a unit circle corresponding to a given angle\nstandard position the position of an angle having the vertex at the origin and the initial side along the positivex-axis\ntangent the quotient of the sine and cosine: on the unit circle,\nterminal side the side of an angle at which rotation ends\nunit circle a circle with a center at and radius 1\nvertex the common endpoint of two rays that form an angle\nKey Equations\narc length\narea of a sector\nangular speed\nlinear speed"
	},
	{
		"chunk_id" : "00002317",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "area of a sector\nangular speed\nlinear speed\nlinear speed related to angular speed\n752 7  Chapter Review\nTrigonometric Functions\nReciprocal Trigonometric Functions\nCofunction Identities\nCosine\nSine\nPythagorean Identity\nTangent function\nSecant function\nCosecant function\nCotangent function\nKey Concepts\n7.1Angles\n An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The\namount of rotation determines the measure of the angle."
	},
	{
		"chunk_id" : "00002318",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " An angle is in standard position if its vertex is at the origin and its initial side lies along the positivex-axis. A positive\nangle is measured counterclockwise from the initial side and a negative angle is measured clockwise.\n To draw an angle in standard position, draw the initial side along the positivex-axis and then place the terminal side\naccording to the fraction of a full rotation the angle represents. SeeExample 1."
	},
	{
		"chunk_id" : "00002319",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " In addition to degrees, the measure of an angle can be described in radians. SeeExample 2.\n To convert between degrees and radians, use the proportion SeeExample 3andExample 4.\n Two angles that have the same terminal side are called coterminal angles.\n We can find coterminal angles by adding or subtracting or SeeExample 5andExample 6.\nAccess for free at openstax.org\n7  Chapter Review 753\n Coterminal angles can be found using radians just as they are for degrees. SeeExample 7."
	},
	{
		"chunk_id" : "00002320",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The length of a circular arc is a fraction of the circumference of the entire circle. SeeExample 8.\n The area of sector is a fraction of the area of the entire circle. SeeExample 9.\n An object moving in a circular path has both linear and angular speed.\n The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a\nunit of time. SeeExample 10."
	},
	{
		"chunk_id" : "00002321",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unit of time. SeeExample 10.\n The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. SeeExample\n11.\n7.2Right Triangle Trigonometry\n We can define trigonometric functions as ratios of the side lengths of a right triangle. SeeExample 1.\n The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle.\nSeeExample 2."
	},
	{
		"chunk_id" : "00002322",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 2.\n We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which\nthey occur. SeeExample 3.\n Any two complementary angles could be the two acute angles of a right triangle.\n If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other\nand vice versa. SeeExample 4.\n We can use trigonometric functions of an angle to find unknown side lengths."
	},
	{
		"chunk_id" : "00002323",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Select the trigonometric function representing the ratio of the unknown side to the known side. SeeExample 5.\n Right-triangle trigonometry facilitates the measurement of inaccessible heights and distances.\n The unknown height or distance can be found by creating a right triangle in which the unknown height or distance\nis one of the sides, and another side and angle are known. SeeExample 6.\n7.3Unit Circle"
	},
	{
		"chunk_id" : "00002324",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7.3Unit Circle\n Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin\nand has a radius of 1 unit.\n Using the unit circle, the sine of an angle equals they-value of the endpoint on the unit circle of an arc of length\nwhereas the cosine of an angle equals thex-value of the endpoint. SeeExample 1.\n The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an\naxis. SeeExample 2."
	},
	{
		"chunk_id" : "00002325",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "axis. SeeExample 2.\n When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity\nis also useful for determining the sines and cosines of special angles. SeeExample 3.\n Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering\ninformation is known. SeeExample 4.\n The domain of the sine and cosine functions is all real numbers.\n The range of both the sine and cosine functions is"
	},
	{
		"chunk_id" : "00002326",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle.\n The signs of the sine and cosine are determined from thex- andy-values in the quadrant of the original angle.\n An angles reference angle is the size angle, formed by the terminal side of the angle and the horizontal axis. See\nExample 5.\n Reference angles can be used to find the sine and cosine of the original angle. SeeExample 6."
	},
	{
		"chunk_id" : "00002327",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Reference angles can also be used to find the coordinates of a point on a circle. SeeExample 7.\n7.4The Other Trigonometric Functions\n The tangent of an angle is the ratio of they-value to thex-value of the corresponding point on the unit circle.\n The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine\nfunction, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine\nfunction."
	},
	{
		"chunk_id" : "00002328",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\n The six trigonometric functions can be found from a point on the unit circle. SeeExample 1.\n Trigonometric functions can also be found from an angle. SeeExample 2.\n Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See\nExample 3.\n A function is said to be even if and odd if for allxin the domain off.\n Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd."
	},
	{
		"chunk_id" : "00002329",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Even and odd properties can be used to evaluate trigonometric functions. SeeExample 4.\n The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.\n Identities can be used to evaluate trigonometric functions. SeeExample 5andExample 6.\n754 7  Exercises\n Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new\nidentities. SeeExample 7.\n The trigonometric functions repeat at regular intervals."
	},
	{
		"chunk_id" : "00002330",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The period of a repeating function is the smallest interval such that for any value of\n The values of trigonometric functions can be found by mathematical analysis. SeeExample 8andExample 9.\n To evaluate trigonometric functions of other angles, we can use a calculator or computer software. SeeExample 10.\nExercises\nReview Exercises\nAngles\nFor the following exercises, convert the angle measures to degrees.\n1. 2.\nFor the following exercises, convert the angle measures to radians."
	},
	{
		"chunk_id" : "00002331",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. 4. 5. Find the length of an arc in a\ncircle of radius 7 meters\nsubtended by the central\nangle of\n6. Find the area of the sector\nof a circle with diameter 32\nfeet and an angle of\nradians.\nFor the following exercises, find the angle between and that is coterminal with the given angle.\n7. 8.\nFor the following exercises, find the angle between 0 and in radians that is coterminal with the given angle.\n9. 10.\nFor the following exercises, draw the angle provided in standard position on the Cartesian plane."
	},
	{
		"chunk_id" : "00002332",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11. 12. 13.\n14. 15. Find the linear speed of a 16. A car wheel with a\npoint on the equator of the diameter of 18 inches spins\nearth if the earth has a at the rate of 10\nradius of 3,960 miles and revolutions per second.\nthe earth rotates on its axis What is the car's speed in\nevery 24 hours. Express miles per hour? Round to\nanswer in miles per hour. the nearest hundredth.\nRound to the nearest\nhundredth.\nAccess for free at openstax.org\n7  Exercises 755\nRight Triangle Trigonometry"
	},
	{
		"chunk_id" : "00002333",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7  Exercises 755\nRight Triangle Trigonometry\nFor the following exercises, use side lengths to evaluate.\n17. 18. 19.\n20. 21.\nFor the following exercises, use the given information to find the lengths of the other two sides of the right triangle.\n22. 23.\nFor the following exercises, useFigure 1to evaluate each trigonometric function.\nFigure1\n24. 25.\nFor the following exercises, solve for the unknown sides of the given triangle.\n26. 27. 28. A 15-ft ladder leans against\na building so that the angle"
	},
	{
		"chunk_id" : "00002334",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a building so that the angle\nbetween the ground and\nthe ladder is How high\ndoes the ladder reach up\nthe side of the building?\nFind the answer to four\ndecimal places.\n29. The angle of elevation to\nthe top of a building in\nBaltimore is found to be 4\ndegrees from the ground\nat a distance of 1 mile from\nthe base of the building.\nUsing this information, find\nthe height of the building.\nFind the answer to four\ndecimal places.\nUnit Circle"
	},
	{
		"chunk_id" : "00002335",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decimal places.\nUnit Circle\n30. Find the exact value of 31. Find the exact value of 32. Find the exact value of\n756 7  Exercises\n33. State the reference angle 34. State the reference angle 35. Compute cosine of\nfor for\n36. Compute sine of 37. State the domain of the 38. State the range of the sine\nsine and cosine functions. and cosine functions.\nThe Other Trigonometric Functions\nFor the following exercises, find the exact value of the given expression.\n39. 40. 41.\n42."
	},
	{
		"chunk_id" : "00002336",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "39. 40. 41.\n42.\nFor the following exercises, use reference angles to evaluate the given expression.\n43. 44. 45. If what is\nthe\n46. If what is the 47. If find 48. If find\nThere are two\npossible solutions.\n49. Which trigonometric 50. Which trigonometric\nfunctions are even? functions are odd?\nPractice Test\n1. Convert radians to 2. Convert to radians. 3. Find the length of a circular\ndegrees. arc with a radius 12\ncentimeters subtended by\nthe central angle of"
	},
	{
		"chunk_id" : "00002337",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "centimeters subtended by\nthe central angle of\n4. Find the area of the sector 5. Find the angle between 6. Find the angle between 0\nwith radius of 8 feet and an and that is coterminal and in radians that is\nangle of radians. with coterminal with\n7. Draw the angle in 8. Draw the angle in 9. A carnival has a Ferris wheel\nstandard position on the standard position on the with a diameter of 80 feet.\nCartesian plane. Cartesian plane. The time for the Ferris wheel\nto make one revolution is 75"
	},
	{
		"chunk_id" : "00002338",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to make one revolution is 75\nseconds. What is the linear\nspeed in feet per second of\na point on the Ferris wheel?\nWhat is the angular speed in\nradians per second?\nAccess for free at openstax.org\n7  Exercises 757\n10. Find the missing sides of 11. Find the missing sides of 12. The angle of elevation to\nthe triangle the triangle. the top of a building in\nChicago is found to be 9\ndegrees from the ground\nat a distance of 2000 feet\nfrom the base of the\nbuilding. Using this\ninformation, find the"
	},
	{
		"chunk_id" : "00002339",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "building. Using this\ninformation, find the\nheight of the building.\n13. Find the exact value of 14. Compute sine of 15. State the domain of the\nsine and cosine functions.\n16. State the range of the sine 17. Find the exact value of 18. Find the exact value of\nand cosine functions.\n19. Use reference angles to 20. Use reference angles to 21. If what is the\nevaluate evaluate\n22. If find 23. Find the missing angle:\n758 7  Exercises\nAccess for free at openstax.org\n8  Introduction 759\n8 PERIODIC FUNCTIONS"
	},
	{
		"chunk_id" : "00002340",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8  Introduction 759\n8 PERIODIC FUNCTIONS\nDawn colors the sky over the Olare Motorgi Conservancy bordering tha Masai Mara National Reserve in Kenya. (Credit:\nModification of \"KenyaLive_Day_#02\"\" by Make it Kenya/flickr)"
	},
	{
		"chunk_id" : "00002341",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "written as Re), as undertaking a two-part daily journey, with one portion in the sky (day) and the other through the\nunderworld (night). Surya, the Hindu sun god, traces a similar path through the sky on a chariot pulled by seven horses.\nWhile their origins and associated narratives are quite different, both Ra and Surya are primary deities and seen as\ncreators and preservers of life. In many Native American cultures, the sun is core to spiritual and religious practice, but is"
	},
	{
		"chunk_id" : "00002342",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "not always a deity. The Sun Dance, practiced differently by many Native American tribes, was a ceremony that generally\npaid homage to the sun and, in many cases, tested or expressed the strength of the tribe's people.\nAs one of the most most prominent natural phenomena and with its close association to giving life, the sun was an\nobvious subject for reverence. And its regularity, even in ancient times, made it the primary determinant of time. Each"
	},
	{
		"chunk_id" : "00002343",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "day, the sun rises in an easterly direction, approaches some maximum height relative to the celestial equator, and sets in\na westerly direction. The celestial equator is an imaginary line that divides the visible universe into two halves in much\nthe same way Earths equator is an imaginary line that divides the planet into two halves. The exact path the sun\nappears to follow depends on the exact location on Earth, but each location observes a predictable pattern over time."
	},
	{
		"chunk_id" : "00002344",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The pattern of the suns motion throughout the course of a year is a periodic function. Creating a visual representation\nof a periodic function in the form of a graph can help us analyze the properties of the function. In this chapter, we will\ninvestigate graphs of sine, cosine, and other trigonometric functions.\n8.1 Graphs of the Sine and Cosine Functions\nLearning Objectives\nIn this section, you will:\nGraph variations of and .\nUse phase shifts of sine and cosine curves.\n760 8  Periodic Functions"
	},
	{
		"chunk_id" : "00002345",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "760 8  Periodic Functions\nFigure1 Light can be separated into colors because of its wavelike properties. (credit: \"wonderferret\"\"/ Flickr)"
	},
	{
		"chunk_id" : "00002346",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Light waves can be represented graphically by the sine function. In the chapter onTrigonometric Functions\n(http://openstax.org/books/precalculus-2e/pages/5-introduction-to-trigonometric-functions), we examined\ntrigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine\nfunctions.\nGraphing Sine and Cosine Functions\nRecall that the sine and cosine functions relate real number values to thex- andy-coordinates of a point on the unit"
	},
	{
		"chunk_id" : "00002347",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "circle. So what do they look like on a graph on a coordinate plane? Lets start with thesine function. We can create a\ntable of values and use them to sketch a graph.Table 1lists some of the values for the sine function on a unit circle.\nTable1\nPlotting the points from the table and continuing along thex-axis gives the shape of the sine function. SeeFigure 2.\nFigure2 The sine function\nNotice how the sine values are positive between 0 and which correspond to the values of the sine function in"
	},
	{
		"chunk_id" : "00002348",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadrants I and II on the unit circle, and the sine values are negative between and which correspond to the values\nof the sine function in quadrants III and IV on the unit circle. SeeFigure 3.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 761\nFigure3 Plotting values of the sine function\nNow lets take a similar look at thecosine function. Again, we can create a table of values and use them to sketch a"
	},
	{
		"chunk_id" : "00002349",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph.Table 2lists some of the values for the cosine function on a unit circle.\nTable2\nAs with the sine function, we can plots points to create a graph of the cosine function as inFigure 4.\nFigure4 The cosine function\nBecause we can evaluate the sine and cosine of any real number, both of these functions are defined for all real\nnumbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the\nrange of both functions must be the interval"
	},
	{
		"chunk_id" : "00002350",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "range of both functions must be the interval\nIn both graphs, the shape of the graph repeats after which means the functions are periodic with a period of A\nperiodic functionis a function for which a specifichorizontal shift,P, results in a function equal to the original function:\nfor all values of in the domain of When this occurs, we call the smallest such horizontal shift with\ntheperiodof the function.Figure 5shows several periods of the sine and cosine functions.\n762 8  Periodic Functions\nFigure5"
	},
	{
		"chunk_id" : "00002351",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "762 8  Periodic Functions\nFigure5\nLooking again at the sine and cosine functions on a domain centered at they-axis helps reveal symmetries. As we can\nsee inFigure 6, thesine functionis symmetric about the origin. Recall fromThe Other Trigonometric Functionsthat we\ndetermined from the unit circle that the sine function is an odd function because Now we can clearly\nsee this property from the graph.\nFigure6 Odd symmetry of the sine function"
	},
	{
		"chunk_id" : "00002352",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6 Odd symmetry of the sine function\nFigure 7shows that the cosine function is symmetric about they-axis. Again, we determined that the cosine function is\nan even function. Now we can see from the graph that\nFigure7 Even symmetry of the cosine function\nCharacteristics of Sine and Cosine Functions\nThe sine and cosine functions have several distinct characteristics:\n They are periodic functions with a period of\n The domain of each function is   and the range is"
	},
	{
		"chunk_id" : "00002353",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The graph of is symmetric about the origin, because it is an odd function.\n The graph of is symmetric about the -axis, because it is an even function.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 763\nInvestigating Sinusoidal Functions\nAs we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond,"
	},
	{
		"chunk_id" : "00002354",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller\nor longer than others. A function that has the same general shape as a sine orcosine functionis known as asinusoidal\nfunction. The general forms of sinusoidal functions are\nDetermining the Period of Sinusoidal Functions\nLooking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions."
	},
	{
		"chunk_id" : "00002355",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can use what we know about transformations to determine the period.\nIn the general formula, is related to the period by If then the period is less than and the function\nundergoes a horizontal compression, whereas if then the period is greater than and the function undergoes\na horizontal stretch. For example, so the period is which we knew. If then\nso the period is and the graph is compressed. If then so the period is and the graph\nis stretched. Notice inFigure 8how the period is indirectly related to"
	},
	{
		"chunk_id" : "00002356",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nPeriod of Sinusoidal Functions\nIf we let and in the general form equations of the sine and cosine functions, we obtain the forms\nThe period is\nEXAMPLE1\nIdentifying the Period of a Sine or Cosine Function\nDetermine the period of the function\nSolution\nLets begin by comparing the equation to the general form\nIn the given equation, so the period will be\n764 8  Periodic Functions\nTRY IT #1 Determine the period of the function\nDetermining Amplitude"
	},
	{
		"chunk_id" : "00002357",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Determining Amplitude\nReturning to the general formula for a sinusoidal function, we have analyzed how the variable relates to the period.\nNow lets turn to the variable so we can analyze how it is related to theamplitude, or greatest distance from rest.\nrepresents the vertical stretch factor, and its absolute value is the amplitude. The local maxima will be a distance\nabove the horizontalmidlineof the graph, which is the line because in this case, the midline is thex-axis."
	},
	{
		"chunk_id" : "00002358",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The local minima will be the same distance below the midline. If the function is stretched. For example, the\namplitude of is twice the amplitude of If the function is compressed.Figure 9\ncompares several sine functions with different amplitudes.\nFigure9\nAmplitude of Sinusoidal Functions\nIf we let and in the general form equations of the sine and cosine functions, we obtain the forms\nTheamplitudeis which is the vertical height from themidline In addition, notice in the example that\nEXAMPLE2"
	},
	{
		"chunk_id" : "00002359",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nIdentifying the Amplitude of a Sine or Cosine Function\nWhat is the amplitude of the sinusoidal function Is the function stretched or compressed vertically?\nSolution\nLets begin by comparing the function to the simplified form\nIn the given function, so the amplitude is The function is stretched.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 765\nAnalysis\nThe negative value of results in a reflection across thex-axis of thesine function, as shown inFigure 10.\nFigure10"
	},
	{
		"chunk_id" : "00002360",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure10\nTRY IT #2 What is the amplitude of the sinusoidal function Is the function stretched or\ncompressed vertically?\nAnalyzing Graphs of Variations ofy= sinxandy= cosx\nNow that we understand how and relate to the general form equation for the sine and cosine functions, we will\nexplore the variables and Recall the general form:\nThe value for a sinusoidal function is called thephase shift, or the horizontal displacement of the basic sine orcosine"
	},
	{
		"chunk_id" : "00002361",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function. If the graph shifts to the right. If the graph shifts to the left. The greater the value of the\nmore the graph is shifted.Figure 11shows that the graph of shifts to the right by units, which is\nmore than we see in the graph of which shifts to the right by units.\nFigure11\nWhile relates to the horizontal shift, indicates the vertical shift from the midline in the general formula for a\nsinusoidal function. SeeFigure 12. The function has its midline at\n766 8  Periodic Functions\nFigure12"
	},
	{
		"chunk_id" : "00002362",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "766 8  Periodic Functions\nFigure12\nAny value of other than zero shifts the graph up or down.Figure 13compares with\nwhich is shifted 2 units up on a graph.\nFigure13\nVariations of Sine and Cosine Functions\nGiven an equation in the form or is thephase shiftand\nis thevertical shift.\nEXAMPLE3\nIdentifying the Phase Shift of a Function\nDetermine the direction and magnitude of the phase shift for\nSolution\nLets begin by comparing the equation to the general form"
	},
	{
		"chunk_id" : "00002363",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the given equation, notice that and So the phase shift is\nor units to the left.\nAnalysis\nWe must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a\nminus sign before Therefore can be rewritten as If the value of\nis negative, the shift is to the left.\nTRY IT #3 Determine the direction and magnitude of the phase shift for\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 767\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002364",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nIdentifying the Vertical Shift of a Function\nDetermine the direction and magnitude of the vertical shift for\nSolution\nLets begin by comparing the equation to the general form\nIn the given equation, so the shift is 3 units downward.\nTRY IT #4 Determine the direction and magnitude of the vertical shift for\n...\nHOW TO\nGiven a sinusoidal function in the form identify the midline, amplitude, period, and\nphase shift.\n1. Determine the amplitude as\n2. Determine the period as"
	},
	{
		"chunk_id" : "00002365",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Determine the period as\n3. Determine the phase shift as\n4. Determine the midline as\nEXAMPLE5\nIdentifying the Variations of a Sinusoidal Function from an Equation\nDetermine the midline, amplitude, period, and phase shift of the function\nSolution\nLets begin by comparing the equation to the general form\nso the amplitude is\nNext, so the period is\nThere is no added constant inside the parentheses, so and the phase shift is\nFinally, so the midline is\nAnalysis"
	},
	{
		"chunk_id" : "00002366",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finally, so the midline is\nAnalysis\nInspecting the graph, we can determine that the period is the midline is and the amplitude is 3. SeeFigure 14.\nFigure14\nTRY IT #5 Determine the midline, amplitude, period, and phase shift of the function\n768 8  Periodic Functions\nEXAMPLE6\nIdentifying the Equation for a Sinusoidal Function from a Graph\nDetermine the formula for the cosine function inFigure 15.\nFigure15\nSolution"
	},
	{
		"chunk_id" : "00002367",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure15\nSolution\nTo determine the equation, we need to identify each value in the general form of a sinusoidal function.\nThe graph could represent either a sine or acosine functionthat is shifted and/or reflected. When the graph has\nan extreme point, Since the cosine function has an extreme point for let us write our equation in terms of a\ncosine function.\nLets start with the midline. We can see that the graph rises and falls an equal distance above and below This"
	},
	{
		"chunk_id" : "00002368",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "value, which is the midline, is in the equation, so\nThe greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the\nminima are 0.5 units below the midline. So Another way we could have determined the amplitude is by\nrecognizing that the difference between the height of local maxima and minima is 1, so Also, the graph is\nreflected about thex-axis so that"
	},
	{
		"chunk_id" : "00002369",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reflected about thex-axis so that\nThe graph is not horizontally stretched or compressed, so and the graph is not shifted horizontally, so\nPutting this all together,\nTRY IT #6 Determine the formula for the sine function inFigure 16.\nFigure16\nEXAMPLE7\nIdentifying the Equation for a Sinusoidal Function from a Graph\nDetermine the equation for the sinusoidal function inFigure 17.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 769\nFigure17\nSolution"
	},
	{
		"chunk_id" : "00002370",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure17\nSolution\nWith the highest value at 1 and the lowest value at the midline will be halfway between at So\nThe distance from the midline to the highest or lowest value gives an amplitude of\nThe period of the graph is 6, which can be measured from the peak at to the next peak at or from the\ndistance between the lowest points. Therefore, Using the positive value for we find that\nSo far, our equation is either or For the shape and shift, we have more"
	},
	{
		"chunk_id" : "00002371",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "than one option. We could write this as any one of the following:\n a cosine shifted to the right\n a negative cosine shifted to the left\n a sine shifted to the left\n a negative sine shifted to the right\nWhile any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because\nthey involve integer values. So our function becomes\nAgain, these functions are equivalent, so both yield the same graph."
	},
	{
		"chunk_id" : "00002372",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #7 Write a formula for the function graphed inFigure 18.\nFigure18\nGraphing Variations ofy= sinxandy= cosx\nThroughout this section, we have learned about types of variations of sine and cosine functions and used that\ninformation to write equations from graphs. Now we can use the same information to create graphs from equations.\nInstead of focusing on the general form equations\n770 8  Periodic Functions\nwe will let and and work with a simplified form of the equations in the following examples.\n..."
	},
	{
		"chunk_id" : "00002373",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the function sketch its graph.\n1. Identify the amplitude,\n2. Identify the period,\n3. Start at the origin, with the function increasing to the right if is positive or decreasing if is negative.\n4. At there is a local maximum for or a minimum for with\n5. The curve returns to thex-axis at\n6. There is a local minimum for (maximum for ) at with\n7. The curve returns again to thex-axis at\nEXAMPLE8\nGraphing a Function and Identifying the Amplitude and Period\nSketch a graph of\nSolution"
	},
	{
		"chunk_id" : "00002374",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketch a graph of\nSolution\nLets begin by comparing the equation to the form\nStep 1.We can see from the equation that so the amplitude is 2.\nStep 2.The equation shows that so the period is\nStep 3.Because is negative, the graph descends as we move to the right of the origin.\nStep 47.Thex-intercepts are at the beginning of one period, the horizontal midpoints are at and at the end\nof one period at\nThe quarter points include the minimum at and the maximum at A local minimum will occur 2 units below"
	},
	{
		"chunk_id" : "00002375",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the midline, at and a local maximum will occur at 2 units above the midline, at Figure 19shows the graph\nof the function.\nFigure19\nTRY IT #8 Sketch a graph of Determine the midline, amplitude, period, and phase\nshift.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 771\n...\nHOW TO\nGiven a sinusoidal function with a phase shift and a vertical shift, sketch its graph.\n1. Express the function in the general form\n2. Identify the amplitude,\n3. Identify the period,"
	},
	{
		"chunk_id" : "00002376",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Identify the period,\n4. Identify the phase shift,\n5. Draw the graph of shifted to the right or left by and up or down by\nEXAMPLE9\nGraphing a Transformed Sinusoid\nSketch a graph of\nSolution\nStep 1.The function is already written in general form: This graph will have the shape of asine\nfunction, starting at the midline and increasing to the right.\nStep 2. The amplitude is 3.\nStep 3.Since we determine the period as follows.\nThe period is 8.\nStep 4.Since the phase shift is\nThe phase shift is 1 unit."
	},
	{
		"chunk_id" : "00002377",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The phase shift is 1 unit.\nStep 5.Figure 20shows the graph of the function.\nFigure20 A horizontally compressed, vertically stretched, and horizontally shifted sinusoid\nTRY IT #9 Draw a graph of Determine the midline, amplitude, period, and phase\nshift.\n772 8  Periodic Functions\nEXAMPLE10\nIdentifying the Properties of a Sinusoidal Function\nGiven determine the amplitude, period, phase shift, and vertical shift. Then graph the\nfunction.\nSolution"
	},
	{
		"chunk_id" : "00002378",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\nSolution\nBegin by comparing the equation to the general form and use the steps outlined inExample 9.\nStep 1.The function is already written in general form.\nStep 2.Since the amplitude is\nStep 3. so the period is The period is 4.\nStep 4. so we calculate the phase shift as The phase shift is\nStep 5. so the midline is and the vertical shift is up 3.\nSince is negative, the graph of the cosine function has been reflected about thex-axis.\nFigure 21shows one cycle of the graph of the function.\nFigure21"
	},
	{
		"chunk_id" : "00002379",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure21\nUsing Transformations of Sine and Cosine Functions\nWe can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning\nof the chapter,circular motioncan be modeled using either the sine orcosine function.\nEXAMPLE11\nFinding the Vertical Component of Circular Motion\nA point rotates around a circle of radius 3 centered at the origin. Sketch a graph of they-coordinate of the point as a\nfunction of the angle of rotation.\nSolution"
	},
	{
		"chunk_id" : "00002380",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function of the angle of rotation.\nSolution\nRecall that, for a point on a circle of radiusr, they-coordinate of the point is so in this case, we get the\nequation The constant 3 causes a vertical stretch of they-values of the function by a factor of 3, which\nwe can see in the graph inFigure 22.\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 773\nFigure22\nAnalysis\nNotice that the period of the function is still as we travel around the circle, we return to the point for"
	},
	{
		"chunk_id" : "00002381",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because the outputs of the graph will now oscillate between and the amplitude of the sine wave\nis\nTRY IT #10 What is the amplitude of the function Sketch a graph of this function.\nEXAMPLE12\nFinding the Vertical Component of Circular Motion\nA circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeledP, as\nshown inFigure 23. Sketch a graph of the height above the ground of the point as the circle is rotated; then find a"
	},
	{
		"chunk_id" : "00002382",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function that gives the height in terms of the angle of rotation.\nFigure23\nSolution\nSketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and\ncontinue to oscillate 3 ft above and below the center value of 4 ft, as shown inFigure 24.\n774 8  Periodic Functions\nFigure24\nAlthough we could use a transformation of either the sine or cosine function, we start by looking for characteristics that"
	},
	{
		"chunk_id" : "00002383",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would make one function easier to use than the other. Lets use a cosine function because it starts at the highest or\nlowest value, while asine functionstarts at the middle value. A standard cosine starts at the highest value, and this\ngraph starts at the lowest value, so we need to incorporate a vertical reflection.\nSecond, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this\ngraph has been vertically stretched by 3, as in the last example."
	},
	{
		"chunk_id" : "00002384",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these\ntransformations together, we find that\nTRY IT #11 A weight is attached to a spring that is then hung from a board, as shown inFigure 25. As the\nspring oscillates up and down, the position of the weight relative to the board ranges from\nin. (at time to in. (at time below the board. Assume the position of is given as a"
	},
	{
		"chunk_id" : "00002385",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sinusoidal function of Sketch a graph of the function, and then find a cosine function that gives\nthe position in terms of\nFigure25\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 775\nEXAMPLE13\nDetermining a Riders Height on a Ferris Wheel\nThe London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30\nminutes. Riders board from a platform 2 meters above the ground. Express a riders height above ground as a function"
	},
	{
		"chunk_id" : "00002386",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of time in minutes.\nSolution\nWith a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and\nbelow the center.\nPassengers board 2 m above ground level, so the center of the wheel must be located m above ground\nlevel. The midline of the oscillation will be at 69.5 m.\nThe wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes."
	},
	{
		"chunk_id" : "00002387",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the\nshape of a vertically reflected cosine curve.\n Amplitude: so\n Midline: so\n Period: so\n Shape:\nAn equation for the riders height would be\nwhere is in minutes and is measured in meters.\nMEDIA\nAccess these online resources for additional instruction and practice with graphs of sine and cosine functions.\nAmplitude and Period of Sine and Cosine(http://openstax.org/l/ampperiod)"
	},
	{
		"chunk_id" : "00002388",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Translations of Sine and Cosine(http://openstax.org/l/translasincos)\nGraphing Sine and Cosine Transformations(http://openstax.org/l/transformsincos)\nGraphing the Sine Function(http://openstax.org/l/graphsinefunc)\n8.1 SECTION EXERCISES\nVerbal\n1. Why are the sine and cosine 2. How does the graph of 3. For the equation\nfunctions called periodic compare with the what\nfunctions? graph of Explain constants affect the range\nhow you could horizontally of the function and how do"
	},
	{
		"chunk_id" : "00002389",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "translate the graph of they affect the range?\nto obtain\n4. How does the range of a 5. How can the unit circle be\ntranslated sine function used to construct the graph\nrelate to the equation of\n776 8  Periodic Functions\nGraphical\nFor the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State\nthe maximum and minimumy-values and their correspondingx-values on one period for Round answers to two\ndecimal places if necessary.\n6. 7. 8.\n9. 10. 11."
	},
	{
		"chunk_id" : "00002390",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decimal places if necessary.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\nFor the following exercises, graph one full period of each function, starting at For each function, state the\namplitude, period, and midline. State the maximum and minimumy-values and their correspondingx-values on one\nperiod for State the phase shift and vertical translation, if applicable. Round answers to two decimal places if\nnecessary.\n18. 19. 20.\n21. 22. 23. Determine the amplitude, midline,\nperiod, and an equation involving"
	},
	{
		"chunk_id" : "00002391",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "period, and an equation involving\nthe sine function for the graph\nshown inFigure 26.\nFigure26\nAccess for free at openstax.org\n8.1  Graphs of the Sine and Cosine Functions 777\n24. Determine the amplitude, period, 25. Determine the amplitude, period, 26. Determine the amplitude, period,\nmidline, and an equation involving midline, and an equation involving midline, and an equation involving\ncosine for the graph shown in cosine for the graph shown in sine for the graph shown inFigure\nFigure 27. Figure 28. 29."
	},
	{
		"chunk_id" : "00002392",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 27. Figure 28. 29.\nFigure28\nFigure29\nFigure27\n27. Determine the amplitude, period, 28. Determine the amplitude, period, 29. Determine the amplitude, period,\nmidline, and an equation involving midline, and an equation involving midline, and an equation involving\ncosine for the graph shown in sine for the graph shown inFigure cosine for the graph shown in\nFigure 30. 31. Figure 32.\nFigure31 Figure32\nFigure30\n30. Determine the amplitude, period,\nmidline, and an equation involving"
	},
	{
		"chunk_id" : "00002393",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "midline, and an equation involving\nsine for the graph shown inFigure\n33.\nFigure33\n778 8  Periodic Functions\nAlgebraic\nFor the following exercises, let\n31. On solve 32. On solve 33. Evaluate\n34. On 35. On the maximum 36. On the minimum\nFind all values of value(s) of the function value(s) of the function\noccur(s) at whatx-value(s)? occur(s) at whatx-value(s)?\n37. Show that\nThis means that\nis an odd\nfunction and possesses\nsymmetry with respect to\n________________.\nFor the following exercises, let"
	},
	{
		"chunk_id" : "00002394",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, let\n38. On solve the 39. On solve 40. On find the\nequation x-intercepts of\n41. On find the 42. On solve the\nx-values at which the\nequation\nfunction has a maximum or\nminimum value.\nTechnology\n43. Graph on 44. Graph on 45. Graph on\nExplain why the Did the graph and verbalize how\ngraph appears as it does. appear as predicted in the the graph varies from the\nprevious exercise? graph of\n46. Graph on 47. Graph on the\nthe window and window and"
	},
	{
		"chunk_id" : "00002395",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the window and window and\nexplain what the graph explain what the graph\nshows. shows.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 779\nReal-World Applications\n48. A Ferris wheel is 25 meters\nin diameter and boarded\nfrom a platform that is 1\nmeter above the ground.\nThe six oclock position on\nthe Ferris wheel is level\nwith the loading platform.\nThe wheel completes 1 full\nrevolution in 10 minutes.\nThe function gives a\npersons height in meters\nabove the groundt"
	},
	{
		"chunk_id" : "00002396",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "persons height in meters\nabove the groundt\nminutes after the wheel\nbegins to turn.\n Find the amplitude,\nmidline, and period of\n Find a formula for the\nheight function\n How high off the\nground is a person after 5\nminutes?\n8.2 Graphs of the Other Trigonometric Functions\nLearning Objectives\nIn this section, you will:\nAnalyze the graph of y=tanx.\nGraph variations of y=tanx.\nAnalyze the graphs of y=secx and y=cscx.\nGraph variations of y=secx and y=cscx.\nAnalyze the graph of y=cotx."
	},
	{
		"chunk_id" : "00002397",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analyze the graph of y=cotx.\nGraph variations of y=cotx.\nWe know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But\nwhat if we want to measure repeated occurrences of distance? Imagine, for example, a fire truck parked next to a\nwarehouse. The rotating light from the truck would travel across the wall of the warehouse in regular intervals. If the"
	},
	{
		"chunk_id" : "00002398",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at\nregular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to\nillustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could\nappear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this"
	},
	{
		"chunk_id" : "00002399",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "section, we will explore the graphs of the tangent and other trigonometric functions.\nAnalyzing the Graph ofy= tanx\nWe will begin with the graph of thetangentfunction, plotting points as we did for the sine and cosine functions. Recall\nthat\nTheperiodof the tangent function is because the graph repeats itself on intervals of where is a constant. If we\ngraph the tangent function on to we can see the behavior of the graph on one complete cycle. If we look at any"
	},
	{
		"chunk_id" : "00002400",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "larger interval, we will see that the characteristics of the graph repeat.\nWe can determine whether tangent is an odd or even function by using the definition of tangent.\n780 8  Periodic Functions\nTherefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking\nat values for some of the special angles, as listed inTable 1.\n0\nundefined 1 0 1 undefined\nTable1"
	},
	{
		"chunk_id" : "00002401",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0\nundefined 1 0 1 undefined\nTable1\nThese points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we\nlook more closely at values when we can use a table to look for a trend. Because and\nwe will evaluate at radian measures as shown inTable 2.\n1.3 1.5 1.55 1.56\n3.6 14.1 48.1 92.6\nTable2\nAs approaches the outputs of the function get larger and larger. Because is an odd function, we see the\ncorresponding table of negative values inTable 3."
	},
	{
		"chunk_id" : "00002402",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "corresponding table of negative values inTable 3.\n1.3 1.5 1.55 1.56\n3.6 14.1 48.1 92.6\nTable3\nWe can see that, as approaches the outputs get smaller and smaller. Remember that there are some values of\nfor which For example, and At these values, thetangent functionis undefined, so\nthe graph of has discontinuities at At these values, the graph of the tangent has vertical\nasymptotes.Figure 1represents the graph of The tangent is positive from 0 to and from to"
	},
	{
		"chunk_id" : "00002403",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "corresponding to quadrants I and III of the unit circle.\nFigure1 Graph of the tangent function\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 781\nGraphing Variations ofy= tanx\nAs with the sine and cosine functions, thetangentfunction can be described by a general equation.\nWe can identify horizontal and vertical stretches and compressions using values of and The horizontal stretch can"
	},
	{
		"chunk_id" : "00002404",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical\nstretch using a point on the graph.\nBecause there are no maximum or minimum values of a tangent function, the termamplitudecannot be interpreted as\nit is for the sine and cosine functions. Instead, we will use the phrasestretching/compressing factorwhen referring to\nthe constant\nFeatures of the Graph ofy=Atan(Bx)\n The stretching factor is\n The period is"
	},
	{
		"chunk_id" : "00002405",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The stretching factor is\n The period is\n The domain is all real numbers where such that is an integer.\n The range is \n The asymptotes occur at where is an integer.\n is an odd function.\nGraphing One Period of a Stretched or Compressed Tangent Function\nWe can use what we know about the properties of thetangent functionto quickly sketch a graph of any stretched and/or\ncompressed tangent function of the form We focus on a singleperiodof the function including the"
	},
	{
		"chunk_id" : "00002406",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "origin, because the periodic property enables us to extend the graph to the rest of the functions domain if we wish. Our\nlimited domain is then the interval and the graph has vertical asymptotes at where On\nthe graph will come up from the left asymptote at cross through the origin, and continue to increase as it\napproaches the right asymptote at To make the function approach the asymptotes at the correct rate, we also"
	},
	{
		"chunk_id" : "00002407",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through.\nFor example, we can use\nbecause\n...\nHOW TO\nGiven the function graph one period.\n1. Identify the stretching factor,\n2. Identify and determine the period,\n3. Draw vertical asymptotes at and\n4. For the graph approaches the left asymptote at negative output values and the right asymptote at\npositive output values (reverse for )."
	},
	{
		"chunk_id" : "00002408",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "positive output values (reverse for ).\n5. Plot reference points at and and draw the graph through these points.\nEXAMPLE1\nSketching a Compressed Tangent\nSketch a graph of one period of the function\n782 8  Periodic Functions\nSolution\nFirst, we identify and\nBecause and we can find thestretching/compressing factorand period. The period is so the\nasymptotes are at At a quarter period from the origin, we have\nThis means the curve must pass through the points and The only inflection point is at the"
	},
	{
		"chunk_id" : "00002409",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "origin.Figure 2shows the graph of one period of the function.\nFigure2\nTRY IT #1 Sketch a graph of\nGraphing One Period of a Shifted Tangent Function\nNow that we can graph atangent functionthat is stretched or compressed, we will add a vertical and/or horizontal (or\nphase) shift. In this case, we add and to the general form of the tangent function.\nThe graph of a transformed tangent function is different from the basic tangent function in several ways:\nFeatures of the Graph ofy=Atan(BxC)+D"
	},
	{
		"chunk_id" : "00002410",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Features of the Graph ofy=Atan(BxC)+D\n The stretching factor is\n The period is\n The domain is where is an integer.\n The range is \n The vertical asymptotes occur at where is an odd integer.\n There is no amplitude.\n is an odd function because it is the quotient of odd and even functions (sine and cosine\nrespectively).\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 783\n...\nHOW TO\nGiven the function sketch the graph of one period."
	},
	{
		"chunk_id" : "00002411",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Express the function given in the form\n2. Identify thestretching/compressing factor,\n3. Identify and determine the period,\n4. Identify and determine the phase shift,\n5. Draw the graph of shifted to the right by and up by\n6. Sketch the vertical asymptotes, which occur at where is an odd integer.\n7. Plot any three reference points and draw the graph through these points.\nEXAMPLE2\nGraphing One Period of a Shifted Tangent Function\nGraph one period of the function\nSolution"
	},
	{
		"chunk_id" : "00002412",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph one period of the function\nSolution\nStep 1.The function is already written in the form\nStep 2. so the stretching factor is\nStep 3. so the period is\nStep 4. so the phase shift is\nStep 5-7.The asymptotes are at and and the three recommended reference points are\nand The graph is shown inFigure 3.\nFigure3\nAnalysis\nNote that this is a decreasing function because\nTRY IT #2 How would the graph inExample 2look different if we made instead of\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002413",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the graph of a tangent function, identify horizontal and vertical stretches.\n1. Find the period from the spacing between successive vertical asymptotes orx-intercepts.\n2. Write\n3. Determine a convenient point on the given graph and use it to determine\n784 8  Periodic Functions\nEXAMPLE3\nIdentifying the Graph of a Stretched Tangent\nFind a formula for the function graphed inFigure 4.\nFigure4 A stretched tangent function\nSolution\nThe graph has the shape of a tangent function."
	},
	{
		"chunk_id" : "00002414",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The graph has the shape of a tangent function.\nStep 1.One cycle extends from 4 to 4, so the period is Since we have\nStep 2.The equation must have the form\nStep 3.To find the vertical stretch we can use the point\nBecause\nThis function would have a formula\nTRY IT #3 Find a formula for the function inFigure 5.\nFigure5\nAnalyzing the Graphs ofy= secxandy= cscx\nThesecantwas defined by thereciprocal identity Notice that the function is undefined when the cosine is"
	},
	{
		"chunk_id" : "00002415",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "0, leading to vertical asymptotes at etc. Because the cosine is never more than 1 in absolute value, the secant,\nbeing the reciprocal, will never be less than 1 in absolute value.\nWe can graph by observing the graph of the cosine function because these two functions are reciprocals of\none another. SeeFigure 6. The graph of the cosine is shown as a dashed orange wave so we can see the relationship."
	},
	{
		"chunk_id" : "00002416",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Where the graph of the cosine function decreases, the graph of thesecant functionincreases. Where the graph of the\ncosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is\nundefined.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 785\nThe secant graph has vertical asymptotes at each value of where the cosine graph crosses thex-axis; we show these in"
	},
	{
		"chunk_id" : "00002417",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the\nsecant and cosecant.\nNote that, because cosine is an even function, secant is also an even function. That is,\nFigure6 Graph of the secant function,\nAs we did for the tangent function, we will again refer to the constant as the stretching factor, not the amplitude.\nFeatures of the Graph ofy=Asec(Bx)\n The stretching factor is\n The period is\n The domain is where is an odd integer."
	},
	{
		"chunk_id" : "00002418",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The domain is where is an odd integer.\n The range is  \n The vertical asymptotes occur at where is an odd integer.\n There is no amplitude.\n is an even function because cosine is an even function.\nSimilar to the secant, thecosecantis defined by the reciprocal identity Notice that the function is\nundefined when the sine is 0, leading to a vertical asymptote in the graph at etc. Since the sine is never more than 1"
	},
	{
		"chunk_id" : "00002419",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.\nWe can graph by observing the graph of the sine function because these two functions are reciprocals of one\nanother. SeeFigure 7. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the\ngraph of the sine function decreases, the graph of thecosecant functionincreases. Where the graph of the sine function\nincreases, the graph of the cosecant function decreases."
	},
	{
		"chunk_id" : "00002420",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The cosecant graph has vertical asymptotes at each value of where the sine graph crosses thex-axis; we show these in\nthe graph below with dashed vertical lines.\nNote that, since sine is an odd function, the cosecant function is also an odd function. That is,\nThe graph of cosecant, which is shown inFigure 7, is similar to the graph of secant.\n786 8  Periodic Functions\nFigure7 The graph of the cosecant function,\nFeatures of the Graph ofy=Acsc(Bx)\n The stretching factor is\n The period is"
	},
	{
		"chunk_id" : "00002421",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The stretching factor is\n The period is\n The domain is where is an integer.\n The range is  \n The asymptotes occur at where is an integer.\n is an odd function because sine is an odd function.\nGraphing Variations ofy= secxandy= cscx\nFor shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods\nto those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions"
	},
	{
		"chunk_id" : "00002422",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for\nthe period in more than one way. The procedure for secant is very similar, because the cofunction identity means that\nthe secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be\napplied to thecosecant functionin the same way as for the secant and other functions.The equations become the\nfollowing."
	},
	{
		"chunk_id" : "00002423",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "following.\nFeatures of the Graph ofy=Asec(BxC)+D\n The stretching factor is\n The period is\n The domain is where is an odd integer.\n The range is  \n The vertical asymptotes occur at where is an odd integer.\n There is no amplitude.\n is an even function because cosine is an even function.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 787\nFeatures of the Graph ofy=Acsc(BxC)+D\n The stretching factor is\n The period is\n The domain is where is an integer."
	},
	{
		"chunk_id" : "00002424",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The domain is where is an integer.\n The range is  \n The vertical asymptotes occur at where is an integer.\n There is no amplitude.\n is an odd function because sine is an odd function.\n...\nHOW TO\nGiven a function of the form graph one period.\n1. Express the function given in the form\n2. Identify the stretching/compressing factor,\n3. Identify and determine the period,\n4. Sketch the graph of\n5. Use the reciprocal relationship between and to draw the graph of\n6. Sketch the asymptotes."
	},
	{
		"chunk_id" : "00002425",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. Sketch the asymptotes.\n7. Plot any two reference points and draw the graph through these points.\nEXAMPLE4\nGraphing a Variation of the Secant Function\nGraph one period of\nSolution\nStep 1.The given function is already written in the general form,\nStep 2. so the stretching factor is\nStep 3. so The period is units.\nStep 4.Sketch the graph of the function\nStep 5.Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function."
	},
	{
		"chunk_id" : "00002426",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Steps 67.Sketch two asymptotes at and We can use two reference points, the local minimum at\nand the local maximum at Figure 8shows the graph.\nFigure8\nTRY IT #4 Graph one period of\n788 8  Periodic Functions\nQ&A Do the vertical shift and stretch/compression affect the secants range?\nYes. The range of is  \n...\nHOW TO\nGiven a function of the form graph one period.\n1. Express the function given in the form\n2. Identify the stretching/compressing factor,\n3. Identify and determine the period,"
	},
	{
		"chunk_id" : "00002427",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Identify and determine the period,\n4. Identify and determine the phase shift,\n5. Draw the graph of , but shift it to the right by and up by\n6. Sketch the vertical asymptotes, which occur at where is an odd integer.\nEXAMPLE5\nGraphing a Variation of the Secant Function\nGraph one period of\nSolution\nStep 1.Express the function given in the form\nStep 2.The stretching/compressing factor is\nStep 3.The period is\nStep 4.The phase shift is\nStep 5.Draw the graph of but shift it to the right by and up by"
	},
	{
		"chunk_id" : "00002428",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Step 6.Sketch the vertical asymptotes, which occur at and There is a local minimum at and a\nlocal maximum at Figure 9shows the graph.\nFigure9\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 789\nTRY IT #5 Graph one period of\nQ&A The domain of was given to be all such that for any integer Would the domain of\nYes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal"
	},
	{
		"chunk_id" : "00002429",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shift and compression or expansion implied by the transformation to the original functions input.\n...\nHOW TO\nGiven a function of the form graph one period.\n1. Express the function given in the form\n2.\n3. Identify and determine the period,\n4. Draw the graph of\n5. Use the reciprocal relationship between and to draw the graph of\n6. Sketch the asymptotes.\n7. Plot any two reference points and draw the graph through these points.\nEXAMPLE6\nGraphing a Variation of the Cosecant Function\nGraph one period of"
	},
	{
		"chunk_id" : "00002430",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph one period of\nSolution\nStep 1.The given function is already written in the general form,\nStep 2. so the stretching factor is 3.\nStep 3. so The period is units.\nStep 4.Sketch the graph of the function\nStep 5.Use the reciprocal relationship of the sine and cosecant functions to draw thecosecant function.\nSteps 67.Sketch three asymptotes at and We can use two reference points, the local maximum at\nand the local minimum at Figure 10shows the graph.\n790 8  Periodic Functions\nFigure10"
	},
	{
		"chunk_id" : "00002431",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "790 8  Periodic Functions\nFigure10\nTRY IT #6 Graph one period of\n...\nHOW TO\nGiven a function of the form graph one period.\n1. Express the function given in the form\n2. Identify the stretching/compressing factor,\n3. Identify and determine the period,\n4. Identify and determine the phase shift,\n5. Draw the graph of but shift it to the right by and up by\n6. Sketch the vertical asymptotes, which occur at where is an integer.\nEXAMPLE7"
	},
	{
		"chunk_id" : "00002432",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nGraphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant\nSketch a graph of What are the domain and range of this function?\nSolution\nStep 1.Express the function given in the form\nStep 2.Identify the stretching/compressing factor,\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 791\nStep 3.The period is\nStep 4.The phase shift is\nStep 5.Draw the graph of but shift it up\nStep 6.Sketch the vertical asymptotes, which occur at"
	},
	{
		"chunk_id" : "00002433",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The graph for this function is shown inFigure 11.\nFigure11 A transformed cosecant function\nAnalysis\nThe vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval\nare shown by dots. Notice how the graph of the transformed cosecant relates to the graph of\nshown as the orange dashed wave.\nTRY IT #7 Given the graph of shown inFigure 12, sketch the graph of\non the same axes.\nFigure12\nAnalyzing the Graph ofy= cotx"
	},
	{
		"chunk_id" : "00002434",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12\nAnalyzing the Graph ofy= cotx\nThe last trigonometric function we need to explore iscotangent. The cotangent is defined by thereciprocal identity\nNotice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in\nthe graph at etc. Since the output of the tangent function is all real numbers, the output of thecotangent functionis\n792 8  Periodic Functions\nalso all real numbers."
	},
	{
		"chunk_id" : "00002435",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "792 8  Periodic Functions\nalso all real numbers.\nWe can graph by observing the graph of the tangent function because these two functions are reciprocals of\none another. SeeFigure 13. Where the graph of the tangent function decreases, the graph of the cotangent function\nincreases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.\nThe cotangent graph has vertical asymptotes at each value of where we show these in the graph below with"
	},
	{
		"chunk_id" : "00002436",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "dashed lines. Since the cotangent is the reciprocal of the tangent, has vertical asymptotes at all values of where\nand at all values of where has its vertical asymptotes.\nFigure13 The cotangent function\nFeatures of the Graph ofy=Acot(Bx)\n The stretching factor is\n The period is\n The domain is where is an integer.\n The range is  \n The asymptotes occur at where is an integer.\n is an odd function.\nGraphing Variations ofy= cotx"
	},
	{
		"chunk_id" : "00002437",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Variations ofy= cotx\nWe can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes\nthe following.\nFeatures of the Graph ofy=Acot(BxC)+D\n The stretching factor is\n The period is\n The domain is where is an integer.\n The range is \n The vertical asymptotes occur at where is an integer.\n There is no amplitude.\n is an odd function because it is the quotient of even and odd functions (cosine and sine,\nrespectively)\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002438",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "respectively)\n...\nHOW TO\nGiven a modified cotangent function of the form graph one period.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 793\n1. Express the function in the form\n2. Identify the stretching factor,\n3. Identify the period,\n4. Draw the graph of\n5. Plot any two reference points.\n6. Use the reciprocal relationship between tangent and cotangent to draw the graph of\n7. Sketch the asymptotes.\nEXAMPLE8\nGraphing Variations of the Cotangent Function"
	},
	{
		"chunk_id" : "00002439",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Variations of the Cotangent Function\nDetermine the stretching factor, period, and phase shift of and then sketch a graph.\nSolution\nStep 1.Expressing the function in the form gives\nStep 2.The stretching factor is\nStep 3.The period is\nStep 4.Sketch the graph of\nStep 5.Plot two reference points. Two such points are and\nStep 6.Use the reciprocal relationship to draw\nStep 7.Sketch the asymptotes,\nThe blue graph inFigure 14shows and the green graph shows\nFigure14\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002440",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure14\n...\nHOW TO\nGiven a modified cotangent function of the form graph one period.\n1. Express the function in the form\n2. Identify the stretching factor,\n3. Identify the period,\n4. Identify the phase shift,\n5. Draw the graph of shifted to the right by and up by\n6. Sketch the asymptotes where is an integer.\n7. Plot any three reference points and draw the graph through these points.\n794 8  Periodic Functions\nEXAMPLE9\nGraphing a Modified Cotangent\nSketch a graph of one period of the function\nSolution"
	},
	{
		"chunk_id" : "00002441",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nStep 1.The function is already written in the general form\nStep 2. so the stretching factor is 4.\nStep 3. so the period is\nStep 4. so the phase shift is\nStep 5.We draw\nStep 6-7.Three points we can use to guide the graph are and We use the reciprocal relationship\nof tangent and cotangent to draw\nStep 8.The vertical asymptotes are and\nThe graph is shown inFigure 15.\nFigure15 One period of a modified cotangent function\nUsing the Graphs of Trigonometric Functions to Solve Real-World Problems"
	},
	{
		"chunk_id" : "00002442",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example,\nlets return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a\nfire truck and wondered about the movement of the light beam itself across the wall? The periodic behavior of the\ndistance the light shines as a function of time is obvious, but how do we determine the distance? We can use thetangent\nfunction.\nEXAMPLE10"
	},
	{
		"chunk_id" : "00002443",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function.\nEXAMPLE10\nUsing Trigonometric Functions to Solve Real-World Scenarios\nSuppose the function marks the distance in the movement of a light beam from the top of a police car\nacross a wall where is the time in seconds and is the distance in feet from a point on the wall directly across from the\npolice car.\n Find and interpret the stretching factor and period.  Graph on the interval\n Evaluate and discuss the functions value at that input.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002444",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 795\nSolution\n We know from the general form of that is the stretching factor and is the period.\nFigure16\nWe see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.\nThe period is This means that every 4 seconds, the beam of light sweeps the wall. The distance from\nthe spot across from the police car grows larger as the police car approaches."
	},
	{
		"chunk_id" : "00002445",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To graph the function, we draw an asymptote at and use the stretching factor and period. SeeFigure 17\nFigure17\n period: after 1 second, the beam of has moved 5 ft from the spot across from\nthe police car.\nMEDIA\nAccess these online resources for additional instruction and practice with graphs of other trigonometric functions.\nGraphing the Tangent(http://openstax.org/l/graphtangent)\nGraphing Cosecant and Secant(http://openstax.org/l/graphcscsec)\nGraphing the Cotangent(http://openstax.org/l/graphcot)"
	},
	{
		"chunk_id" : "00002446",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8.2 SECTION EXERCISES\nVerbal\n1. Explain how the graph of 2. How can the graph of 3. Explain why the period of\nthe sine function can be be used to is equal to\nused to graph construct the graph of\n4. Why are there no intercepts 5. How does the period of\non the graph of compare with the\nperiod of\n796 8  Periodic Functions\nAlgebraic\nFor the following exercises, match each trigonometric function with one of the following graphs.\nFigure18\n6. 7. 8.\n9."
	},
	{
		"chunk_id" : "00002447",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure18\n6. 7. 8.\n9.\nFor the following exercises, find the period and horizontal shift of each of the functions.\n10. 11. 12.\n13. If find 14. If find 15. If find\n16. If find\nFor the following exercises, rewrite each expression such that the argument is positive.\n17. 18.\nGraphical\nFor the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching\nfactor, period, and asymptotes.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30."
	},
	{
		"chunk_id" : "00002448",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 797\n31. 32. 33.\n34. 35. 36.\nFor the following exercises, find and graph two periods of the periodic function with the given stretching factor,\nperiod, and phase shift.\n37. A tangent curve, period of and phase 38. A tangent curve, period of and phase\nshift shift\nFor the following exercises, find an equation for the graph of each function.\n39. 40.\n41. 42."
	},
	{
		"chunk_id" : "00002449",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "39. 40.\n41. 42.\n798 8  Periodic Functions\n43. 44.\n45.\nTechnology\nFor the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing\ncalculators do not have a cosecant button; therefore, you will need to input as\n46. 47. 48.\n49. 50. Graph 51.\nWhat is the function shown in\nthe graph?\n52. 53.\nAccess for free at openstax.org\n8.2  Graphs of the Other Trigonometric Functions 799\nReal-World Applications\n54. The function marks the"
	},
	{
		"chunk_id" : "00002450",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "54. The function marks the\ndistance in the movement of a light beam from a\npolice car across a wall for time in seconds, and\ndistance in feet.\n Graph on the interval\n Find and interpret the stretching factor,\nperiod, and asymptote.\n Evaluate and and discuss the\nfunctions values at those inputs.\n55. Standing on the shore of a lake, a fisherman sights a 56. A laser rangefinder is locked on a comet\nboat far in the distance to his left. Let measured in approaching Earth. The distance in"
	},
	{
		"chunk_id" : "00002451",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radians, be the angle formed by the line of sight to the kilometers, of the comet after days, for in the\nship and a line due north from his position. Assume due interval 0 to 30 days, is given by\nnorth is 0 and is measured negative to the left and\npositive to the right. (SeeFigure 19.) The boat travels\nfrom due west to due east and, ignoring the curvature  Graph on the interval\nof the Earth, the distance in kilometers, from the  Evaluate and interpret the information."
	},
	{
		"chunk_id" : "00002452",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fisherman to the boat is given by the function  What is the minimum distance between the\ncomet and Earth? When does this occur? To which\n What is a reasonable domain for constant in the equation does this correspond?\n Find and discuss the meaning of any vertical\n Graph on this domain.\nasymptotes.\n Find and discuss the meaning of any vertical\nasymptotes on the graph of\n Calculate and interpret Round to the\nsecond decimal place.\n Calculate and interpret Round to the second\ndecimal place."
	},
	{
		"chunk_id" : "00002453",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decimal place.\n What is the minimum distance between the\nfisherman and the boat? When does this occur?\nFigure19\n800 8  Periodic Functions\n57. A video camera is focused on a rocket on a\nlaunching pad 2 miles from the camera. The angle\nof elevation from the ground to the rocket after\nseconds is\n Write a function expressing the altitude\nin miles, of the rocket above the ground after\nseconds. Ignore the curvature of the Earth.\n Graph on the interval\n Evaluate and interpret the values and"
	},
	{
		"chunk_id" : "00002454",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Evaluate and interpret the values and\n What happens to the values of as\napproaches 60 seconds? Interpret the meaning of\nthis in terms of the problem.\n8.3 Inverse Trigonometric Functions\nLearning Objectives\nIn this section, you will:\nUnderstand and use the inverse sine, cosine, and tangent functions.\nFind the exact value of expressions involving the inverse sine, cosine, and tangent functions.\nUse a calculator to evaluate inverse trigonometric functions."
	},
	{
		"chunk_id" : "00002455",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find exact values of composite functions with inverse trigonometric functions.\nFor anyright triangle, given one other angle and the length of one side, we can figure out what the other angles and\nsides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of\nsides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we\nwill explore theinverse trigonometric functions."
	},
	{
		"chunk_id" : "00002456",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will explore theinverse trigonometric functions.\nUnderstanding and Using the Inverse Sine, Cosine, and Tangent Functions\nIn order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function undoes\nwhat the original trigonometric function does, as is the case with any other function and its inverse. In other words,\nthe domain of the inverse function is the range of the original function, and vice versa, as summarized inFigure 1.\nFigure1"
	},
	{
		"chunk_id" : "00002457",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nFor example, if then we would write Be aware that does not mean The\nfollowing examples illustrate the inverse trigonometric functions:\n Since then\n Since then\n Since then\nIn previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what\nangle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for aone-to-\none function, if then an inverse function would satisfy"
	},
	{
		"chunk_id" : "00002458",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would\nfail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds\nto at least one input in every period, and there are an infinite number of periods. As with other functions that are not\none-to-one, we will need to restrict thedomainof each function to yield a new function that is one-to-one. We choose a"
	},
	{
		"chunk_id" : "00002459",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "domain for each function that includes the number 0.Figure 2shows the graph of the sine function limited to\nand the graph of the cosine function limited to\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 801\nFigure2 (a) Sine function on a restricted domain of (b) Cosine function on a restricted domain of\nFigure 3shows the graph of the tangent function limited to\nFigure3 Tangent function on a restricted domain of"
	},
	{
		"chunk_id" : "00002460",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful\ncharacteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-\nto-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the\nuseful property that it extends from onevertical asymptoteto the next instead of being divided into two parts by an\nasymptote."
	},
	{
		"chunk_id" : "00002461",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptote.\nOn these restricted domains, we can define theinverse trigonometric functions.\n Theinverse sine function means The inverse sine function is sometimes called thearcsine\nfunction, and notated\n Theinverse cosine function means The inverse cosine function is sometimes called the\narccosinefunction, and notated\n Theinverse tangent function means The inverse tangent function is sometimes called the\narctangentfunction, and notated\n"
	},
	{
		"chunk_id" : "00002462",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "arctangentfunction, and notated\n\nThe graphs of the inverse functions are shown inFigure 4,Figure 5, andFigure 6. Notice that the output of each of these\ninverse functions is anumber,an angle in radian measure. We see that has domain and range\nhas domain and range and has domain of all real numbers and range To find the\ndomainandrangeof inverse trigonometric functions, switch the domain and range of the original functions. Each graph"
	},
	{
		"chunk_id" : "00002463",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the inverse trigonometric function is a reflection of the graph of the original function about the line\n802 8  Periodic Functions\nFigure4 The sine function and inverse sine (or arcsine) function\nFigure5 The cosine function and inverse cosine (or arccosine) function\nFigure6 The tangent function and inverse tangent (or arctangent) function\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 803\nRelations for Inverse Sine, Cosine, and Tangent Functions\nFor angles in the interval if then"
	},
	{
		"chunk_id" : "00002464",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For angles in the interval if then\nFor angles in the interval if then\nFor angles in the interval if then\nEXAMPLE1\nWriting a Relation for an Inverse Function\nGiven write a relation involving the inverse sine.\nSolution\nUse the relation for the inverse sine. If then .\nIn this problem, and\nTRY IT #1 Given write a relation involving the inverse cosine.\nFinding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and\nTangent Functions"
	},
	{
		"chunk_id" : "00002465",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Tangent Functions\nNow that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must\nevaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other\nnumerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse\nfunctions when we are using the special angles, specifically (30), (45), and (60), and their reflections into other\nquadrants.\n..."
	},
	{
		"chunk_id" : "00002466",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadrants.\n...\nHOW TO\nGiven a special input value, evaluate an inverse trigonometric function.\n1. Find angle for which the original trigonometric function has an output equal to the given input for the inverse\ntrigonometric function.\n2. If is not in the defined range of the inverse, find another angle that is in the defined range and has the same\nsine, cosine, or tangent as depending on which corresponds to the given inverse function.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00002467",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nEvaluating Inverse Trigonometric Functions for Special Input Values\nEvaluate each of the following.\n   \nSolution\n Evaluating is the same as determining the angle that would have a sine value of In other words,\nwhat angle would satisfy There are multiple values that would satisfy this relationship, such as and\nbut we know we need the angle in the interval so the answer will be Remember that\nthe inverse is a function, so for each input, we will get exactly one output.\n804 8  Periodic Functions"
	},
	{
		"chunk_id" : "00002468",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "804 8  Periodic Functions\n To evaluate we know that and both have a sine value of but neither is in the\ninterval For that, we need the negative angle coterminal with\n To evaluate we are looking for an angle in the interval with a cosine value of The\nangle that satisfies this is\n Evaluating we are looking for an angle in the interval with a tangent value of 1. The correct\nangle is\nTRY IT #2 Evaluate each of the following.\n   \nUsing a Calculator to Evaluate Inverse Trigonometric Functions"
	},
	{
		"chunk_id" : "00002469",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To evaluateinverse trigonometric functionsthat do not involve the special angles discussed previously, we will need to\nuse a calculator or other type of technology. Most scientific calculators and calculator-emulating applications have\nspecific keys or buttons for the inverse sine, cosine, and tangent functions. These may be labeled, for example,SIN ,\nARCSIN, orASIN.\nIn the previous chapter, we worked with trigonometry on a right triangle to solve for the sides of a triangle given one"
	},
	{
		"chunk_id" : "00002470",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "side and an additional angle. Using the inverse trigonometric functions, we can solve for the angles of a right triangle\ngiven two sides, and we can use a calculator to find the values to several decimal places.\nIn these examples and exercises, the answers will be interpreted as angles and we will use as the independent\nvariable. The value displayed on the calculator may be in degrees or radians, so be sure to set the mode appropriate to\nthe application.\nEXAMPLE3\nEvaluating the Inverse Sine on a Calculator"
	},
	{
		"chunk_id" : "00002471",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Evaluating the Inverse Sine on a Calculator\nEvaluate using a calculator.\nSolution\nBecause the output of the inverse function is an angle, the calculator will give us a degree value if in degree mode and a\nradian value if in radian mode. Calculators also use the same domain restrictions on the angles as we are using.\nIn radian mode, In degree mode, Note that in calculus and beyond we will\nuse radians in almost all cases.\nTRY IT #3 Evaluate using a calculator.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002472",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #3 Evaluate using a calculator.\n...\nHOW TO\nGiven two sides of a right triangle like the one shown inFigure 7, find an angle.\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 805\nFigure7\n1. If one given side is the hypotenuse of length and the side of length adjacent to the desired angle is given, use\nthe equation\n2. If one given side is the hypotenuse of length and the side of length opposite to the desired angle is given, use\nthe equation"
	},
	{
		"chunk_id" : "00002473",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equation\n3. If the two legs (the sides adjacent to the right angle) are given, then use the equation\nEXAMPLE4\nApplying the Inverse Cosine to a Right Triangle\nSolve the triangle inFigure 8for the angle\nFigure8\nSolution\nBecause we know the hypotenuse and the side adjacent to the angle, it makes sense for us to use the cosine function.\nTRY IT #4 Solve the triangle inFigure 9for the angle\nFigure9\nFinding Exact Values of Composite Functions with Inverse Trigonometric\nFunctions"
	},
	{
		"chunk_id" : "00002474",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Functions\nThere are times when we need to compose a trigonometric function with an inverse trigonometric function. In these\ncases, we can usually find exact values for the resulting expressions without resorting to a calculator. Even when the\n806 8  Periodic Functions\ninput to the composite function is a variable or an expression, we can often find an expression for the output. To help\nsort out different cases, let and be two different trigonometric functions belonging to the set"
	},
	{
		"chunk_id" : "00002475",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and let and be their inverses.\nEvaluating Compositions of the Formf(f1(y)) andf1(f(x))\nFor any trigonometric function, for all in the proper domain for the given function. This follows from\nthe definition of the inverse and from the fact that the range of was defined to be identical to the domain of\nHowever, we have to be a little more careful with expressions of the form\nCompositions of a trigonometric function and its inverse\n \nQ&A Is it correct that"
	},
	{
		"chunk_id" : "00002476",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nQ&A Is it correct that\nNo. This equation is correct if belongs to the restricted domain but sine is defined for all real\ninput values, and for outside the restricted interval, the equation is not correct because its inverse\nalways returns a value in The situation is similar for cosine and tangent and their inverses. For\nexample,\n...\nHOW TO\nGiven an expression of the form f1(f()) where evaluate.\n1. If is in the restricted domain of"
	},
	{
		"chunk_id" : "00002477",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. If is in the restricted domain of\n2. If not, then find an angle within the restricted domain of such that Then\nEXAMPLE5\nUsing Inverse Trigonometric Functions\nEvaluate the following:\n   \nSolution\n so\n but so\n so\n but because cosine is an even function. so\nTRY IT #5 Evaluate\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 807\nEvaluating Compositions of the Formf1(g(x))"
	},
	{
		"chunk_id" : "00002478",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Evaluating Compositions of the Formf1(g(x))\nNow that we can compose a trigonometric function with its inverse, we can explore how to evaluate a composition of a\ntrigonometric function and the inverse of another trigonometric function. We will begin with compositions of the form\nFor special values of we can exactly evaluate the inner function and then the outer, inverse function.\nHowever, we can find a more general approach by considering the relation between the two acute angles of a right"
	},
	{
		"chunk_id" : "00002479",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "triangle where one is making the other Consider the sine and cosine of each angle of the right triangle in\nFigure 10.\nFigure10 Right triangle illustrating the cofunction relationships\nBecause we have if If is not in this domain, then we need\nto find another angle that has the same cosine as and does belong to the restricted domain; we then subtract this\nangle from Similarly, so if These are just the function-\ncofunction relationships presented in another way.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002480",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven functions of the form and evaluate them.\n1. If then\n2. If then find another angle such that\n3. If then\n4. If then find another angle such that\nEXAMPLE6\nEvaluating the Composition of an Inverse Sine with a Cosine\nEvaluate\n by direct evaluation.  by the method described previously.\nSolution\n Here, we can directly evaluate the inside of the composition.\nNow, we can evaluate the inverse function as we did earlier.  We have and\n808 8  Periodic Functions\nTRY IT #6 Evaluate"
	},
	{
		"chunk_id" : "00002481",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "808 8  Periodic Functions\nTRY IT #6 Evaluate\nEvaluating Compositions of the Formf(g1(x))\nTo evaluate compositions of the form where and are any two of the functions sine, cosine, or tangent\nand is any input in the domain of we have exact formulas, such as When we need to use\nthem, we can derive these formulas by using the trigonometric relations between the angles and sides of a right\ntriangle, together with the use of Pythagorass relation between the lengths of the sides. We can use the Pythagorean"
	},
	{
		"chunk_id" : "00002482",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "identity, to solve for one when given the other. We can also use theinverse trigonometric functions\nto find compositions involving algebraic expressions.\nEXAMPLE7\nEvaluating the Composition of a Sine with an Inverse Cosine\nFind an exact value for\nSolution\nBeginning with the inside, we can say there is some angle such that which means and we\nare looking for We can use the Pythagorean identity to do this.\nSince is in quadrant I, must be positive, so the solution is SeeFigure 11."
	},
	{
		"chunk_id" : "00002483",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure11 Right triangle illustrating that if then\nWe know that the inverse cosine always gives an angle on the interval so we know that the sine of that angle must\nbe positive; therefore\nTRY IT #7 Evaluate\nEXAMPLE8\nEvaluating the Composition of a Sine with an Inverse Tangent\nFind an exact value for\nSolution\nWhile we could use a similar technique as inExample 6, we will demonstrate a different technique here. From the inside,"
	},
	{
		"chunk_id" : "00002484",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we know there is an angle such that We can envision this as the opposite and adjacent sides on a right\ntriangle, as shown inFigure 12.\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 809\nFigure12 A right triangle with two sides known\nUsing the Pythagorean Theorem, we can find the hypotenuse of this triangle.\nNow, we can evaluate the sine of the angle as the opposite side divided by the hypotenuse.\nThis gives us our desired composition.\nTRY IT #8 Evaluate\nEXAMPLE9"
	},
	{
		"chunk_id" : "00002485",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #8 Evaluate\nEXAMPLE9\nFinding the Cosine of the Inverse Sine of an Algebraic Expression\nFind a simplified expression for for\nSolution\nWe know there is an angle such that\nBecause we know that the inverse sine must give an angle on the interval we can deduce that the cosine of\nthat angle must be positive.\nTRY IT #9 Find a simplified expression for for\nMEDIA\nAccess this online resource for additional instruction and practice with inverse trigonometric functions.\n810 8  Periodic Functions"
	},
	{
		"chunk_id" : "00002486",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "810 8  Periodic Functions\nEvaluate Expressions Involving Inverse Trigonometric Functions(http://openstax.org/l/evalinverstrig)\n8.3 SECTION EXERCISES\nVerbal\n1. Why do the functions 2. Since the functions 3. Explain the meaning of\nand and\nhave are inverse functions, why is\ndifferent ranges? not\nequal to\n4. Most calculators do not 5. Why must the domain of the 6. Discuss why this statement\nhave a key to evaluate sine function, be is incorrect:\nExplain how this restricted to for the for all"
	},
	{
		"chunk_id" : "00002487",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Explain how this restricted to for the for all\ncan be done using the inverse sine function to\ncosine function or the exist?\ninverse cosine function.\n7. Determine whether the\nfollowing statement is true\nor false and explain your\nanswer:\nAlgebraic\nFor the following exercises, evaluate the expressions.\n8. 9. 10.\n11. 12. 13.\n14. 15. 16.\nFor the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.\n17. 18. 19.\n20. 21.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002488",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "20. 21.\nAccess for free at openstax.org\n8.3  Inverse Trigonometric Functions 811\nFor the following exercises, find the angle in the given right triangle. Round answers to the nearest hundredth.\n22. 23.\nFor the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why.\n24. 25. 26.\n27. 28. 29.\n30. 31. 32.\n33. 34. 35.\n36.\nFor the following exercises, find the exact value of the expression in terms of with the help of a reference triangle.\n37. 38. 39."
	},
	{
		"chunk_id" : "00002489",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "37. 38. 39.\n40. 41.\nExtensions\nFor the following exercises, evaluate the expression without using a calculator. Give the exact value.\n42.\nFor the following exercises, find the function if\n43. 44. 45.\n46. 47.\n812 8  Periodic Functions\nGraphical\n48. Graph and 49. Graph and 50. Graph one cycle of\nstate the domain and state the domain and and state the\nrange of the function. range of the function. domain and range of the\nfunction.\n51. For what value of does 52. For what value of does\nUse a Use a"
	},
	{
		"chunk_id" : "00002490",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use a Use a\ngraphing calculator to graphing calculator to\napproximate the answer. approximate the answer.\nReal-World Applications\n53. Suppose a 13-foot ladder is 54. Suppose you drive 0.6 55. An isosceles triangle has\nleaning against a building, miles on a road so that the two congruent sides of\nreaching to the bottom of a vertical distance changes length 9 inches. The\nsecond-floor window 12 from 0 to 150 feet. What is remaining side has a"
	},
	{
		"chunk_id" : "00002491",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "feet above the ground. the angle of elevation of length of 8 inches. Find the\nWhat angle, in radians, the road? angle that a side of 9\ndoes the ladder make with inches makes with the\nthe building? 8-inch side.\n56. Without using a calculator, 57. A truss (interior beam 58. The line passes\napproximate the value of structure) for the roof of a through the origin in the\nExplain house is constructed from x,y-plane. What is the\nwhy your answer is two identical right measure of the angle that"
	},
	{
		"chunk_id" : "00002492",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reasonable. triangles. Each has a base the line makes with the\nof 12 feet and height of 4 positivex-axis?\nfeet. Find the measure of\nthe acute angle adjacent to\nthe 4-foot side.\n59. The line passes 60. What percentage grade 61. A 20-foot ladder leans up\nthrough the origin in the should a road have if the against the side of a\nx,y-plane. What is the angle of elevation of the building so that the foot of\nmeasure of the angle that road is 4 degrees? (The the ladder is 10 feet from"
	},
	{
		"chunk_id" : "00002493",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the line makes with the percentage grade is the base of the building. If\nnegativex-axis? defined as the change in specifications call for the\nthe altitude of the road ladder's angle of elevation\nover a 100-foot horizontal to be between 35 and 45\ndistance. For example a 5% degrees, does the\ngrade means that the road placement of this ladder\nrises 5 feet for every 100 satisfy safety\nfeet of horizontal distance.) specifications?\n62. Suppose a 15-foot ladder\nleans against the side of a"
	},
	{
		"chunk_id" : "00002494",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "leans against the side of a\nhouse so that the angle of\nelevation of the ladder is 42\ndegrees. How far is the\nfoot of the ladder from the\nside of the house?\nAccess for free at openstax.org\n8  Chapter Review 813\nChapter Review\nKey Terms\namplitude the vertical height of a function; the constant appearing in the definition of a sinusoidal function\narccosine another name for the inverse cosine;\narcsine another name for the inverse sine;\narctangent another name for the inverse tangent;"
	},
	{
		"chunk_id" : "00002495",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "arctangent another name for the inverse tangent;\ninverse cosine function the function which is the inverse of the cosine function and the angle that has a\ncosine equal to a given number\ninverse sine function the function which is the inverse of the sine function and the angle that has a sine equal\nto a given number\ninverse tangent function the function which is the inverse of the tangent function and the angle that has a\ntangent equal to a given number"
	},
	{
		"chunk_id" : "00002496",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tangent equal to a given number\nmidline the horizontal line where appears in the general form of a sinusoidal function\nperiodic function a function that satisfies for a specific constant and any value of\nphase shift the horizontal displacement of the basic sine or cosine function; the constant\nsinusoidal function any function that can be expressed in the form or\nKey Equations\nSinusoidal functions\nShifted, compressed, and/or stretched tangent function\nShifted, compressed, and/or stretched secant function"
	},
	{
		"chunk_id" : "00002497",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Shifted, compressed, and/or stretched cosecant function\nShifted, compressed, and/or stretched cotangent function\nKey Concepts\n8.1Graphs of the Sine and Cosine Functions\n Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine\nfunctions have a period of\n The function is odd, so its graph is symmetric about the origin. The function is even, so its graph is\nsymmetric about they-axis."
	},
	{
		"chunk_id" : "00002498",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "symmetric about they-axis.\n The graph of a sinusoidal function has the same general shape as a sine or cosine function.\n In the general formula for a sinusoidal function, the period is SeeExample 1.\n In the general formula for a sinusoidal function, represents amplitude. If the function is stretched,\nwhereas if the function is compressed. SeeExample 2.\n The value in the general formula for a sinusoidal function indicates the phase shift. SeeExample 3."
	},
	{
		"chunk_id" : "00002499",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The value in the general formula for a sinusoidal function indicates the vertical shift from the midline. See\nExample 4.\n Combinations of variations of sinusoidal functions can be detected from an equation. SeeExample 5.\n The equation for a sinusoidal function can be determined from a graph. SeeExample 6andExample 7.\n A function can be graphed by identifying its amplitude and period. SeeExample 8andExample 9."
	},
	{
		"chunk_id" : "00002500",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. SeeExample\n10.\n Sinusoidal functions can be used to solve real-world problems. SeeExample 11,Example 12, andExample 13.\n814 8  Exercises\n8.2Graphs of the Other Trigonometric Functions\n The tangent function has period\n is a tangent with vertical and/or horizontal stretch/compression and shift. SeeExample\n1,Example 2, andExample 3."
	},
	{
		"chunk_id" : "00002501",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1,Example 2, andExample 3.\n The secant and cosecant are both periodic functions with a period of gives a shifted,\ncompressed, and/or stretched secant function graph. SeeExample 4andExample 5.\n gives a shifted, compressed, and/or stretched cosecant function graph. SeeExample 6\nandExample 7.\n The cotangent function has period and vertical asymptotes at\n The range of cotangent is   and the function is decreasing at each point in its range.\n The cotangent is zero at"
	},
	{
		"chunk_id" : "00002502",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The cotangent is zero at\n is a cotangent with vertical and/or horizontal stretch/compression and shift. See\nExample 8andExample 9.\n Real-world scenarios can be solved using graphs of trigonometric functions. SeeExample 10.\n8.3Inverse Trigonometric Functions\n An inverse function is one that undoes another function. The domain of an inverse function is the range of the\noriginal function and the range of an inverse function is the domain of the original function."
	},
	{
		"chunk_id" : "00002503",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions\nare defined for restricted domains.\n For any trigonometric function if then However, only implies if is\nin the restricted domain of SeeExample 1.\n Special angles are the outputs of inverse trigonometric functions for special input values; for example,\nSeeExample 2.\n A calculator will return an angle within the restricted domain of the original trigonometric function. SeeExample 3."
	},
	{
		"chunk_id" : "00002504",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Inverse functions allow us to find an angle when given two sides of a right triangle. SeeExample 4.\n In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions;\nfor example, SeeExample 5.\n If the inside function is a trigonometric function, then the only possible combinations are if\nand if SeeExample 6andExample 7.\n When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a"
	},
	{
		"chunk_id" : "00002505",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric\nfunction. SeeExample 8.\n When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use\ntrig identities to assist in determining the ratio of sides. SeeExample 9.\nExercises\nReview Exercises\nGraphs of the Sine and Cosine Functions\nFor the following exercises, graph the functions for two periods and determine the amplitude or stretching factor,"
	},
	{
		"chunk_id" : "00002506",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "period, midline equation, and asymptotes.\n1. 2. 3.\n4. 5. 6.\n7. 8.\nAccess for free at openstax.org\n8  Exercises 815\nGraphs of the Other Trigonometric Functions\nFor the following exercises, graph the functions for two periods and determine the amplitude or stretching factor,\nperiod, midline equation, and asymptotes.\n9. 10. 11.\n12.\nFor the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.\n13. 14. 15.\n16. 17. 18."
	},
	{
		"chunk_id" : "00002507",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13. 14. 15.\n16. 17. 18.\nFor the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its\npopulation may be modeled by the following function: where the domain is the years\nsince 1980 and the range is the population of the city.\n19. What is the largest and 20. Graph the function on the 21. What are the amplitude,\nsmallest population the city domain of . period, and phase shift for\nmay have? the function?"
	},
	{
		"chunk_id" : "00002508",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "may have? the function?\n22. Over this domain, when 23. What is the predicted\ndoes the population reach population in 2007? 2010?\n18,000? 13,000?\n816 8  Exercises\nFor the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.\n24. Suppose the graph of the displacement function is 25. At time = 0, what is the displacement of the\nshown inFigure 1, where the values on thex-axis weight?\nrepresent the time in seconds and they-axis"
	},
	{
		"chunk_id" : "00002509",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represent the time in seconds and they-axis\nrepresents the displacement in inches. Give the\nequation that models the vertical displacement of\nthe weight on the spring.\nFigure1\n26. At what time does the displacement from the 27. What is the time required for the weight to return\nequilibrium point equal zero? to its initial height of 5 inches? In other words,\nwhat is the period for the displacement function?\nInverse Trigonometric Functions"
	},
	{
		"chunk_id" : "00002510",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Inverse Trigonometric Functions\nFor the following exercises, find the exact value without the aid of a calculator.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39. Graph and\non the\ninterval and explain\nany observations.\nAccess for free at openstax.org\n8  Exercises 817\n40. Graph and 41. Graph the function\nand explain\nany observations.\non the interval and\ncompare the graph to the\ngraph of on\nthe same interval. Describe\nany observations.\nPractice Test"
	},
	{
		"chunk_id" : "00002511",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "any observations.\nPractice Test\nFor the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period,\nand the equation for the midline.\n1. 2. 3.\n4. 5. 6.\n7. 8. 9.\n10. 11. 12.\n13.\nFor the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the\nfunction.\n14. Give in terms of a sine function. 15. Give in terms of a sine function.\n818 8  Exercises\n16. Give in terms of a tangent\nfunction."
	},
	{
		"chunk_id" : "00002512",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "16. Give in terms of a tangent\nfunction.\nFor the following exercises, find the amplitude, period, phase shift, and midline.\n17. 18.\n19. The outside temperature over the course of a day 20. Water is pumped into a storage bin and empties\ncan be modeled as a sinusoidal function. Suppose according to a periodic rate. The depth of the\nyou know the temperature is 68F at midnight and water is 3 feet at its lowest at 2:00 a.m. and 71 feet"
	},
	{
		"chunk_id" : "00002513",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the high and low temperatures during the day are at its highest, which occurs every 5 hours. Write a\n80F and 56F, respectively. Assuming is the cosine function that models the depth of the\nnumber of hours since midnight, find a function water as a function of time, and then graph the\nfor the temperature, in terms of function for one period.\nFor the following exercises, find the period and horizontal shift of each function.\n21. 22.\n23. Write the equation for the graph inFigure 1in 24. If find"
	},
	{
		"chunk_id" : "00002514",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms of the secant function and give the period\nand phase shift.\nFigure1\n25. If find\nAccess for free at openstax.org\n8  Exercises 819\nFor the following exercises, graph the functions on the specified window and answer the questions.\n26. Graph 27. Graph 28. Graph on\non and explain any\non the viewing window the following domains in observations.\nby and\nApproximate the graphs Suppose this function\nperiod. models sound waves. Why\nwould these views look so\ndifferent?\nFor the following exercises, let"
	},
	{
		"chunk_id" : "00002515",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "different?\nFor the following exercises, let\n29. What is the largest possible 30. What is the smallest 31. Where is the function\nvalue for possible value for increasing on the interval\nFor the following exercises, find and graph one period of the periodic function with the given amplitude, period, and\nphase shift.\n32. Sine curve with amplitude 33. Cosine curve with\n3, period and phase amplitude 2, period and\nshift phase shift"
	},
	{
		"chunk_id" : "00002516",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shift phase shift\nFor the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior,\namplitude, asymptotes, or undefined points.\n34. 35.\nFor the following exercises, find the exact value.\n36. 37. 38.\n39. 40. 41.\n42. 43.\nFor the following exercises, suppose Evaluate the following expressions.\n44. 45.\n820 8  Exercises\n46. GivenFigure 2, find the measure of angle to\nthree decimal places. Answer in radians.\nFigure2"
	},
	{
		"chunk_id" : "00002517",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "three decimal places. Answer in radians.\nFigure2\nFor the following exercises, determine whether the equation is true or false.\n47. 48. 49. The grade of a road is 7%.\nThis means that for every\nhorizontal distance of 100\nfeet on the road, the\nvertical rise is 7 feet. Find\nthe angle the road makes\nwith the horizontal in\nradians.\nAccess for free at openstax.org\n9  Introduction 821\n9 TRIGONOMETRIC IDENTITIES AND EQUATIONS"
	},
	{
		"chunk_id" : "00002518",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9 TRIGONOMETRIC IDENTITIES AND EQUATIONS\nTennis players can create advantages by changing the angles of their shots. The technology used to decide close calls\nalso relies heavily on mathematics. (credit: modification of \"From the 2013 US Open\"\" by Edwin Martinez/flickr)"
	},
	{
		"chunk_id" : "00002519",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.4Sum-to-Product and Product-to-Sum Formulas\n9.5Solving Trigonometric Equations\nIntroduction to Trigonometric Identities and Equations\nWhen we think of tennis as a game of angles, we may imagine players racing up to the net, creating options to deliver\npowerful cross shots that will leave their opponent stumbling toward the line. This is an exciting and effective method of\nplay, though it brings greater risk."
	},
	{
		"chunk_id" : "00002520",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "play, though it brings greater risk.\nBut while the excitement of the game interplays with all types of geometry, some of the newest innovations make even\nmore use of mathematics. With balls traveling well over 100 miles per hour judges cannot always discern the centimeter\nor millimeters of difference between a ball that is in or out of bounds. Professional tennis was among the first sports to"
	},
	{
		"chunk_id" : "00002521",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rely on an advanced tracking system called Hawk-Eye to help make close calls. The system uses several high-resolution\ncameras that are able to monitor and the ball's movement and its position on the court. Using the images from several\ncameras at once, the system's computers use trigonometric calculations to triangulate the ball's exact position and,\nessentially, turn a series of two-dimensional images into a three-dimensional one. Also, since the ball travels faster than"
	},
	{
		"chunk_id" : "00002522",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the cameras' frame rate, the system also must make predictions to show where a ball is at all times. These technologies\ngenerally provide a more accurate game that builds more confidence and fairness. Similar technologies are used for\nbaseball, and automated strike-calling is under discussion.\n9.1 Verifying Trigonometric Identities and Using Trigonometric\nIdentities to Simplify Trigonometric Expressions\nLearning Objectives\nIn this section, you will:\nVerify the fundamental trigonometric identities."
	},
	{
		"chunk_id" : "00002523",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Verify the fundamental trigonometric identities.\nSimplify trigonometric expressions using algebra and the identities.\n822 9  Trigonometric Identities and Equations\nFigure1 International passports and travel documents\nIn espionage movies, we see international spies with multiple passports, each claiming a different identity. However, we\nknow that each of those passports represents the same person. The trigonometric identities act in a similar manner to"
	},
	{
		"chunk_id" : "00002524",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiple passportsthere are many ways to represent the same trigonometric expression. Just as a spy will choose an\nItalian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a\ntrigonometric equation.\nIn this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify\nthem and how we can use them to simplify trigonometric expressions.\nVerifying the Fundamental Trigonometric Identities"
	},
	{
		"chunk_id" : "00002525",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving\ntrigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools\nof solving algebraic equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions.\nBasic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will"
	},
	{
		"chunk_id" : "00002526",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "simplify the work involved with trigonometric expressions and equations. We already know that all of the trigonometric\nfunctions are related because they all are defined in terms of the unit circle. Consequently, any trigonometric identity\ncan be written in many ways.\nTo verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially\nrewrite the expression until it has been transformed into the same expression as the other side of the equation."
	},
	{
		"chunk_id" : "00002527",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic\nstrategies to obtain the desired result. In this first section, we will work with the fundamental identities: thePythagorean\nidentities, the even-odd identities, the reciprocal identities, and the quotient identities.\nWe will begin with thePythagorean identities(seeTable 1), which are equations involving trigonometric functions"
	},
	{
		"chunk_id" : "00002528",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "based on the properties of a right triangle. We have already seen and used the first of these identifies, but now we will\nalso use additional identities.\nPythagorean Identities\nTable1\nThe second and third identities can be obtained by manipulating the first. The identity is found by\nrewriting the left side of the equation in terms of sine and cosine.\nProve:\nAccess for free at openstax.org\n9.1  Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions 823"
	},
	{
		"chunk_id" : "00002529",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, can be obtained by rewriting the left side of this identity in terms of sine and cosine. This\ngives\nRecall that we determined which trigonometric functions are odd and which are even. The next set of fundamental\nidentities is the set ofeven-odd identities.Theeven-odd identitiesrelate the value of a trigonometric function at a\ngiven angle to the value of the function at the opposite angle. (SeeTable 2).\nEven-Odd Identities\nTable2"
	},
	{
		"chunk_id" : "00002530",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Even-Odd Identities\nTable2\nRecall that anodd functionis one in which for all in the domain of Thesinefunction is an odd\nfunction because The graph of an odd function is symmetric about the origin. For example, consider\ncorresponding inputs of and The output of is opposite the output of Thus,\nThis is shown inFigure 2.\nFigure2 Graph of\nRecall that aneven functionis one in which\nThe graph of an even function is symmetric about they-axis. The cosine function is an even function because"
	},
	{
		"chunk_id" : "00002531",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For example, consider corresponding inputs and The output of is the same as the\n824 9  Trigonometric Identities and Equations\noutput of Thus,\nSeeFigure 3.\nFigure3 Graph of\nFor all in the domain of the sine and cosine functions, respectively, we can state the following:\n Since sine is an odd function.\n Since, cosine is an even function.\nThe other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example,"
	},
	{
		"chunk_id" : "00002532",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "consider the tangent identity, We can interpret the tangent of a negative angle as\nTangent is therefore an odd function, which means that for\nall in the domain of thetangent function.\nThe cotangent identity, also follows from the sine and cosine identities. We can interpret the\ncotangent of a negative angle as Cotangent is therefore an odd function, which\nmeans that for all in the domain of thecotangent function."
	},
	{
		"chunk_id" : "00002533",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thecosecant functionis the reciprocal of the sine function, which means that the cosecant of a negative angle will be\ninterpreted as The cosecant function is therefore odd.\nFinally, the secant function is the reciprocal of the cosine function, and the secant of a negative angle is interpreted as\nThe secant function is therefore even.\nTo sum up, only two of the trigonometric functions, cosine and secant, are even. The other four functions are odd,\nverifying the even-odd identities."
	},
	{
		"chunk_id" : "00002534",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "verifying the even-odd identities.\nThe next set of fundamental identities is the set ofreciprocal identities, which, as their name implies, relate\ntrigonometric functions that are reciprocals of each other. SeeTable 3. Recall that we first encountered these identities\nwhen defining trigonometric functions from right angles inRight Angle Trigonometry.\nReciprocal Identities\nTable3\nThe final set of identities is the set ofquotient identities, which define relationships among certain trigonometric"
	},
	{
		"chunk_id" : "00002535",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions and can be very helpful in verifying other identities. SeeTable 4.\nAccess for free at openstax.org\n9.1  Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions 825\nQuotient Identities\nTable4\nThe reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.\nSummarizing Trigonometric Identities\nThePythagorean identitiesare based on the properties of a right triangle."
	},
	{
		"chunk_id" : "00002536",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Theeven-odd identitiesrelate the value of a trigonometric function at a given angle to the value of the function at\nthe opposite angle.\nThereciprocal identitiesdefine reciprocals of the trigonometric functions.\nThequotient identitiesdefine the relationship among the trigonometric functions.\nEXAMPLE1\nGraphing the Equations of an Identity\nGraph both sides of the identity In other words, on the graphing calculator, graph and\n826 9  Trigonometric Identities and Equations\nSolution\nSeeFigure 4.\nFigure4\nAnalysis"
	},
	{
		"chunk_id" : "00002537",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSeeFigure 4.\nFigure4\nAnalysis\nWe see only one graph because both expressions generate the same image. One is on top of the other. This is a good\nway to confirm an identity verified with analytical means. If both expressions give the same graph, then they are most\nlikely identities.\n...\nHOW TO\nGiven a trigonometric identity, verify that it is true.\n1. Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to\nsimplify than to build."
	},
	{
		"chunk_id" : "00002538",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "simplify than to build.\n2. Look for opportunities to factor expressions, square a binomial, or add fractions.\n3. Noting which functions are in the final expression, look for opportunities to use the identities and make the\nproper substitutions.\n4. If these steps do not yield the desired result, try converting all terms to sines and cosines.\nEXAMPLE2\nVerifying a Trigonometric Identity\nVerify\nSolution\nWe will start on the left side, as it is the more complicated side:\nAnalysis"
	},
	{
		"chunk_id" : "00002539",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nThis identity was fairly simple to verify, as it only required writing in terms of and\nTRY IT #1 Verify the identity\nAccess for free at openstax.org\n9.1  Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions 827\nEXAMPLE3\nVerifying the Equivalency Using the Even-Odd Identities\nVerify the following equivalency using the even-odd identities:\nSolution\nWorking on the left side of the equation, we have\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002540",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nVerifying a Trigonometric Identity Involvingsec2\nVerify the identity\nSolution\nAs the left side is more complicated, lets begin there.\nThere is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side.\nAnalysis\nIn the first method, we used the identity and continued to simplify. In the second method, we split the\nfraction, putting both terms in the numerator over the common denominator. This problem illustrates that there are"
	},
	{
		"chunk_id" : "00002541",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiple ways we can verify an identity. Employing some creativity can sometimes simplify a procedure. As long as the\nsubstitutions are correct, the answer will be the same.\nTRY IT #2 Show that\nEXAMPLE5\nCreating and Verifying an Identity\nCreate an identity for the expression by rewriting strictly in terms of sine.\n828 9  Trigonometric Identities and Equations\nSolution\nThere are a number of ways to begin, but here we will use the quotient and reciprocal identities to rewrite the\nexpression:\nThus,\nEXAMPLE6"
	},
	{
		"chunk_id" : "00002542",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expression:\nThus,\nEXAMPLE6\nVerifying an Identity Using Algebra and Even/Odd Identities\nVerify the identity:\nSolution\nLets start with the left side and simplify:\nTRY IT #3 Verify the identity\nEXAMPLE7\nVerifying an Identity Involving Cosines and Cotangents\nVerify the identity:\nSolution\nWe will work on the left side of the equation.\nAccess for free at openstax.org\n9.1  Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions 829"
	},
	{
		"chunk_id" : "00002543",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Algebra to Simplify Trigonometric Expressions\nWe have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying\ntrigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the\ndifference of squares formula, the perfect square formula, or substitution, will simplify the work involved with\ntrigonometric expressions and equations."
	},
	{
		"chunk_id" : "00002544",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "trigonometric expressions and equations.\nFor example, the equation resembles the equation which uses the\nfactored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can\nset each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric\nexpressions or equations.\nAnother example is the difference of squares formula, which is widely used in many areas"
	},
	{
		"chunk_id" : "00002545",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by\ncontinually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas\nmakes many trigonometric equations easier to understand and solve.\nEXAMPLE8\nWriting the Trigonometric Expression as an Algebraic Expression\nWrite the following trigonometric expression as an algebraic expression:\nSolution"
	},
	{
		"chunk_id" : "00002546",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nNotice that the pattern displayed has the same form as a standard quadratic expression, Letting\nwe can rewrite the expression as follows:\nThis expression can be factored as If it were set equal to zero and we wanted to solve the equation, we\nwould use the zero factor property and solve each factor for At this point, we would replace with and solve for\nEXAMPLE9\nRewriting a Trigonometric Expression Using the Difference of Squares\nRewrite the trigonometric expression using the difference of squares:"
	},
	{
		"chunk_id" : "00002547",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nNotice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the\nnumber 1 is 1. This is the difference of squares.\nAnalysis\nIf this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero\nfactor property. We could also use substitution like we did in the previous problem and let rewrite the\nexpression as and factor Then replace with and solve for the angle."
	},
	{
		"chunk_id" : "00002548",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #4 Rewrite the trigonometric expression using the difference of squares:\nEXAMPLE10\nSimplify by Rewriting and Using Substitution\nSimplify the expression by rewriting and using identities:\n830 9  Trigonometric Identities and Equations\nSolution\nWe can start with the Pythagorean identity.\nNow we can simplify by substituting for We have\nTRY IT #5 Use algebraic techniques to verify the identity:\n(Hint: Multiply the numerator and denominator on the left side by\nMEDIA"
	},
	{
		"chunk_id" : "00002549",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with the fundamental trigonometric identities.\nFundamental Trigonometric Identities(http://openstax.org/l/funtrigiden)\nVerifying Trigonometric Identities(http://openstax.org/l/verifytrigiden)\n9.1 SECTION EXERCISES\nVerbal\n1. We know is an 2. Examine the graph of 3. After examining the\neven function, and on the interval reciprocal identity for\nand How can we tell explain why the function is"
	},
	{
		"chunk_id" : "00002550",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and How can we tell explain why the function is\nare odd whether the function is even undefined at certain points.\nfunctions. What about or odd by only observing the\ngraph of\nand Are they\neven, odd, or neither? Why?\n4. All of the Pythagorean\nidentities are related.\nDescribe how to manipulate\nthe equations to get from\nto the\nother forms.\nAlgebraic\nFor the following exercises, use the fundamental identities to fully simplify the expression.\n5. 6. 7.\n8. 9.\n10. 11.\n12. 13.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002551",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10. 11.\n12. 13.\nAccess for free at openstax.org\n9.1  Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions 831\n14. 15.\nFor the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the\nsecond expression.\n16. 17. 18.\n19. 20.\n21. 22.\n23. 24.\n25. 26. 27.\n28.\nFor the following exercises, verify the identity.\n29. 30.\n31. 32.\n33.\nExtensions\nFor the following exercises, prove or disprove the identity."
	},
	{
		"chunk_id" : "00002552",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. 35.\n36. 37.\n38. 39.\nFor the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent\nexpression.\n40. 41. 42.\n832 9  Trigonometric Identities and Equations\n9.2 Sum and Difference Identities\nLearning Objectives\nIn this section, you will:\nUse sum and difference formulas for cosine.\nUse sum and difference formulas for sine.\nUse sum and difference formulas for tangent.\nUse sum and difference formulas for cofunctions."
	},
	{
		"chunk_id" : "00002553",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use sum and difference formulas for cofunctions.\nUse sum and difference formulas to verify identities.\nFigure1 Mount McKinley, in Denali National Park, Alaska, rises 20,237 feet (6,168 m) above sea level. It is the highest\npeak in North America. (credit: Daniel A. Leifheit, Flickr)\nHow can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly\nimpossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly"
	},
	{
		"chunk_id" : "00002554",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long\ndistances.\nThe trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950\nAD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are\nspecial equations or postulates, true for all values input to the equations, and with innumerable applications."
	},
	{
		"chunk_id" : "00002555",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The\nformulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this\nsection, the termformulais used synonymously with the wordidentity.\nUsing the Sum and Difference Formulas for Cosine\nFinding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in"
	},
	{
		"chunk_id" : "00002556",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms of two angles that have known trigonometric values. We can use thespecial angles, which we can review in the\nunit circle shown inFigure 2.\nFigure2 The Unit Circle\nAccess for free at openstax.org\n9.2  Sum and Difference Identities 833\nWe will begin with thesum and difference formulas for cosine, so that we can find the cosine of a given angle if we can\nbreak it up into the sum or difference of two of the special angles. SeeTable 1.\nSum formula for cosine\nDifference formula for cosine\nTable1"
	},
	{
		"chunk_id" : "00002557",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Difference formula for cosine\nTable1\nFirst, we will prove the difference formula for cosines. Lets consider two points on the unit circle. SeeFigure 3. Point is\nat an angle from the positivex-axis with coordinates and point is at an angle of from the positive\nx-axis with coordinates Note the measure of angle is\nLabel two more points: at an angle of from the positivex-axis with coordinates and\npoint with coordinates Triangle is a rotation of triangle and thus the distance from to is the"
	},
	{
		"chunk_id" : "00002558",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same as the distance from to\nFigure3\nWe can find the distance from to using thedistance formula.\nThen we apply thePythagorean identityand simplify.\nSimilarly, using the distance formula we can find the distance from to\nApplying the Pythagorean identity and simplifying we get:\nBecause the two distances are the same, we set them equal to each other and simplify.\n834 9  Trigonometric Identities and Equations\nFinally we subtract from both sides and divide both sides by"
	},
	{
		"chunk_id" : "00002559",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two\nangles.\nSum and Difference Formulas for Cosine\nThese formulas can be used to calculate the cosine of sums and differences of angles.\n...\nHOW TO\nGiven two angles, find the cosine of the difference between the angles.\n1. Write the difference formula for cosine.\n2. Substitute the values of the given angles into the formula.\n3. Simplify.\nEXAMPLE1"
	},
	{
		"chunk_id" : "00002560",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Simplify.\nEXAMPLE1\nFinding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles\nUsing the formula for the cosine of the difference of two angles, find the exact value of\nSolution\nBegin by writing the formula for the cosine of the difference of two angles. Then substitute the given values.\nKeep in mind that we can always check the answer using a graphing calculator in radian mode.\nTRY IT #1 Find the exact value of\nEXAMPLE2"
	},
	{
		"chunk_id" : "00002561",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #1 Find the exact value of\nEXAMPLE2\nFinding the Exact Value Using the Formula for the Sum of Two Angles for Cosine\nFind the exact value of\nSolution\nAs we can evaluate as\nAccess for free at openstax.org\n9.2  Sum and Difference Identities 835\nKeep in mind that we can always check the answer using a graphing calculator in degree mode.\nAnalysis\nNote that we could have also solved this problem using the fact that\nTRY IT #2 Find the exact value of\nUsing the Sum and Difference Formulas for Sine"
	},
	{
		"chunk_id" : "00002562",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using the Sum and Difference Formulas for Sine\nThesum and difference formulas for sinecan be derived in the same manner as those for cosine, and they resemble the\ncosine formulas.\nSum and Difference Formulas for Sine\nThese formulas can be used to calculate the sines of sums and differences of angles.\n...\nHOW TO\nGiven two angles, find the sine of the difference between the angles.\n1. Write the difference formula for sine.\n2. Substitute the given angles into the formula.\n3. Simplify.\nEXAMPLE3"
	},
	{
		"chunk_id" : "00002563",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Simplify.\nEXAMPLE3\nUsing Sum and Difference Identities to Evaluate the Difference of Angles\nUse the sum and difference identities to evaluate the difference of the angles and show that partaequals partb.\n \n836 9  Trigonometric Identities and Equations\nSolution\n Lets begin by writing the formula and substitute the given angles.\nNext, we need to find the values of the trigonometric expressions.\nNow we can substitute these values into the equation and simplify."
	},
	{
		"chunk_id" : "00002564",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Again, we write the formula and substitute the given angles.\nNext, we find the values of the trigonometric expressions.\nNow we can substitute these values into the equation and simplify.\nEXAMPLE4\nFinding the Exact Value of an Expression Involving an Inverse Trigonometric Function\nFind the exact value of Then check the answer with a graphing calculator.\nSolution\nThe pattern displayed in this problem is Let and Then we can write\nWe will use the Pythagorean identities to find and"
	},
	{
		"chunk_id" : "00002565",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9.2  Sum and Difference Identities 837\nUsing the sum formula for sine,\nUsing the Sum and Difference Formulas for Tangent\nFinding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a\nmatter of recognizing the pattern.\nFinding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and\nsimplifying. Recall,\nLets derive the sum formula for tangent."
	},
	{
		"chunk_id" : "00002566",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Lets derive the sum formula for tangent.\nWe can derive the difference formula for tangent in a similar way.\nSum and Difference Formulas for Tangent\nThesum and difference formulas for tangentare:\n838 9  Trigonometric Identities and Equations\n...\nHOW TO\nGiven two angles, find the tangent of the sum of the angles.\n1. Write the sum formula for tangent.\n2. Substitute the given angles into the formula.\n3. Simplify.\nEXAMPLE5\nFinding the Exact Value of an Expression Involving Tangent\nFind the exact value of"
	},
	{
		"chunk_id" : "00002567",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the exact value of\nSolution\nLets first write the sum formula for tangent and then substitute the given angles into the formula.\nNext, we determine the individual function values within the formula:\nSo we have\nTRY IT #3 Find the exact value of\nEXAMPLE6\nFinding Multiple Sums and Differences of Angles\nGiven find\n   \nAccess for free at openstax.org\n9.2  Sum and Difference Identities 839\nSolution"
	},
	{
		"chunk_id" : "00002568",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.2  Sum and Difference Identities 839\nSolution\nWe can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or\ntangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help\nfind each component of the sum and difference formulas.\n To find we begin with and The side opposite has length 3, the hypotenuse has"
	},
	{
		"chunk_id" : "00002569",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "length 5, and is in the first quadrant. SeeFigure 4. Using the Pythagorean Theorem, we can find the length of side\nFigure4\nSince and the side adjacent to is the hypotenuse is 13, and is in the third quadrant.\nSeeFigure 5. Again, using the Pythagorean Theorem, we have\nSince is in the third quadrant,\n840 9  Trigonometric Identities and Equations\nFigure5\nThe next step is finding the cosine of and the sine of The cosine of is the adjacent side over the hypotenuse. We"
	},
	{
		"chunk_id" : "00002570",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can find it from the triangle inFigure 5: We can also find the sine of from the triangle inFigure 5, as\nopposite side over the hypotenuse: Now we are ready to evaluate\n We can find in a similar manner. We substitute the values according to the formula.\n For if and then\nIf and then\nThen,\nAccess for free at openstax.org\n9.2  Sum and Difference Identities 841\n To find we have the values we need. We can substitute them in and evaluate.\nAnalysis"
	},
	{
		"chunk_id" : "00002571",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nA common mistake when addressing problems such as this one is that we may be tempted to think that and are\nangles in the same triangle, which of course, they are not. Also note that\nUsing Sum and Difference Formulas for Cofunctions\nNow that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to\ndo the same for their cofunctions. You may recall fromRight Triangle Trigonometrythat, if the sum of two positive"
	},
	{
		"chunk_id" : "00002572",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angles is those two angles are complements, and the sum of the two acute angles in a right triangle is so they are\nalso complements. InFigure 6, notice that if one of the acute angles is labeled as then the other acute angle must be\nlabeled\nNotice also that which is opposite over hypotenuse. Thus, when two angles are complementary,\nwe can say that the sine of equals thecofunctionof the complement of Similarly, tangent and cotangent are\ncofunctions, and secant and cosecant are cofunctions.\nFigure6"
	},
	{
		"chunk_id" : "00002573",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nFrom these relationships, thecofunction identitiesare formed. Recall that you first encountered these identities inThe\nUnit Circle: Sine and Cosine Functions.\n842 9  Trigonometric Identities and Equations\nCofunction Identities\nThe cofunction identities are summarized inTable 2.\nTable2\nNotice that the formulas in the table may also justified algebraically using the sum and difference formulas. For example,\nusing\nwe can write\nEXAMPLE7\nFinding a Cofunction with the Same Value as the Given Expression"
	},
	{
		"chunk_id" : "00002574",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write in terms of its cofunction.\nSolution\nThe cofunction of Thus,\nTRY IT #4 Write in terms of its cofunction.\nUsing the Sum and Difference Formulas to Verify Identities\nVerifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very\nfamiliar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules\npresented earlier may help simplify the process of verifying an identity.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002575",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an identity, verify using sum and difference formulas.\n1. Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until\nit matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only\none side is the most efficient.\n2. Look for opportunities to use the sum and difference formulas.\n3. Rewrite sums or differences of quotients as single quotients."
	},
	{
		"chunk_id" : "00002576",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.\nAccess for free at openstax.org\n9.2  Sum and Difference Identities 843\nEXAMPLE8\nVerifying an Identity Involving Sine\nVerify the identity\nSolution\nWe see that the left side of the equation includes the sines of the sum and the difference of angles.\nWe can rewrite each using the sum and difference formulas.\nWe see that the identity is verified.\nEXAMPLE9\nVerifying an Identity Involving Tangent"
	},
	{
		"chunk_id" : "00002577",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nVerifying an Identity Involving Tangent\nVerify the following identity.\nSolution\nWe can begin by rewriting the numerator on the left side of the equation.\nWe see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by\nwriting the tangent in terms of sine and cosine.\nTRY IT #5 Verify the identity:\nEXAMPLE10\nUsing Sum and Difference Formulas to Solve an Application Problem"
	},
	{
		"chunk_id" : "00002578",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Let and denote two non-vertical intersecting lines, and let denote the acute angle between and See\nFigure 7. Show that\nwhere and are the slopes of and respectively. (Hint:Use the fact that and )\n844 9  Trigonometric Identities and Equations\nFigure7\nSolution\nUsing the difference formula for tangent, this problem does not seem as daunting as it might.\nEXAMPLE11\nInvestigating a Guy-wire Problem"
	},
	{
		"chunk_id" : "00002579",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE11\nInvestigating a Guy-wire Problem\nFor a climbing wall, a guy-wire is attached 47 feet high on a vertical pole. Added support is provided by another guy-\nwire attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole,\nfind the angle between the wires. SeeFigure 8.\nFigure8\nSolution\nLets first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are"
	},
	{
		"chunk_id" : "00002580",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "known, we can use the tangent function. Notice that and We can then use difference\nformula for tangent.\nNow, substituting the values we know into the formula, we have\nUse the distributive property, and then simplify the functions.\nAccess for free at openstax.org\n9.2  Sum and Difference Identities 845\nNow we can calculate the angle in degrees.\nAnalysis\nOccasionally, when an application appears that includes a right triangle, we may think that solving is a matter of"
	},
	{
		"chunk_id" : "00002581",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what\ninformation is given.\nMEDIA\nAccess these online resources for additional instruction and practice with sum and difference identities.\nSum and Difference Identities for Cosine(http://openstax.org/l/sumdifcos)\nSum and Difference Identities for Sine(http://openstax.org/l/sumdifsin)\nSum and Difference Identities for Tangent(http://openstax.org/l/sumdiftan)\n9.2 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00002582",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.2 SECTION EXERCISES\nVerbal\n1. Explain the basis for the 2. Is there only one way to 3. Explain to someone who has\ncofunction identities and evaluate Explain forgotten the even-odd\nwhen they apply. how to set up the solution in properties of sinusoidal\ntwo different ways, and then functions how the addition\ncompute to make sure they and subtraction formulas\ngive the same answer. can determine this\ncharacteristic for\nand\n(Hint:\n)\nAlgebraic\nFor the following exercises, find the exact value.\n4. 5. 6."
	},
	{
		"chunk_id" : "00002583",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. 5. 6.\n7. 8. 9.\nFor the following exercises, rewrite in terms of and\n10. 11. 12.\n846 9  Trigonometric Identities and Equations\n13.\nFor the following exercises, simplify the given expression.\n14. 15. 16.\n17. 18. 19.\nFor the following exercises, find the requested information.\n20. Given that and 21. Given that and\nwith and with and\nboth in the interval both in the interval\nfind and find and\nFor the following exercises, find the exact value of each expression.\n22. 23. 24.\nGraphical"
	},
	{
		"chunk_id" : "00002584",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "22. 23. 24.\nGraphical\nFor the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs\nare identical. Confirm your answer using a graphing calculator.\n25. 26. 27.\n28. 29. 30.\n31. 32.\nFor the following exercises, use a graph to determine whether the functions are the same or different. If they are the\nsame, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think\n)\n33.\n34.\n35. 36.\n37. 38.\n39."
	},
	{
		"chunk_id" : "00002585",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : ")\n33.\n34.\n35. 36.\n37. 38.\n39.\nAccess for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 847\n40. 41.\nTechnology\nFor the following exercises, find the exact value algebraically, and then confirm the answer with a calculator to the fourth\ndecimal point.\n42. 43. 44.\n45. 46.\nExtensions\nFor the following exercises, prove the identities provided.\n47. 48. 49.\n50. 51.\nFor the following exercises, prove or disprove the statements.\n52. 53."
	},
	{
		"chunk_id" : "00002586",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "52. 53.\n54. 55. If and are angles in the same triangle, then\nprove or disprove\n56. If and are angles in the same\ntriangle, then prove or disprove\n9.3 Double-Angle, Half-Angle, and Reduction Formulas\nLearning Objectives\nIn this section, you will:\nUse double-angle formulas to find exact values.\nUse double-angle formulas to verify identities.\nUse reduction formulas to simplify an expression.\nUse half-angle formulas to find exact values.\n848 9  Trigonometric Identities and Equations"
	},
	{
		"chunk_id" : "00002587",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "848 9  Trigonometric Identities and Equations\nFigure1 Bicycle and skateboard ramps for advanced riders have a steeper incline than those designed for novices.\nBicycle and skateboard ramps made for competition (seeFigure 1) must vary in height depending on the skill level of the\ncompetitors. For advanced competitors, the angle formed by the ramp and the ground should be such that\nThe angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate"
	},
	{
		"chunk_id" : "00002588",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "three additional categories of identities that we can use to answer questions such as this one.\nUsing Double-Angle Formulas to Find Exact Values\nIn the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another\nlook at those same formulas. Thedouble-angle formulasare a special case of the sum formulas, where Deriving\nthe double-angle formula for sine begins with the sum formula,\nIf we let then we have"
	},
	{
		"chunk_id" : "00002589",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If we let then we have\nDeriving the double-angle for cosine gives us three options. First, starting from the sum formula,\nand letting we have\nUsing the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The\nfirst variation is:\nThe second variation is:\nSimilarly, to derive the double-angle formula for tangent, replacing in the sum formula gives\nDouble-Angle Formulas\nThedouble-angle formulasare summarized as follows:\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002590",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 849\n...\nHOW TO\nGiven the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find\nthe exact value.\n1. Draw a triangle to reflect the given information.\n2. Determine the correct double-angle formula.\n3. Substitute values into the formula based on the triangle.\n4. Simplify.\nEXAMPLE1\nUsing a Double-Angle Formula to Find the Exact Value Involving Tangent"
	},
	{
		"chunk_id" : "00002591",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given that and is in quadrant II, find the following:\n  \nSolution\nIf we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image.\nWe are given such that is in quadrant II. The tangent of an angle is equal to the opposite side over the\nadjacent side, and because is in the second quadrant, the adjacent side is on thex-axis and is negative. Use the\nPythagorean Theoremto find the length of the hypotenuse:"
	},
	{
		"chunk_id" : "00002592",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now we can draw a triangle similar to the one shown inFigure 2.\nFigure2\n850 9  Trigonometric Identities and Equations\n Lets begin by writing the double-angle formula for sine.\nWe see that we to need to find and Based onFigure 2, we see that the hypotenuse equals 5, so\nand Substitute these values into the equation, and simplify.\nThus,\n Write the double-angle formula for cosine.\nAgain, substitute the values of the sine and cosine into the equation, and simplify."
	},
	{
		"chunk_id" : "00002593",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Write the double-angle formula for tangent.\nIn this formula, we need the tangent, which we were given as Substitute this value into the equation,\nand simplify.\nTRY IT #1 Given with in quadrant I, find\nEXAMPLE2\nUsing the Double-Angle Formula for Cosine without Exact Values\nUse the double-angle formula for cosine to write in terms of\nSolution\nAnalysis\nThis example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the"
	},
	{
		"chunk_id" : "00002594",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric\nfunction.\nUsing Double-Angle Formulas to Verify Identities\nEstablishing identities using the double-angle formulas is performed using the same steps we used to derive the sum\nAccess for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 851\nand difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.\nEXAMPLE3"
	},
	{
		"chunk_id" : "00002595",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nUsing the Double-Angle Formulas to Verify an Identity\nVerify the following identity using double-angle formulas:\nSolution\nWe will work on the right side of the equal sign and rewrite the expression until it matches the left side.\nAnalysis\nThis process is not complicated, as long as we recall the perfect square formula from algebra:\nwhere and Part of being successful in mathematics is the ability to recognize patterns. While the\nterms or symbols may change, the algebra remains consistent."
	},
	{
		"chunk_id" : "00002596",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Verify the identity:\nEXAMPLE4\nVerifying a Double-Angle Identity for Tangent\nVerify the identity:\nSolution\nIn this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the\nequation.\nAnalysis\nHere is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the\nleft side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the"
	},
	{
		"chunk_id" : "00002597",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equivalency. For example, suppose that we wanted to show\nLets work on the right side.\n852 9  Trigonometric Identities and Equations\nWhen using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually\nseveral paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin\nwith the guidelines set forth earlier.\nTRY IT #3 Verify the identity:\nUse Reduction Formulas to Simplify an Expression"
	},
	{
		"chunk_id" : "00002598",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use Reduction Formulas to Simplify an Expression\nThe double-angle formulas can be used to derive thereduction formulas, which are formulas we can use to reduce the\npower of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or\ncosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses,"
	},
	{
		"chunk_id" : "00002599",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from\nthe double-angle formulas.\nWe can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Lets\nbegin with Solve for\nNext, we use the formula Solve for\nThe last reduction formula is derived by writing tangent in terms of sine and cosine:\nReduction Formulas\nThereduction formulasare summarized as follows:\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002600",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 853\nEXAMPLE5\nWriting an Equivalent Expression Not Containing Powers Greater Than 1\nWrite an equivalent expression for that does not involve any powers of sine or cosine greater than 1.\nSolution\nWe will apply the reduction formula for cosine twice.\nAnalysis\nThe solution is found by using the reduction formula twice, as noted, and the perfect square formula from algebra.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00002601",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nUsing the Power-Reducing Formulas to Prove an Identity\nUse the power-reducing formulas to prove\nSolution\nWe will work on simplifying the left side of the equation:\nAnalysis\nNote that in this example, we substituted\nfor The formula states\nWe let so\nTRY IT #4 Use the power-reducing formulas to prove that\nUsing Half-Angle Formulas to Find Exact Values\nThe next set of identities is the set ofhalf-angle formulas, which can be derived from the reduction formulas and we"
	},
	{
		"chunk_id" : "00002602",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can use when we have an angle that is half the size of a special angle. If we replace with the half-angle formula for\nsine is found by simplifying the equation and solving for Note that the half-angle formulas are preceded by a\n854 9  Trigonometric Identities and Equations\nsign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in\nwhich terminates.\nThe half-angle formula for sine is derived as follows:"
	},
	{
		"chunk_id" : "00002603",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To derive the half-angle formula for cosine, we have\nFor the tangent identity, we have\nHalf-Angle Formulas\nThehalf-angle formulasare as follows:\nEXAMPLE7\nUsing a Half-Angle Formula to Find the Exact Value of a Sine Function\nFind using a half-angle formula.\nSolution\nSince we use the half-angle formula for sine:\nAccess for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 855\nRemember that we can check the answer with a graphing calculator.\nAnalysis"
	},
	{
		"chunk_id" : "00002604",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNotice that we used only the positive root because is positive.\n...\nHOW TO\nGiven the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric\nfunctions of half of the angle.\n1. Draw a triangle to represent the given information.\n2. Determine the correct half-angle formula.\n3. Substitute values into the formula based on the triangle.\n4. Simplify.\nEXAMPLE8\nFinding Exact Values Using Half-Angle Identities"
	},
	{
		"chunk_id" : "00002605",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Exact Values Using Half-Angle Identities\nGiven that and lies in quadrant III, find the exact value of the following:\n  \nSolution\nUsing the given information, we can draw the triangle shown inFigure 3. Using the Pythagorean Theorem, we find the\nhypotenuse to be 17. Therefore, we can calculate and\nFigure3\n856 9  Trigonometric Identities and Equations\n Before we start, we must remember that if is in quadrant III, then so This\nmeans that the terminal side of is in quadrant II, since"
	},
	{
		"chunk_id" : "00002606",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To find we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we\nfound from the triangle inFigure 3and simplify.\nWe choose the positive value of because the angle terminates in quadrant II and sine is positive in quadrant II.\n To find we will write the half-angle formula for cosine, substitute the value of the cosine we found from\nthe triangle inFigure 3, and simplify."
	},
	{
		"chunk_id" : "00002607",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the triangle inFigure 3, and simplify.\nWe choose the negative value of because the angle is in quadrant II because cosine is negative in quadrant II.\n To find we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found\nfrom the triangle inFigure 3and simplify.\nWe choose the negative value of because lies in quadrant II, and tangent is negative in quadrant II.\nAccess for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 857"
	},
	{
		"chunk_id" : "00002608",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Given that and lies in quadrant IV, find the exact value of\nEXAMPLE9\nFinding the Measurement of a Half Angle\nNow, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level\ncompetition with an angle of formed by the ramp and the ground. Another ramp is to be constructed half as steep for\nnovice competition. If for higher-level competition, what is the measurement of the angle for novice\ncompetition?\nSolution"
	},
	{
		"chunk_id" : "00002609",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "competition?\nSolution\nSince the angle for novice competition measures half the steepness of the angle for the high level competition, and\nfor high competition, we can find from the right triangle and the Pythagorean theorem so that we can\nuse the half-angle identities. SeeFigure 4.\nFigure4\nWe see that We can use the half-angle formula for tangent: Since is in\nthe first quadrant, so is\nWe can take the inverse tangent to find the angle: So the angle of the ramp for novice competition\nis\nMEDIA"
	},
	{
		"chunk_id" : "00002610",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is\nMEDIA\nAccess these online resources for additional instruction and practice with double-angle, half-angle, and reduction\nformulas.\nDouble-Angle Identities(http://openstax.org/l/doubleangiden)\nHalf-Angle Identities(http://openstax.org/l/halfangleident)\n858 9  Trigonometric Identities and Equations\n9.3 SECTION EXERCISES\nVerbal\n1. Explain how to determine the reduction identities 2. Explain how to determine the double-angle\nfrom the double-angle identity formula for using the double-angle\nformulas for and"
	},
	{
		"chunk_id" : "00002611",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "formulas for and\n3. We can determine the half-angle formula for 4. For the half-angle formula given in the previous\nexercise for explain why dividing by 0 is\nby dividing the formula for\nnot a concern. (Hint: examine the values of\nby Explain how to determine two necessary for the denominator to be 0.)\nformulas for that do not involve any square\nroots.\nAlgebraic\nFor the following exercises, find the exact values of a) b) and c) without solving for\n5. If and is in quadrant I. 6. If and is in quadrant I."
	},
	{
		"chunk_id" : "00002612",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7. If and is in quadrant III. 8. If and is in quadrant IV.\nFor the following exercises, find the values of the six trigonometric functions if the conditions provided hold.\n9. and 10. and\nFor the following exercises, simplify to one trigonometric expression.\n11. 12.\nFor the following exercises, find the exact value using half-angle formulas.\n13. 14. 15.\n16. 17. 18.\n19.\nFor the following exercises, find the exact values of a) b) and c) without solving for when"
	},
	{
		"chunk_id" : "00002613",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "20. If and is in 21. If and is in 22. If and is in\nquadrant IV. quadrant III. quadrant II.\nAccess for free at openstax.org\n9.3  Double-Angle, Half-Angle, and Reduction Formulas 859\n23. If and is in\nquadrant II.\nFor the following exercises, useFigure 5to find the requested half and double angles.\nFigure5\n24. Find and 25. Find and\n26. Find and 27. Find and\nFor the following exercises, simplify each expression. Do not evaluate.\n28. 29. 30.\n31. 32. 33.\nFor the following exercises, prove the given identity."
	},
	{
		"chunk_id" : "00002614",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. 35. 36.\n37.\nFor the following exercises, rewrite the expression with an exponent no higher than 1.\n38. 39. 40.\n41. 42. 43.\n44.\nTechnology\nFor the following exercises, reduce the equations to powers of one, and then check the answer graphically.\n45. 46. 47.\n48. 49. 50.\n51. 52.\n860 9  Trigonometric Identities and Equations\nFor the following exercises, algebraically find an equivalent function, only in terms of and/or and then check\nthe answer by graphing both functions.\n53. 54.\nExtensions"
	},
	{
		"chunk_id" : "00002615",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "53. 54.\nExtensions\nFor the following exercises, prove the identities.\n55. 56. 57.\n58. 59.\n60. 61.\n62.\n63.\n9.4 Sum-to-Product and Product-to-Sum Formulas\nLearning Objectives\nIn this section, you will:\nExpress products as sums.\nExpress sums as products.\nFigure1 The UCLA marching band (credit: Eric Chan, Flickr).\nA band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that"
	},
	{
		"chunk_id" : "00002616",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "can be interpreted using trigonometric functions. For example,Figure 2represents a sound wave for the musical note A.\nIn this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as\nsound waves.\nAccess for free at openstax.org\n9.4  Sum-to-Product and Product-to-Sum Formulas 861\nFigure2\nExpressing Products as Sums\nWe have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but"
	},
	{
		"chunk_id" : "00002617",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sometimes we may need to express the product of cosine and sine as a sum. We can use theproduct-to-sum formulas,\nwhich express products of trigonometric functions as sums. Lets investigate the cosine identity first and then the sine\nidentity.\nExpressing Products as Sums for Cosine\nWe can derive the product-to-sum formula from the sum and difference identities forcosine. If we add the two\nequations, we get:\nThen, we divide by to isolate the product of cosines:\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002618",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a product of cosines, express as a sum.\n1. Write the formula for the product of cosines.\n2. Substitute the given angles into the formula.\n3. Simplify.\nEXAMPLE1\nWriting the Product as a Sum Using the Product-to-Sum Formula for Cosine\nWrite the following product of cosines as a sum:\nSolution\nWe begin by writing the formula for the product of cosines:\nWe can then substitute the given angles into the formula and simplify."
	},
	{
		"chunk_id" : "00002619",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #1 Use the product-to-sum formula to write the product as a sum or difference:\nExpressing the Product of Sine and Cosine as a Sum\nNext, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas forsine. If we\nadd the sum and difference identities, we get:\n862 9  Trigonometric Identities and Equations\nThen, we divide by 2 to isolate the product of cosine and sine:\nEXAMPLE2\nWriting the Product as a Sum Containing only Sine or Cosine"
	},
	{
		"chunk_id" : "00002620",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Express the following product as a sum containing only sine or cosine and no products:\nSolution\nWrite the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.\nTRY IT #2 Use the product-to-sum formula to write the product as a sum:\nExpressing Products of Sines in Terms of Cosine\nExpressing the product of sines in terms ofcosineis also derived from the sum and difference identities for cosine. In\nthis case, we will first subtract the two cosine formulas:"
	},
	{
		"chunk_id" : "00002621",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then, we divide by 2 to isolate the product of sines:\nSimilarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.\nThe Product-to-Sum Formulas\nTheproduct-to-sum formulasare as follows:\nEXAMPLE3\nExpress the Product as a Sum or Difference\nWrite as a sum or difference.\nAccess for free at openstax.org\n9.4  Sum-to-Product and Product-to-Sum Formulas 863\nSolution"
	},
	{
		"chunk_id" : "00002622",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and\nsimplify.\nTRY IT #3 Use the product-to-sum formula to evaluate\nExpressing Sums as Products\nSome problems require the reverse of the process we just used. Thesum-to-product formulasallow us to express sums\nof sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few"
	},
	{
		"chunk_id" : "00002623",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitutions, we can derive the sum-to-product identity forsine. Let and\nThen,\nThus, replacing and in the product-to-sum formula with the substitute expressions, we have\nThe other sum-to-product identities are derived similarly.\nSum-to-Product Formulas\nThesum-to-product formulasare as follows:\nEXAMPLE4\nWriting the Difference of Sines as a Product\nWrite the following difference of sines expression as a product:\n864 9  Trigonometric Identities and Equations\nSolution"
	},
	{
		"chunk_id" : "00002624",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe begin by writing the formula for the difference of sines.\nSubstitute the values into the formula, and simplify.\nTRY IT #4 Use the sum-to-product formula to write the sum as a product:\nEXAMPLE5\nEvaluating Using the Sum-to-Product Formula\nEvaluate Check the answer with a graphing calculator.\nSolution\nWe begin by writing the formula for the difference of cosines.\nThen we substitute the given angles and simplify.\nEXAMPLE6\nProving an Identity\nProve the identity:\nSolution"
	},
	{
		"chunk_id" : "00002625",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Proving an Identity\nProve the identity:\nSolution\nWe will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the\nright side.\nAccess for free at openstax.org\n9.4  Sum-to-Product and Product-to-Sum Formulas 865\nAnalysis\nRecall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the"
	},
	{
		"chunk_id" : "00002626",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make\nsubstitutions until that side is transformed into the other side.\nEXAMPLE7\nVerifying the Identity Using Double-Angle Formulas and Reciprocal Identities\nVerify the identity\nSolution\nFor verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and"
	},
	{
		"chunk_id" : "00002627",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.\nTRY IT #5 Verify the identity\nMEDIA\nAccess these online resources for additional instruction and practice with the product-to-sum and sum-to-product\nidentities.\nSum to Product Identities(http://openstax.org/l/sumtoprod)\nSum to Product and Product to Sum Identities(http://openstax.org/l/sumtpptsum)\n9.4 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00002628",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.4 SECTION EXERCISES\nVerbal\n1. Starting with the product to sum formula 2. Provide two different methods of calculating\nexplain one of which uses the product\nhow to determine the formula for to sum. Which method is easier?\n3. Describe a situation where we would convert an 4. Describe a situation where we would convert an\nequation from a sum to a product and give an equation from a product to a sum, and give an\nexample. example.\nAlgebraic"
	},
	{
		"chunk_id" : "00002629",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "example. example.\nAlgebraic\nFor the following exercises, rewrite the product as a sum or difference.\n5. 6. 7.\n8. 9. 10.\n866 9  Trigonometric Identities and Equations\nFor the following exercises, rewrite the sum or difference as a product.\n11. 12. 13.\n14. 15. 16.\nFor the following exercises, evaluate the product for the following using a sum or difference of two functions. Evaluate\nexactly.\n17. 18. 19.\n20. 21."
	},
	{
		"chunk_id" : "00002630",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "exactly.\n17. 18. 19.\n20. 21.\nFor the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and\ncosine.\n22. 23. 24.\n25. 26.\nFor the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.\n27. 28. 29.\n30. 31.\nFor the following exercises, prove the identity.\n32. 33.\n34. 35.\n36. 37.\n38.\nNumeric"
	},
	{
		"chunk_id" : "00002631",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "32. 33.\n34. 35.\n36. 37.\n38.\nNumeric\nFor the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give\nyour answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.\n39. 40. 41.\n42. 43.\nAccess for free at openstax.org\n9.4  Sum-to-Product and Product-to-Sum Formulas 867\nTechnology"
	},
	{
		"chunk_id" : "00002632",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Technology\nFor the following exercises, algebraically determine whether each of the given equation is an identity. If it is not an\nidentity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both\nexpressions on a calculator.\n44. 45.\n46. 47.\n48.\nFor the following exercises, simplify the expression to one term, then graph the original function and your simplified\nversion to verify they are identical.\n49. 50.\n51. 52.\n53.\nExtensions"
	},
	{
		"chunk_id" : "00002633",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "49. 50.\n51. 52.\n53.\nExtensions\nFor the following exercises, prove the following sum-to-product formulas.\n54. 55.\nFor the following exercises, prove the identity.\n56. 57.\n58. 59.\n60. 61.\n62.\n63.\n868 9  Trigonometric Identities and Equations\n9.5 Solving Trigonometric Equations\nLearning Objectives\nIn this section, you will:\nSolve linear trigonometric equations in sine and cosine.\nSolve equations involving a single trigonometric function.\nSolve trigonometric equations using a calculator."
	},
	{
		"chunk_id" : "00002634",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve trigonometric equations using a calculator.\nSolve trigonometric equations that are quadratic in form.\nSolve trigonometric equations using fundamental identities.\nSolve trigonometric equations with multiple angles.\nSolve right triangle problems.\nFigure1 Egyptian pyramids standing near a modern city. (credit: Oisin Mulvihill)\nThales of Miletus (circa 625547 BC) is known as the founder of geometry. The legend is that he calculated the height of"
	},
	{
		"chunk_id" : "00002635",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the Great Pyramid of Giza in Egypt using the theory ofsimilar triangles, which he developed by measuring the shadow of\nhis staff. He reasoned that when the height of his staff's shadow was exactly equal to the actual height of the staff, then\nthe height of the nearby pyramid's shadow must also be equal to the height of the actual pyramid. Since the structures\nand their shadows were creating a right triangle with two equal sides, they were similar triangles. By measuring the"
	},
	{
		"chunk_id" : "00002636",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "length of the pyramid's shadow at that moment, he could obtain the height of the pyramid. Based on proportions, this\ntheory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle\nof elevation and the angle of depression are found using similar triangles.\nIn earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of"
	},
	{
		"chunk_id" : "00002637",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the\nfinding the dimensions of the pyramids.\nSolving Linear Trigonometric Equations in Sine and Cosine\nTrigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways\nto solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are"
	},
	{
		"chunk_id" : "00002638",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be\nasked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each\nperiod. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational\nequations, the domain of the function must be considered before we assume that any solution is valid. Theperiodof"
	},
	{
		"chunk_id" : "00002639",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both the sine function and the cosine function is In other words, every units, they-values repeat. If we need to\nfind all possible solutions, then we must add where is an integer, to the initial solution. Recall the rule that gives\nthe format for stating all possible solutions for a function where the period is\nThere are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric"
	},
	{
		"chunk_id" : "00002640",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations requires the same techniques as solving algebraic equations. We read the equation from left to right,\nhorizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain\nexpressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we\nalso have the advantage of using the identities we developed in the previous sections.\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 869"
	},
	{
		"chunk_id" : "00002641",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.5  Solving Trigonometric Equations 869\nEXAMPLE1\nSolving a Linear Trigonometric Equation Involving the Cosine Function\nFind all possible exact solutions for the equation\nSolution\nFrom theunit circle, we know that\nThese are the solutions in the interval All possible solutions are given by\nwhere is an integer.\nEXAMPLE2\nSolving a Linear Equation Involving the Sine Function\nFind all possible exact solutions for the equation\nSolution"
	},
	{
		"chunk_id" : "00002642",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSolving for all possible values oftmeans that solutions include angles beyond the period of FromFigure 2, we can\nsee that the solutions are and But the problem is asking for all possible values that solve the equation.\nTherefore, the answer is\nwhere is an integer.\n...\nHOW TO\nGiven a trigonometric equation, solve using algebra.\n1. Look for a pattern that suggests an algebraic property, such as the difference of squares or a factoring\nopportunity."
	},
	{
		"chunk_id" : "00002643",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opportunity.\n2. Substitute the trigonometric expression with a single variable, such as or\n3. Solve the equation the same way an algebraic equation would be solved.\n4. Substitute the trigonometric expression back in for the variable in the resulting expressions.\n5. Solve for the angle.\nEXAMPLE3\nSolve the Linear Trigonometric Equation\nSolve the equation exactly:\nSolution\nUse algebraic techniques to solve the equation.\n870 9  Trigonometric Identities and Equations"
	},
	{
		"chunk_id" : "00002644",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "870 9  Trigonometric Identities and Equations\nTRY IT #1 Solve exactly the following linear equation on the interval\nSolving Equations Involving a Single Trigonometric Function\nWhen we are given equations that involve only one of the six trigonometric functions, their solutions involve using\nalgebraic techniques and the unit circle (seeFigure 2). We need to make several considerations when the equation"
	},
	{
		"chunk_id" : "00002645",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary\ntrigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal\nfunction, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different\nfrom one containing a sine or cosine function. First, as we know, the period of tangent is not Further, the domain"
	},
	{
		"chunk_id" : "00002646",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of tangent is all real numbers with the exception of odd integer multiples of unless, of course, a problem places its\nown restrictions on the domain.\nEXAMPLE4\nSolving a Problem Involving a Single Trigonometric Function\nSolve the problem exactly:\nSolution\nAs this problem is not easily factored, we will solve using the square root property. First, we use algebra to isolate\nThen we will find the angles.\nEXAMPLE5\nSolving a Trigonometric Equation Involving Cosecant\nSolve the following equation exactly:\nSolution"
	},
	{
		"chunk_id" : "00002647",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the following equation exactly:\nSolution\nWe want all values of for which over the interval\nAnalysis\nAs notice that all four solutions are in the third and fourth quadrants.\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 871\nEXAMPLE6\nSolving an Equation Involving Tangent\nSolve the equation exactly:\nSolution\nRecall that the tangent function has a period of On the interval and at the angle of the tangent has a value\nof 1. However, the angle we want is Thus, if then"
	},
	{
		"chunk_id" : "00002648",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 1. However, the angle we want is Thus, if then\nOver the interval we have two solutions:\nTRY IT #2 Find all solutions for\nEXAMPLE7\nIdentify all Solutions to the Equation Involving Tangent\nIdentify all exact solutions to the equation\nSolution\nWe can solve this equation using only algebra. Isolate the expression on the left side of the equals sign.\nThere are two angles on the unit circle that have a tangent value of and\nSolve Trigonometric Equations Using a Calculator"
	},
	{
		"chunk_id" : "00002649",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve Trigonometric Equations Using a Calculator\nNot all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle\nother than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either\ndegrees or radians, depending on the criteria of the given problem.\nEXAMPLE8\nUsing a Calculator to Solve a Trigonometric Equation Involving Sine\nUse a calculator to solve the equation where is in radians.\nSolution"
	},
	{
		"chunk_id" : "00002650",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nMake sure mode is set to radians. To find use the inverse sine function. On most calculators, you will need to push the\n2NDbutton and then the SIN button to bring up the function. What is shown on the screen is The\ncalculator is ready for the input within the parentheses. For this problem, we enter and press ENTER. Thus,\nto four decimals places,\nThe solution is\nThe angle measurement in degrees is\n872 9  Trigonometric Identities and Equations\nAnalysis"
	},
	{
		"chunk_id" : "00002651",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNote that a calculator will only return an angle in quadrants I or IV for the sine function, since that is the range of the\ninverse sine. The other angle is obtained by using Thus, the additional solution is\nEXAMPLE9\nUsing a Calculator to Solve a Trigonometric Equation Involving Secant\nUse a calculator to solve the equation giving your answer in radians.\nSolution\nWe can begin with some algebra.\nCheck that the MODE is in radians. Now use the inverse cosine function."
	},
	{
		"chunk_id" : "00002652",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Since and 1.8235 is between these two numbers, thus is in quadrant II. Cosine is also\nnegative in quadrant III. Note that a calculator will only return an angle in quadrants I or II for the cosine function, since\nthat is the range of the inverse cosine. SeeFigure 2.\nFigure2\nSo, we also need to find the measure of the angle in quadrant III. In quadrant II, the reference angle is\nThe other solution in quadrant III is\nThe solutions are and\nTRY IT #3 Solve\nSolving Trigonometric Equations in Quadratic Form"
	},
	{
		"chunk_id" : "00002653",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Trigonometric Equations in Quadratic Form\nSolving aquadratic equationmay be more complicated, but once again, we can use algebra as we would for any\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 873\nquadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation,\nor is there only one? Which trigonometric function is squared? If there is only one function represented and one of the"
	},
	{
		"chunk_id" : "00002654",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such\nas or If substitution makes the equation look like a quadratic equation, then we can use the same methods for\nsolving quadratics to solve the trigonometric equations.\nEXAMPLE10\nSolving a Trigonometric Equation in Quadratic Form\nSolve the equation exactly:\nSolution\nWe begin by using substitution and replacing cos with It is not necessary to use substitution, but it may make the"
	},
	{
		"chunk_id" : "00002655",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "problem easier to solve visually. Let We have\nThe equation cannot be factored, so we will use thequadratic formula\nReplace with and solve.\nNote that only the + sign is used. This is because we get an error when we solve on a calculator,\nsince the domain of the inverse cosine function is However, there is a second solution:\nThis terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is\nEXAMPLE11"
	},
	{
		"chunk_id" : "00002656",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE11\nSolving a Trigonometric Equation in Quadratic Form by Factoring\nSolve the equation exactly:\nSolution\nUsing grouping, this quadratic can be factored. Either make the real substitution, or imagine it, as we factor:\nNow set each factor equal to zero.\n874 9  Trigonometric Identities and Equations\nNext solve for as the range of the sine function is However, giving the solution\nAnalysis\nMake sure to check all solutions on the given domain as some factors have no solution."
	},
	{
		"chunk_id" : "00002657",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #4 Solve [Hint: Make a substitution to express the equation only in\nterms of cosine.]\nEXAMPLE12\nSolving a Trigonometric Equation Using Algebra\nSolve exactly:\nSolution\nThis problem should appear familiar as it is similar to a quadratic. Let The equation becomes\nWe begin by factoring:\nSet each factor equal to zero.\nThen, substitute back into the equation the original expression for Thus,\nThe solutions within the domain are"
	},
	{
		"chunk_id" : "00002658",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solutions within the domain are\nIf we prefer not to substitute, we can solve the equation by following the same pattern of factoring and setting each\nfactor equal to zero.\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 875\nAnalysis\nWe can see the solutions on the graph inFigure 3. On the interval the graph crosses thex-axis four times, at\nthe solutions noted. Notice that trigonometric equations that are in quadratic form can yield up to four solutions instead"
	},
	{
		"chunk_id" : "00002659",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the expected two that are found with quadratic equations. In this example, each solution (angle) corresponding to a\npositive sine value will yield two angles that would result in that value.\nFigure3\nWe can verify the solutions on the unit circle inFigure 2as well.\nEXAMPLE13\nSolving a Trigonometric Equation Quadratic in Form\nSolve the equation quadratic in form exactly:\nSolution\nWe can factor using grouping. Solution values of can be found on the unit circle.\nTRY IT #5 Solve the quadratic equation"
	},
	{
		"chunk_id" : "00002660",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Solve the quadratic equation\nSolving Trigonometric Equations Using Fundamental Identities\nWhile algebra can be used to solve a number of trigonometric equations, we can also use the fundamental identities\nbecause they make solving equations simpler. Remember that the techniques we use for solving are not the same as\n876 9  Trigonometric Identities and Equations\nthose for verifying identities. The basic rules of algebra apply here, as opposed to rewriting one side of the identity to"
	},
	{
		"chunk_id" : "00002661",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "match the other side. In the next example, we use two identities to simplify the equation.\nEXAMPLE14\nUse Identities to Solve an Equation\nUse identities to solve exactly the trigonometric equation over the interval\nSolution\nNotice that the left side of the equation is the difference formula for cosine.\nFrom the unit circle inFigure 2, we see that when\nEXAMPLE15\nSolving the Equation Using a Double-Angle Formula\nSolve the equation exactly using a double-angle formula:\nSolution"
	},
	{
		"chunk_id" : "00002662",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe have three choices of expressions to substitute for the double-angle of cosine. As it is simpler to solve for one\ntrigonometric function at a time, we will choose the double-angle identity involving only cosine:\nSo, if then and if then\nEXAMPLE16\nSolving an Equation Using an Identity\nSolve the equation exactly using an identity:\nSolution\nIf we rewrite the right side, we can write the equation in terms of cosine:\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 877"
	},
	{
		"chunk_id" : "00002663",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.5  Solving Trigonometric Equations 877\nOur solutions are\nSolving Trigonometric Equations with Multiple Angles\nSometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as\nor When confronted with these equations, recall that is ahorizontal compressionby a factor of 2 of\nthe function On an interval of we can graph two periods of as opposed to one cycle of"
	},
	{
		"chunk_id" : "00002664",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This compression of the graph leads us to believe there may be twice as manyx-intercepts or solutions to\ncompared to This information will help us solve the equation.\nEXAMPLE17\nSolving a Multiple Angle Trigonometric Equation\nSolve exactly: on\nSolution\nWe can see that this equation is the standard equation with a multiple of an angle. If we know is in\nquadrants I and IV. While will only yield solutions in quadrants I and II, we recognize that the solutions to\nthe equation will be in quadrants I and IV."
	},
	{
		"chunk_id" : "00002665",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the equation will be in quadrants I and IV.\nTherefore, the possible angles are and So, or which means that or Does\nthis make sense? Yes, because\nAre there any other possible answers? Let us return to our first step.\nIn quadrant I, so as noted. Let us revolve around the circle again:\nso\nOne more rotation yields\nso this value for is larger than so it is not a solution on\nIn quadrant IV, so as noted. Let us revolve around the circle again:\n878 9  Trigonometric Identities and Equations\nso"
	},
	{
		"chunk_id" : "00002666",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "878 9  Trigonometric Identities and Equations\nso\nOne more rotation yields\nso this value for is larger than so it is not a solution on\nOur solutions are . Note that whenever we solve a problem in the form of we must\ngo around the unit circle times.\nSolving Right Triangle Problems\nWe can now use all of the methods we have learned to solve problems that involve applying the properties of right\ntriangles and thePythagorean Theorem. We begin with the familiar Pythagorean Theorem, and model an"
	},
	{
		"chunk_id" : "00002667",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation to fit a situation.\nEXAMPLE18\nUsing the Pythagorean Theorem to Model an Equation\nUse the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.\nOne of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of\nthe Ferris wheel is 69.5 meters above the ground, and the second anchor on the ground is 23 meters from the base of"
	},
	{
		"chunk_id" : "00002668",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the Ferris wheel. Approximately how long is the cable, and what is the angle of elevation (from ground up to the center\nof the Ferris wheel)? SeeFigure 4.\nFigure4\nSolution\nUsing the information given, we can draw a right triangle. We can find the length of the cable with the Pythagorean\nTheorem.\nThe angle of elevation is formed by the second anchor on the ground and the cable reaching to the center of the\nwheel. We can use the tangent function to find its measure. Round to two decimal places."
	},
	{
		"chunk_id" : "00002669",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9.5  Solving Trigonometric Equations 879\nThe angle of elevation is approximately and the length of the cable is 73.2 meters.\nEXAMPLE19\nUsing the Pythagorean Theorem to Model an Abstract Problem\nOSHA safety regulations require that the base of a ladder be placed 1 foot from the wall for every 4 feet of ladder length.\nFind the angle that a ladder of any length forms with the ground and the height at which the ladder touches the wall.\nSolution"
	},
	{
		"chunk_id" : "00002670",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nFor any length of ladder, the base needs to be a distance from the wall equal to one fourth of the ladders length.\nEquivalently, if the base of the ladder is afeet from the wall, the length of the ladder will be 4afeet. SeeFigure 5.\nFigure5\nThe side adjacent to isaand the hypotenuse is Thus,\nThe elevation of the ladder forms an angle of with the ground. The height at which the ladder touches the wall can\nbe found using the Pythagorean Theorem:"
	},
	{
		"chunk_id" : "00002671",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "be found using the Pythagorean Theorem:\nThus, the ladder touches the wall at feet from the ground.\nMEDIA\nAccess these online resources for additional instruction and practice with solving trigonometric equations.\nSolving Trigonometric Equations I(http://openstax.org/l/solvetrigeqI)\nSolving Trigonometric Equations II(http://openstax.org/l/solvetrigeqII)\nSolving Trigonometric Equations III(http://openstax.org/l/solvetrigeqIII)\nSolving Trigonometric Equations IV(http://openstax.org/l/solvetrigeqIV)"
	},
	{
		"chunk_id" : "00002672",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Trigonometric Equations V(http://openstax.org/l/solvetrigeqV)\nSolving Trigonometric Equations VI(http://openstax.org/l/solvetrigeqVI)\n880 9  Trigonometric Identities and Equations\n9.5 SECTION EXERCISES\nVerbal\n1. Will there always be 2. When solving a 3. When solving linear trig\nsolutions to trigonometric trigonometric equation equations in terms of only\nfunction equations? If not, involving more than one trig sine or cosine, how do we"
	},
	{
		"chunk_id" : "00002673",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "describe an equation that function, do we always want know whether there will be\nwould not have a solution. to try to rewrite the solutions?\nExplain why or why not. equation so it is expressed\nin terms of one\ntrigonometric function? Why\nor why not?\nAlgebraic\nFor the following exercises, find all solutions exactly on the interval\n4. 5. 6.\n7. 8. 9.\n10. 11. 12.\nFor the following exercises, solve exactly on\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22.\nFor the following exercises, find all exact solutions on"
	},
	{
		"chunk_id" : "00002674",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. 24. 25.\n26. 27.\n28. 29. 30.\n31. 32.\nFor the following exercises, solve with the methods shown in this section exactly on the interval\n33. 34.\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 881\n35. 36.\n37. 38. 39.\n40.\nFor the following exercises, solve exactly on the interval Use the quadratic formula if the equations do not factor.\n41. 42. 43.\n44. 45. 46.\n47. 48. 49.\nFor the following exercises, find exact solutions on the interval Look for opportunities to use trigonometric"
	},
	{
		"chunk_id" : "00002675",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "identities.\n50. 51. 52.\n53. 54. 55.\n56. 57. 58.\n59. 60. 61.\n62. 63. 64.\n65.\nGraphical\nFor the following exercises, algebraically determine all solutions of the trigonometric equation exactly, then verify the\nresults by graphing the equation and finding the zeros.\n66. 67. 68.\n69. 70.\n71. 72.\nTechnology\nFor the following exercises, use a calculator to find all solutions to four decimal places.\n73. 74. 75.\n882 9  Trigonometric Identities and Equations\n76."
	},
	{
		"chunk_id" : "00002676",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "76.\nFor the following exercises, solve the equations algebraically, and then use a calculator to find the values on the interval\nRound to four decimal places.\n77. 78. 79.\n80. 81. 82.\nExtensions\nFor the following exercises, find all solutions exactly to the equations on the interval\n83. 84.\n85. 86.\n87. 88.\n89. 90.\n91. 92.\nReal-World Applications\n93. An airplane has only 94. If a loading ramp is placed 95. If a loading ramp is placed"
	},
	{
		"chunk_id" : "00002677",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "enough gas to fly to a city next to a truck, at a height next to a truck, at a height\n200 miles northeast of its of 4 feet, and the ramp is of 2 feet, and the ramp is\ncurrent location. If the pilot 15 feet long, what angle 20 feet long, what angle\nknows that the city is 25 does the ramp make with does the ramp make with\nmiles north, how many the ground? the ground?\ndegrees north of east\nshould the airplane fly?\n96. A woman is watching a 97. An astronaut is in a 98. A woman is standing 8"
	},
	{
		"chunk_id" : "00002678",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "launched rocket currently launched rocket currently meters away from a\n11 miles in altitude. If she 15 miles in altitude. If a 10-meter tall building. At\nis standing 4 miles from man is standing 2 miles what angle is she looking\nthe launch pad, at what from the launch pad, at to the top of the building?\nangle is she looking up what angle is the astronaut\nfrom horizontal? looking down at him from\nhorizontal? (Hint: this is\ncalled the angle of\ndepression.)\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002679",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "depression.)\nAccess for free at openstax.org\n9.5  Solving Trigonometric Equations 883\n99. Issa is standing 10 meters 100. A 20-foot tall building has 101. A 90-foot tall building has\naway from a 6-meter tall a shadow that is 55 feet a shadow that is 2 feet\nbuilding. Travis is at the top long. What is the angle of long. What is the angle of\nof the building looking elevation of the sun? elevation of the sun?\ndown at Issa. At what angle\nis Travis looking at Issa?"
	},
	{
		"chunk_id" : "00002680",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is Travis looking at Issa?\n102. A spotlight on the ground 103. A spotlight on the ground\n3 meters from a 2-meter 3 feet from a 5-foot tall\ntall man casts a 6 meter woman casts a 15-foot tall\nshadow on a wall 6 shadow on a wall 6 feet\nmeters from the man. At from the woman. At what\nwhat angle is the light? angle is the light?\nFor the following exercises, find a solution to the following word problem algebraically. Then use a calculator to verify\nthe result. Round the answer to the nearest tenth of a degree."
	},
	{
		"chunk_id" : "00002681",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "104. A person does a 105. A person does a 106. A 23-foot ladder is\nhandstand with their feet handstand with her feet positioned next to a\ntouching a wall and their touching a wall and her house. If the ladder slips\nhands 1.5 feet away from hands 3 feet away from at 7 feet from the house\nthe wall. If the person is 6 the wall. If the person is 5 when there is not enough\nfeet tall, what angle do feet tall, what angle do traction, what angle"
	},
	{
		"chunk_id" : "00002682",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "their feet make with the her feet make with the should the ladder make\nwall? wall? with the ground to avoid\nslipping?\n884 9  Chapter Review\nChapter Review\nKey Terms\ndouble-angle formulas identities derived from the sum formulas for sine, cosine, and tangent in which the angles are\nequal\neven-odd identities set of equations involving trigonometric functions such that if the identity is\nodd, and if the identity is even"
	},
	{
		"chunk_id" : "00002683",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "odd, and if the identity is even\nhalf-angle formulas identities derived from the reduction formulas and used to determine half-angle values of\ntrigonometric functions\nproduct-to-sum formula a trigonometric identity that allows the writing of a product of trigonometric functions as a\nsum or difference of trigonometric functions\nPythagorean identities set of equations involving trigonometric functions based on the right triangle properties"
	},
	{
		"chunk_id" : "00002684",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quotient identities pair of identities based on the fact that tangent is the ratio of sine and cosine, and cotangent is the\nratio of cosine and sine\nreciprocal identities set of equations involving the reciprocals of basic trigonometric definitions\nreduction formulas identities derived from the double-angle formulas and used to reduce the power of a\ntrigonometric function\nsum-to-product formula a trigonometric identity that allows, by using substitution, the writing of a sum of"
	},
	{
		"chunk_id" : "00002685",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "trigonometric functions as a product of trigonometric functions\nKey Equations\nPythagorean identities\nEven-odd identities\nReciprocal identities\nQuotient identities\nSum Formula for Cosine\nDifference Formula for Cosine\nSum Formula for Sine\nDifference Formula for Sine\nAccess for free at openstax.org\n9  Chapter Review 885\nSum Formula for Tangent\nDifference Formula for Tangent\nCofunction identities\nDouble-angle formulas\nReduction formulas\nHalf-angle formulas\nProduct-to-sum Formulas\nSum-to-product Formulas"
	},
	{
		"chunk_id" : "00002686",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Product-to-sum Formulas\nSum-to-product Formulas\n886 9  Chapter Review\nKey Concepts\n9.1Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify\nTrigonometric Expressions\n There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions\ncan be rewritten to simplify a problem.\n Graphing both sides of an identity will verify it. SeeExample 1."
	},
	{
		"chunk_id" : "00002687",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Simplifying one side of the equation to equal the other side is another method for verifying an identity. SeeExample\n2andExample 3.\n The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more\ncomplex side of the equation. SeeExample 4.\n We can create an identity and then verify it. SeeExample 5.\n Verifying an identity may involve algebra with the fundamental identities. SeeExample 6andExample 7."
	},
	{
		"chunk_id" : "00002688",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout\nthis text, as they consist of the fundamental rules of mathematics. SeeExample 8,Example 9, andExample 10.\n9.2Sum and Difference Identities\n The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the\nangles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the"
	},
	{
		"chunk_id" : "00002689",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.\n The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle. See\nExample 1andExample 2.\n The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first\nangle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second"
	},
	{
		"chunk_id" : "00002690",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the\nsine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine\nof the second angle. SeeExample 3.\n The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See\nExample 4."
	},
	{
		"chunk_id" : "00002691",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 4.\n The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the\nangles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that\nthe tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the\nproduct of the tangents of the angles. SeeExample 5."
	},
	{
		"chunk_id" : "00002692",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and\ndifferences of angles. SeeExample 6.\n The cofunction identities apply to complementary angles and pairs of reciprocal functions. SeeExample 7.\n Sum and difference formulas are useful in verifying identities. SeeExample 8andExample 9.\n Application problems are often easier to solve by using sum and difference formulas. SeeExample 10andExample\n11.\n9.3Double-Angle, Half-Angle, and Reduction Formulas"
	},
	{
		"chunk_id" : "00002693",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine,\ncosine, and tangent. SeeExample 1,Example 2,Example 3, andExample 4.\n Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term.\nSeeExample 5andExample 6.\n Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original\nangle is known or not. SeeExample 7,Example 8, andExample 9."
	},
	{
		"chunk_id" : "00002694",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9.4Sum-to-Product and Product-to-Sum Formulas\n From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product\nformulas for sine and cosine.\n We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and\ncosine as sums or differences of sines and cosines. SeeExample 1,Example 2, andExample 3.\n We can also derive the sum-to-product identities from the product-to-sum identities using substitution."
	},
	{
		"chunk_id" : "00002695",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " We can use the sum-to-product formulas to rewrite sum or difference of sines, cosines, or products sine and cosine\nas products of sines and cosines. SeeExample 4.\n Trigonometric expressions are often simpler to evaluate using the formulas. SeeExample 5.\n The identities can be verified using other formulas or by converting the expressions to sines and cosines. To verify\nAccess for free at openstax.org\n9  Exercises 887"
	},
	{
		"chunk_id" : "00002696",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9  Exercises 887\nan identity, we choose the more complicated side of the equals sign and rewrite it until it is transformed into the\nother side. SeeExample 6andExample 7.\n9.5Solving Trigonometric Equations\n When solving linear trigonometric equations, we can use algebraic techniques just as we do solving algebraic\nequations. Look for patterns, like the difference of squares, quadratic form, or an expression that lends itself well to"
	},
	{
		"chunk_id" : "00002697",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitution. SeeExample 1,Example 2, andExample 3.\n Equations involving a single trigonometric function can be solved or verified using the unit circle. SeeExample 4,\nExample 5, andExample 6, andExample 7.\n We can also solve trigonometric equations using a graphing calculator. SeeExample 8andExample 9.\n Many equations appear quadratic in form. We can use substitution to make the equation appear simpler, and then"
	},
	{
		"chunk_id" : "00002698",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "use the same techniques we use solving an algebraic quadratic: factoring, the quadratic formula, etc. SeeExample\n10,Example 11,Example 12, andExample 13.\n We can also use the identities to solve trigonometric equation. SeeExample 14,Example 15, andExample 16.\n We can use substitution to solve a multiple-angle trigonometric equation, which is a compression of a standard\ntrigonometric function. We will need to take the compression into account and verify that we have found all"
	},
	{
		"chunk_id" : "00002699",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions on the given interval. SeeExample 17.\n Real-world scenarios can be modeled and solved using the Pythagorean Theorem and trigonometric functions. See\nExample 18.\nExercises\nReview Exercises\nSolving Trigonometric Equations with Identities\nFor the following exercises, find all solutions exactly that exist on the interval\n1. 2. 3.\n4. 5. 6.\nFor the following exercises, use basic identities to simplify the expression.\n7. 8.\nFor the following exercises, determine if the given identities are equivalent."
	},
	{
		"chunk_id" : "00002700",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9. 10.\nSum and Difference Identities\nFor the following exercises, find the exact value.\n11. 12.\n13. 14.\nFor the following exercises, prove the identity.\n15. 16.\n888 9  Exercises\nFor the following exercise, simplify the expression.\n17.\nFor the following exercises, find the exact value.\n18. 19.\nDouble-Angle, Half-Angle, and Reduction Formulas\nFor the following exercises, find the exact value.\n20. Find and given 21. Find and given\nand is in the interval and is in the interval\n22. 23."
	},
	{
		"chunk_id" : "00002701",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "22. 23.\nFor the following exercises, useFigure 1to find the desired quantities.\nFigure1\n24.\n25.\nFor the following exercises, prove the identity.\n26. 27.\nFor the following exercises, rewrite the expression with no powers.\n28. 29.\nSum-to-Product and Product-to-Sum Formulas\nFor the following exercises, evaluate the product for the given expression using a sum or difference of two functions.\nWrite the exact answer.\n30. 31. 32."
	},
	{
		"chunk_id" : "00002702",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write the exact answer.\n30. 31. 32.\nFor the following exercises, evaluate the sum by using a product formula. Write the exact answer.\n33. 34.\nAccess for free at openstax.org\n9  Exercises 889\nFor the following exercises, change the functions from a product to a sum or a sum to a product.\n35. 36. 37.\n38.\nSolving Trigonometric Equations\nFor the following exercises, find all exact solutions on the interval\n39. 40.\nFor the following exercises, find all exact solutions on the interval\n41. 42. 43.\n44. 45."
	},
	{
		"chunk_id" : "00002703",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "41. 42. 43.\n44. 45.\nFor the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the\nsolutions on the interval Round to four decimal places.\n46. 47.\nFor the following exercises, graph each side of the equation to find the approximate solutions on the interval\n48. 49.\nPractice Test\nFor the following exercises, simplify the given expression.\n1. 2.\n3. 4.\nFor the following exercises, find the exact value.\n5. 6. 7.\n8. 9. 10."
	},
	{
		"chunk_id" : "00002704",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. 6. 7.\n8. 9. 10.\nFor the following exercises, simplify each expression. Do not evaluate.\n11. 12.\nFor the following exercises, find all exact solutions to the equation on\n13. 14.\n890 9  Exercises\n15. 16.\n17. Rewrite the expression as a product instead of a\nsum:\nFor the following exercise, rewrite the product as a sum or difference.\n18.\nFor the following exercise, rewrite the sum or difference as a product.\n19. 20. Find all solutions of 21. Find the solutions of\non\nthe interval\nalgebraically; then graph"
	},
	{
		"chunk_id" : "00002705",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on\nthe interval\nalgebraically; then graph\nboth sides of the equation\nto determine the answer.\nFor the following exercises, find all solutions exactly on the interval\n22. 23. 24. Find and\ngiven\nand is on the interval\n25. Find and 26. Rewrite the expression\ngiven with no powers\ngreater than 1.\nand is in quadrant IV.\nFor the following exercises, prove the identity.\n27. 28.\n29. 30. Plot the points and find a function of the form\nthat fits the given data.\nAccess for free at openstax.org\n9  Exercises 891"
	},
	{
		"chunk_id" : "00002706",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n9  Exercises 891\n31. The displacement in 32. A woman is standing 300 33. Two frequencies of sound are\ncentimeters of a mass feet away from a 2000-foot played on an instrument\nsuspended by a spring is building. If she looks to the governed by the equation\nmodeled by the function top of the building, at what\nwhere angle above horizontal is What are the period and\nis measured in seconds. she looking? A worker frequency of the fast and"
	},
	{
		"chunk_id" : "00002707",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the amplitude, period, looks down at her from the slow oscillations? What is the\nand frequency of this 15thfloor (1500 feet above amplitude?\ndisplacement. her). At what angle is he\nlooking down at her?\nRound to the nearest tenth\nof a degree.\n34. The average monthly 35. A spring attached to a 36. Water levels near a glacier\nsnowfall in a small village ceiling is pulled down 20 currently average 9 feet,\nin the Himalayas is 6 cm. After 3 seconds, varying seasonally by 2"
	},
	{
		"chunk_id" : "00002708",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inches, with the low of 1 wherein it completes 6 full inches above and below\ninch occurring in July. periods, the amplitude is the average and reaching\nConstruct a function that only 15 cm. Find the their highest point in\nmodels this behavior. function modeling the January. Due to global\nDuring what period is there position of the spring warming, the glacier has\nmore than 10 inches of seconds after being begun melting faster than\nsnowfall? released. At what time will normal. Every year, the"
	},
	{
		"chunk_id" : "00002709",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the spring come to rest? In water levels rise by a\nthis case, use 1 cm steady 3 inches. Find a\namplitude as rest. function modeling the\ndepth of the water\nmonths from now. If the\ndocks are 2 feet above\ncurrent water levels, at\nwhat point will the water\nfirst rise above the docks?\n892 9  Exercises\nAccess for free at openstax.org\n10  Introduction 893\n10 FURTHER APPLICATIONS OF TRIGONOMETRY\nGeneral Sherman, the worlds largest living tree. (credit: Mike Baird, Flickr)\nChapter Outline"
	},
	{
		"chunk_id" : "00002710",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Chapter Outline\n10.1Non-right Triangles: Law of Sines\n10.2Non-right Triangles: Law of Cosines\n10.3Polar Coordinates\n10.4Polar Coordinates: Graphs\n10.5Polar Form of Complex Numbers\n10.6Parametric Equations\n10.7Parametric Equations: Graphs\n10.8Vectors\nIntroduction to Further Applications of Trigonometry\nThe worlds largest tree by volume, named General Sherman, stands 274.9 feet tall and resides in Northern California.1"
	},
	{
		"chunk_id" : "00002711",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Just how do scientists know its true height? A common way to measure the height involves determining the angle of\nelevation, which is formed by the tree and the ground at a point some distance away from the base of the tree. This\nmethod is much more practical than climbing the tree and dropping a very long tape measure.\nIn this chapter, we will explore applications of trigonometry that will enable us to solve many different kinds of"
	},
	{
		"chunk_id" : "00002712",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "problems, including finding the height of a tree. We extend topics we introduced inTrigonometric Functions\n(http://openstax.org/books/precalculus-2e/pages/5-introduction-to-trigonometric-functions)and investigate\napplications more deeply and meaningfully.\n10.1 Non-right Triangles: Law of Sines\nLearning Objectives\nIn this section, you will:\nUse the Law of Sines to solve oblique triangles.\nFind the area of an oblique triangle using the sine function.\nSolve applied problems using the Law of Sines."
	},
	{
		"chunk_id" : "00002713",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve applied problems using the Law of Sines.\nTo ensure the safety of over 5,000 U.S. aircraft flying simultaneously during peak times, air traffic controllers monitor and\ncommunicate with them after receiving data from the robust radar beacon system. Suppose two radar stations located\n20 miles apart each detect an aircraft between them. The angle of elevation measured by the first station is 35 degrees,"
	},
	{
		"chunk_id" : "00002714",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 Source: National Park Service. \"The General Sherman Tree.\"\" http://www.nps.gov/seki/naturescience/sherman.htm. Accessed April 25"
	},
	{
		"chunk_id" : "00002715",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "cannot use what we know about right triangles. In this section, we will find out how to solve problems involvingnon-\nright triangles.\nFigure1\nUsing the Law of Sines to Solve Oblique Triangles\nIn any triangle, we can draw analtitude, a perpendicular line from one vertex to the opposite side, forming two right\ntriangles. It would be preferable, however, to have methods that we can apply directly to non-right triangles without first\nhaving to create right triangles."
	},
	{
		"chunk_id" : "00002716",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "having to create right triangles.\nAny triangle that is not a right triangle is anoblique triangle. Solving an oblique triangle means finding the\nmeasurements of all three angles and all three sides. To do so, we need to start with at least three of these values,\nincluding at least one of the sides. We will investigate three possible oblique triangle problem situations:\n1. ASA (angle-side-angle)We know the measurements of two angles and the included side. SeeFigure 2.\nFigure2"
	},
	{
		"chunk_id" : "00002717",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\n2. AAS (angle-angle-side)We know the measurements of two angles and a side that is not between the known\nangles. SeeFigure 3.\nFigure3\n3. SSA (side-side-angle)We know the measurements of two sides and an angle that is not between the known sides.\nSeeFigure 4.\nFigure4\nKnowing how to approach each of these situations enables us to solve oblique triangles without having to drop a\nperpendicular to form two right triangles. Instead, we can use the fact that the ratio of the measurement of one of the"
	},
	{
		"chunk_id" : "00002718",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angles to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. Lets see\nhow this statement is derived by considering the triangle shown inFigure 5.\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 895\nFigure5\nUsing the right triangle relationships, we know that and Solving both equations for gives two\ndifferent expressions for\nWe then set the expressions equal to each other.\nSimilarly, we can compare the other ratios."
	},
	{
		"chunk_id" : "00002719",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, we can compare the other ratios.\nCollectively, these relationships are called theLaw of Sines.\nNote the standard way of labeling triangles: angle (alpha) is opposite side angle (beta) is opposite side and\nangle (gamma) is opposite side SeeFigure 6.\nWhile calculating angles and sides, be sure to carry the exact values through to the final answer. Generally, final answers\nare rounded to the nearest tenth, unless otherwise specified.\nFigure6\nLaw of Sines"
	},
	{
		"chunk_id" : "00002720",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nLaw of Sines\nGiven a triangle with angles and opposite sides labeled as inFigure 6, the ratio of the measurement of an angle to the\nlength of its opposite side will be equal to the other two ratios of angle measure to opposite side. All proportions will\nbe equal. TheLaw of Sinesis based on proportions and is presented symbolically two ways.\nTo solve an oblique triangle, use any pair of applicable ratios.\n896 10  Further Applications of Trigonometry\nEXAMPLE1"
	},
	{
		"chunk_id" : "00002721",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nSolving for Two Unknown Sides and Angle of an AAS Triangle\nSolve the triangle shown inFigure 7to the nearest tenth.\nFigure7\nSolution\nThe three angles must add up to 180 degrees. From this, we can determine that\nTo find an unknown side, we need to know the corresponding angle and a known ratio. We know that angle and\nits corresponding side We can use the following proportion from the Law of Sines to find the length of\nSimilarly, to solve for we set up another proportion."
	},
	{
		"chunk_id" : "00002722",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Therefore, the complete set of angles and sides is\nTRY IT #1 Solve the triangle shown inFigure 8to the nearest tenth.\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 897\nFigure8\nUsing The Law of Sines to Solve SSA Triangles\nWe can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. In some\ncases, more than one triangle may satisfy the given criteria, which we describe as anambiguous case. Triangles"
	},
	{
		"chunk_id" : "00002723",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "classified as SSA, those in which we know the lengths of two sides and the measurement of the angle opposite one of\nthe given sides, may result in one or two solutions, or even no solution.\nPossible Outcomes for SSA Triangles\nOblique triangles in the category SSA may have four different outcomes.Figure 9illustrates the solutions with the\nknown sides and and known angle\nFigure9\nEXAMPLE2\nSolving an Oblique SSA Triangle"
	},
	{
		"chunk_id" : "00002724",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nEXAMPLE2\nSolving an Oblique SSA Triangle\nSolve the triangle inFigure 10for the missing side and find the missing angle measures to the nearest tenth.\nFigure10\nSolution\nUse the Law of Sines to find angle and angle and then side Solving for we have the proportion\n898 10  Further Applications of Trigonometry\nHowever, in the diagram, angle appears to be an obtuse angle and may be greater than 90. How did we get an acute"
	},
	{
		"chunk_id" : "00002725",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle, and how do we find the measurement of Lets investigate further. Dropping a perpendicular from and\nviewing the triangle from a right angle perspective, we haveFigure 11. It appears that there may be a second triangle\nthat will fit the given criteria.\nFigure11\nThe angle supplementary to is approximately equal to 49.9, which means that (Remember\nthat the sine function is positive in both the first and second quadrants.) Solving for we have"
	},
	{
		"chunk_id" : "00002726",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can then use these measurements to solve the other triangle. Since is supplementary to the sum of and we\nhave\nNow we need to find and\nWe have\nFinally,\nTo summarize, there are two triangles with an angle of 35, an adjacent side of 8, and an opposite side of 6, as shown in\nFigure 12.\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 899\nFigure12\nHowever, we were looking for the values for the triangle with an obtuse angle We can see them in the first triangle (a)\ninFigure 12."
	},
	{
		"chunk_id" : "00002727",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inFigure 12.\nTRY IT #2 Given and find the missing side and angles. If there is more than one\npossible solution, show both.\nEXAMPLE3\nSolving for the Unknown Sides and Angles of a SSA Triangle\nIn the triangle shown inFigure 13, solve for the unknown side and angles. Round your answers to the nearest tenth.\nFigure13\nSolution\nIn choosing the pair of ratios from the Law of Sines to use, look at the information given. In this case, we know the angle"
	},
	{
		"chunk_id" : "00002728",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and its corresponding side and we know side We will use this proportion to solve for\nTo find apply the inverse sine function. The inverse sine will produce a single result, but keep in mind that there may\nbe two values for It is important to verify the result, as there may be two viable solutions, only one solution (the usual\ncase), or no solutions.\nIn this case, if we subtract from 180, we find that there may be a second possible solution. Thus,"
	},
	{
		"chunk_id" : "00002729",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To check the solution, subtract both angles, 131.7 and 85, from 180. This gives\nwhich is impossible, and so\n900 10  Further Applications of Trigonometry\nTo find the remaining missing values, we calculate Now, only side is needed. Use the\nLaw of Sines to solve for by one of the proportions.\nThe complete set of solutions for the given triangle is\nTRY IT #3 Given find the missing side and angles. If there is more than one\npossible solution, show both. Round your answers to the nearest tenth.\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002730",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nFinding the Triangles That Meet the Given Criteria\nFind all possible triangles if one side has length 4 opposite an angle of 50, and a second side has length 10.\nSolution\nUsing the given information, we can solve for the angle opposite the side of length 10. SeeFigure 14.\nFigure14\nWe can stop here without finding the value of Because the range of the sine function is it is impossible for the\nsine value to be 1.915. In fact, inputting in a graphing calculator generates an ERROR DOMAIN. Therefore,"
	},
	{
		"chunk_id" : "00002731",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "no triangles can be drawn with the provided dimensions.\nTRY IT #4 Determine the number of triangles possible given\nFinding the Area of an Oblique Triangle Using the Sine Function\nNow that we can solve a triangle for missing values, we can use some of those values and the sine function to find the\narea of an oblique triangle. Recall that the area formula for a triangle is given as where is base and is"
	},
	{
		"chunk_id" : "00002732",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "height. For oblique triangles, we must find before we can use the area formula. Observing the two triangles inFigure\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 901\n15, one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric\nproperty to write an equation for area in oblique triangles. In the acute triangle, we have\nor However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base to"
	},
	{
		"chunk_id" : "00002733",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form a right triangle. The angle used in calculation is or\nFigure15\nThus,\nSimilarly,\nArea of an Oblique Triangle\nThe formula for the area of an oblique triangle is given by\nThis is equivalent to one-half of the product of two sides and the sine of their included angle.\nEXAMPLE5\nFinding the Area of an Oblique Triangle\nFind the area of a triangle with sides and angle Round the area to the nearest integer.\nSolution\nUsing the formula, we have"
	},
	{
		"chunk_id" : "00002734",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUsing the formula, we have\nTRY IT #5 Find the area of the triangle given Round the area to the nearest\ntenth.\nSolving Applied Problems Using the Law of Sines\nThe more we study trigonometric applications, the more we discover that the applications are countless. Some are flat,\ndiagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and\nmotion.\n902 10  Further Applications of Trigonometry\nEXAMPLE6\nFinding an Altitude"
	},
	{
		"chunk_id" : "00002735",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nFinding an Altitude\nFind the altitude of the aircraft in the problem introduced at the beginning of this section, shown inFigure 16. Round the\naltitude to the nearest tenth of a mile.\nFigure16\nSolution\nTo find the elevation of the aircraft, we first find the distance from one station to the aircraft, such as the side and then\nuse right triangle relationships to find the height of the aircraft,"
	},
	{
		"chunk_id" : "00002736",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Because the angles in the triangle add up to 180 degrees, the unknown angle must be 1801535=130. This angle is\nopposite the side of length 20, allowing us to set up a Law of Sines relationship.\nThe distance from one station to the aircraft is about 14.98 miles.\nNow that we know we can use right triangle relationships to solve for\nThe aircraft is at an altitude of approximately 3.9 miles.\nTRY IT #6 The diagram shown inFigure 17represents the height of a blimp flying over a football stadium."
	},
	{
		"chunk_id" : "00002737",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the height of the blimp if the angle of elevation at the southern end zone, point A, is 70, the\nangle of elevation from the northern end zone, point is 62, and the distance between the\nviewing points of the two end zones is 145 yards.\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 903\nFigure17\nMEDIA\nAccess these online resources for additional instruction and practice with trigonometric applications.\nLaw of Sines: The Basics(http://openstax.org/l/sinesbasic)"
	},
	{
		"chunk_id" : "00002738",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Law of Sines: The Ambiguous Case(http://openstax.org/l/sinesambiguous)\n10.1 SECTION EXERCISES\nVerbal\n1. Describe the altitude of a 2. Compare right triangles and 3. When can you use the Law\ntriangle. oblique triangles. of Sines to find a missing\nangle?\n4. In the Law of Sines, what is 5. What type of triangle results\nthe relationship between in an ambiguous case?\nthe angle in the numerator\nand the side in the\ndenominator?\nAlgebraic"
	},
	{
		"chunk_id" : "00002739",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and the side in the\ndenominator?\nAlgebraic\nFor the following exercises, assume is opposite side is opposite side and is opposite side Solve each triangle,\nif possible. Round each answer to the nearest tenth.\n6. 7. 8.\n9. 10.\n904 10  Further Applications of Trigonometry\nFor the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each\nanswer to the nearest hundredth. Assume that angle is opposite side angle is opposite side and angle is\nopposite side"
	},
	{
		"chunk_id" : "00002740",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opposite side\n11. Find side when 12. Find side when 13. Find side when\nFor the following exercises, assume is opposite side is opposite side and is opposite side Determine whether\nthere is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the\nnearest tenth.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23.\nFor the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or"
	},
	{
		"chunk_id" : "00002741",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "triangles in the ambiguous case. Round each answer to the nearest tenth.\n24. Find angle when 25. Find angle when 26. Find angle when\nFor the following exercises, find the area of the triangle with the given measurements. Round each answer to the\nnearest tenth.\n27. 28. 29.\n30.\nGraphical\nFor the following exercises, find the length of side Round to the nearest tenth.\n31. 32. 33.\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 905\n34. 35. 36."
	},
	{
		"chunk_id" : "00002742",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. 35. 36.\nFor the following exercises, find the measure of angle if possible. Round to the nearest tenth.\n37. 38. 39.\n40. 41. Notice that is an obtuse 42.\nangle.\nFor the following exercise, solve the triangle. Round each answer to the nearest tenth.\n43. 44. For the following exercises, find 45.\nthe area of each triangle. Round\neach answer to the nearest tenth.\n906 10  Further Applications of Trigonometry\n46. 47. 48.\n49.\nExtensions"
	},
	{
		"chunk_id" : "00002743",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47. 48.\n49.\nExtensions\n50. Find the radius of the circle 51. Find the diameter of the 52. Find inFigure 20.\ninFigure 18. Round to the circle inFigure 19. Round Round to the nearest tenth.\nnearest tenth. to the nearest tenth.\nFigure20\nFigure18 Figure19\n53. Find inFigure 21. Round to the54. Solve both triangles inFigure 22. 55. Find in the\nnearest tenth. Round each answer to the nearest parallelogram shown in\ntenth. Figure 23.\nFigure21\nFigure23\nFigure22\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002744",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure23\nFigure22\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 907\n56. Solve the triangle inFigure 24. (Hint5:7. Solve the triangle inFigure 58. InFigure 26, is not a\nDraw a perpendicular from to 25. (Hint: Draw a parallelogram. is obtuse. Solve\nRound each answer to the perpendicular from to both triangles. Round each answer\nnearest tenth. Round each answer to the nearest tenth.\nto the nearest tenth.\nFigure24\nFigure25\nFigure26\nReal-World Applications"
	},
	{
		"chunk_id" : "00002745",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure25\nFigure26\nReal-World Applications\n59. A pole leans away from the sun at an angle of 60. To determine how far a boat is from shore, two\nto the vertical, as shown inFigure 27. When the radar stations 500 feet apart find the angles out to\nelevation of the sun is the pole casts a the boat, as shown inFigure 28. Determine the\nshadow 42 feet long on the level ground. How distance of the boat from station and the\nlong is the pole? Round the answer to the nearest distance of the boat from shore. Round your"
	},
	{
		"chunk_id" : "00002746",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tenth. answers to the nearest whole foot.\nFigure27\nFigure28\n908 10  Further Applications of Trigonometry\n61. Figure 29shows a satellite orbiting Earth. The 62. A communications tower is located at the top of a\nsatellite passes directly over two tracking stations steep hill, as shown inFigure 30. The angle of\nand which are 69miles apart. When the inclination of the hill is A guy wire is to be\nsatellite is on one side of the two stations, the attached to the top of the tower and to the"
	},
	{
		"chunk_id" : "00002747",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angles of elevation at and are measured to be ground, 165 meters downhill from the base of the\nand respectively. How far is the tower. The angle formed by the guy wire and the\nsatellite from station and how high is the hill is Find the length of the cable required for\nsatellite above the ground? Round answers to the the guy wire to the nearest whole meter.\nnearest whole mile.\nFigure30\nFigure29\n63. The roof of a house is at a angle. An 8-foot solar\npanel is to be mounted on the roof and should be"
	},
	{
		"chunk_id" : "00002748",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "panel is to be mounted on the roof and should be\nangled relative to the horizontal for optimal\nresults. (SeeFigure 31). How long does the vertical\nsupport holding up the back of the panel need to be?\nRound to the nearest tenth.\nFigure31\nAccess for free at openstax.org\n10.1  Non-right Triangles: Law of Sines 909\n64. Similar to an angle of elevation, anangle of 65. A pilot is flying over a straight highway. He\ndepressionis the acute angle formed by a determines the angles of depression to two"
	},
	{
		"chunk_id" : "00002749",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal line and an observers line of sight to mileposts, 4.3 km apart, to be 32 and 56, as\nan object below the horizontal. A pilot is flying shown inFigure 33. Find the distance of the plane\nover a straight highway. He determines the angles from point to the nearest tenth of a kilometer.\nof depression to two mileposts, 6.6 km apart, to\nbe and as shown inFigure 32. Find the\ndistance of the plane from point to the nearest\ntenth of a kilometer.\nFigure33\nFigure32"
	},
	{
		"chunk_id" : "00002750",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tenth of a kilometer.\nFigure33\nFigure32\n66. In order to estimate the height of a building, two 67. In order to estimate the\nstudents stand at a certain distance from the height of a building, two\nbuilding at street level. From this point, they find students stand at a certain\nthe angle of elevation from the street to the top of distance from the building\nthe building to be 39. They then move 300 feet at street level. From this\ncloser to the building and find the angle of point, they find the angle"
	},
	{
		"chunk_id" : "00002751",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "elevation to be 50. Assuming that the street is of elevation from the street\nlevel, estimate the height of the building to the to the top of the building\nnearest foot. to be 35. They then move\n250 feet closer to the\nbuilding and find the angle\nof elevation to be 53.\nAssuming that the street is\nlevel, estimate the height\nof the building to the\nnearest foot.\n68. Points and are on 69. A man and a woman 70. Two search teams spot a\nopposite sides of a lake. standing miles apart stranded climber on a"
	},
	{
		"chunk_id" : "00002752",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Point is 97 meters from spot a hot air balloon at mountain. The first search\nThe measure of angle the same time. If the angle team is 0.5 miles from the\nis determined to be of elevation from the man second search team, and\n101, and the measure of to the balloon is 27, and both teams are at an\nangle is determined the angle of elevation from altitude of 1 mile. The\nto be 53. What is the the woman to the balloon angle of elevation from the"
	},
	{
		"chunk_id" : "00002753",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distance from to is 41, find the altitude of first search team to the\nrounded to the nearest the balloon to the nearest stranded climber is 15.\nwhole meter? foot. The angle of elevation from\nthe second search team to\nthe climber is 22. What is\nthe altitude of the climber?\nRound to the nearest tenth\nof a mile.\n910 10  Further Applications of Trigonometry\n71. A street light is mounted 72. Three cities, and\non a pole. A 6-foot-tall man are located so that city is"
	},
	{
		"chunk_id" : "00002754",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is standing on the street a due east of city If city\nshort distance from the is located 35 west of north\npole, casting a shadow. The from city and is 100 miles\nangle of elevation from the from city and 70 miles\ntip of the mans shadow to from city how far is city\nthe top of his head of 28. from city Round the\nA 6-foot-tall woman is distance to the nearest\nstanding on the same tenth of a mile.\nstreet on the opposite side\nof the pole from the man.\nThe angle of elevation from\nthe tip of her shadow to"
	},
	{
		"chunk_id" : "00002755",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the tip of her shadow to\nthe top of her head is 28.\nIf the man and woman are\n20 feet apart, how far is the\nstreet light from the tip of\nthe shadow of each\nperson? Round the\ndistance to the nearest\ntenth of a foot.\n73. Two streets meet at an 80 angle. At the corner, a\npark is being built in the shape of a triangle. Find\nthe area of the park if, along one road, the park\nmeasures 180 feet, and along the other road, the\npark measures 215 feet."
	},
	{
		"chunk_id" : "00002756",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "park measures 215 feet.\n74. Brians house is on a corner lot. Find the area of the 75. The Bermuda triangle is a\nfront yard if the edges measure 40 and 56 feet, as region of the Atlantic\nshown inFigure 34. Ocean that connects\nBermuda, Florida, and\nPuerto Rico. Find the area\nof the Bermuda triangle if\nthe distance from Florida\nto Bermuda is 1030 miles,\nthe distance from Puerto\nFigure34\nRico to Bermuda is 980\nmiles, and the angle\ncreated by the two\ndistances is 62.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002757",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distances is 62.\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 911\n76. A yield sign measures 30 77. Naomi bought a dining table whose\ninches on all three sides. top is in the shape of a triangle. Find\nWhat is the area of the the area of the table top if two of\nsign? the sides measure 4 feet and 4.5\nfeet, and the smaller angles\nmeasure 32 and 42, as shown in\nFigure 35.\nFigure35\n10.2 Non-right Triangles: Law of Cosines\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00002758",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nUse the Law of Cosines to solve oblique triangles.\nSolve applied problems using the Law of Cosines.\nUse Herons formula to find the area of a triangle.\nSuppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown inFigure 1. How far\nfrom port is the boat?\nFigure1\nUnfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with"
	},
	{
		"chunk_id" : "00002759",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "triangles where the known angle is between two known sides, aSAS (side-angle-side) triangle, or when all three sides\nare known, but no angles are known, aSSS (side-side-side) triangle. In this section, we will investigate another tool for\nsolving oblique triangles described by these last two cases.\nUsing the Law of Cosines to Solve Oblique Triangles\nThe tool we need to solve the problem of the boats distance from the port is theLaw of Cosines, which defines the"
	},
	{
		"chunk_id" : "00002760",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of\nCosines. At first glance, the formulas may appear complicated because they include many variables. However, once the\n912 10  Further Applications of Trigonometry\npattern is understood, the Law of Cosines is easier to work with than most formulas at this mathematical level.\nUnderstanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the"
	},
	{
		"chunk_id" : "00002761",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Generalized Pythagorean Theorem, which is an extension of thePythagorean Theoremto non-right triangles. Here is\nhow it works: An arbitrary non-right triangle is placed in the coordinate plane with vertex at the origin, side\ndrawn along thex-axis, and vertex located at some point in the plane, as illustrated inFigure 2. Generally,\ntriangles exist anywhere in the plane, but for this explanation we will place the triangle as noted.\nFigure2"
	},
	{
		"chunk_id" : "00002762",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nWe can drop a perpendicular from to thex-axis (this is the altitude or height). Recalling the basictrigonometric\nidentities, we know that\nIn terms of and The point located at has coordinates Using the\nside as one leg of a right triangle and as the second leg, we can find the length of hypotenuse using the\nPythagorean Theorem. Thus,\nThe formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar\nfashion."
	},
	{
		"chunk_id" : "00002763",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fashion.\nKeep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, try\nto draw a diagram of the situation. As more information emerges, the diagram may have to be altered. Make those\nalterations to the diagram and, in the end, the problem will be easier to solve.\nLaw of Cosines\nTheLaw of Cosinesstates that the square of any side of a triangle is equal to the sum of the squares of the other two"
	},
	{
		"chunk_id" : "00002764",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sides minus twice the product of the other two sides and the cosine of the included angle. For triangles labeled as in\nFigure 3, with angles and and opposite corresponding sides and respectively, the Law of Cosines is\ngiven as three equations.\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 913\nFigure3\nTo solve for a missing side measurement, the corresponding opposite angle measure is needed."
	},
	{
		"chunk_id" : "00002765",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "When solving for an angle, the corresponding opposite side measure is needed. We can use another version of the\nLaw of Cosines to solve for an angle.\n...\nHOW TO\nGiven two sides and the angle between them (SAS), find the measures of the remaining side and angles of a\ntriangle.\n1. Sketch the triangle. Identify the measures of the known sides and angles. Use variables to represent the\nmeasures of the unknown sides and angles.\n2. Apply the Law of Cosines to find the length of the unknown side or angle."
	},
	{
		"chunk_id" : "00002766",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Apply theLaw of Sinesor Cosines to find the measure of a second angle.\n4. Compute the measure of the remaining angle.\nEXAMPLE1\nFinding the Unknown Side and Angles of a SAS Triangle\nFind the unknown side and angles of the triangle inFigure 4.\nFigure4\nSolution\nFirst, make note of what is given: two sides and the angle between them. This arrangement is classified as SAS and\nsupplies the data needed to apply the Law of Cosines.\n914 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002767",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "914 10  Further Applications of Trigonometry\nEach one of the three laws of cosines begins with the square of an unknown side opposite a known angle. For this\nexample, the first side to solve for is side as we know the measurement of the opposite angle\nBecause we are solving for a length, we use only the positive square root. Now that we know the length we can use\nthe Law of Sines to fill in the remaining angles of the triangle. Solving for angle we have"
	},
	{
		"chunk_id" : "00002768",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The other possibility for would be In the original diagram, is adjacent to the longest side,\nso is an acute angle and, therefore, does not make sense. Notice that if we choose to apply theLaw of Cosines,\nwe arrive at a unique answer. We do not have to consider the other possibilities, as cosine is unique for angles between\nand Proceeding with we can then find the third angle of the triangle.\nThe complete set of angles and sides is\nTRY IT #1 Find the missing side and angles of the given triangle:\nEXAMPLE2"
	},
	{
		"chunk_id" : "00002769",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nSolving for an Angle of a SSS Triangle\nFind the angle for the given triangle if side side and side\nSolution\nFor this example, we have no angles. We can solve for any angle using the Law of Cosines. To solve for angle we have\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 915\nSeeFigure 5.\nFigure5\nAnalysis\nBecause the inverse cosine can return any angle between 0 and 180 degrees, there will not be any ambiguous cases\nusing this method."
	},
	{
		"chunk_id" : "00002770",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "using this method.\nTRY IT #2 Given and find the missing angles.\nSolving Applied Problems Using the Law of Cosines\nJust as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is\napplicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation,\nsurveying, astronomy, and geometry, just to name a few.\nEXAMPLE3\nUsing the Law of Cosines to Solve a Communication Problem"
	},
	{
		"chunk_id" : "00002771",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "On many cell phones with GPS, an approximate location can be given before the GPS signal is received. This is\naccomplished through a process called triangulation, which works by using the distances from two known points.\nSuppose there are two cell phone towers within range of a cell phone. The two towers are located 6000 feet apart along\na straight highway, running east to west, and the cell phone is north of the highway. Based on the signal delay, it can be"
	},
	{
		"chunk_id" : "00002772",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determined that the signal is 5,050 feet from the first tower and 2,420 feet from the second tower. Determine the\nposition of the cell phone north and east of the first tower, and determine how far it is from the highway.\nSolution\nFor simplicity, we start by drawing a diagram similar toFigure 6and labeling our given information.\n916 10  Further Applications of Trigonometry\nFigure6\nUsing the Law of Cosines, we can solve for the angle Remember that the Law of Cosines uses the square of one side to"
	},
	{
		"chunk_id" : "00002773",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "find the cosine of the opposite angle. For this example, let and Thus, corresponds to\nthe opposite side\nTo answer the questions about the phones position north and east of the tower, and the distance to the highway, drop a\nperpendicular from the position of the cell phone, as inFigure 7. This forms two right triangles, although we only need\nthe right triangle that includes the first tower for this problem.\nFigure7\nUsing the angle and the basic trigonometric identities, we can find the solutions. Thus"
	},
	{
		"chunk_id" : "00002774",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The cell phone is approximately 4,638 feet east and 1998 feet north of the first tower, and 1998 feet from the highway.\nEXAMPLE4\nCalculating Distance Traveled Using a SAS Triangle\nReturning to our problem at the beginning of this section, suppose a boat leaves port, travels 10 miles, turns 20 degrees,\nand travels another 8 miles. How far from port is the boat? The diagram is repeated here inFigure 8.\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 917\nFigure8\nSolution"
	},
	{
		"chunk_id" : "00002775",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nSolution\nThe boat turned 20 degrees, so the obtuse angle of the non-right triangle is the supplemental angle,\nWith this, we can utilize the Law of Cosines to find the missing side of the obtuse trianglethe distance of the boat to\nthe port.\nThe boat is about 17.7 miles from port.\nUsing Herons Formula to Find the Area of a Triangle\nWe already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the"
	},
	{
		"chunk_id" : "00002776",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use\nHerons formulainstead of finding the height.Heron of Alexandriawas a geometer who lived during the first century\nA.D. He discovered a formula for finding the area of oblique triangles when three sides are known.\nHerons Formula\nHerons formula finds the area of oblique triangles in which sides and are known."
	},
	{
		"chunk_id" : "00002777",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where is one half of the perimeter of the triangle, sometimes called the semi-perimeter.\nEXAMPLE5\nUsing Herons Formula to Find the Area of a Given Triangle\nFind the area of the triangle inFigure 9using Herons formula.\n918 10  Further Applications of Trigonometry\nFigure9\nSolution\nFirst, we calculate\nThen we apply the formula.\nThe area is approximately 29.4 square units.\nTRY IT #3 Use Herons formula to find the area of a triangle with sides of lengths\nand\nEXAMPLE6"
	},
	{
		"chunk_id" : "00002778",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and\nEXAMPLE6\nApplying Herons Formula to a Real-World Problem\nA Chicago city developer wants to construct a building consisting of artists lofts on a triangular lot bordered by Rush\nStreet, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash\nAvenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square\nmeters are available to the developer? SeeFigure 10for a view of the city property.\nFigure10"
	},
	{
		"chunk_id" : "00002779",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure10\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 919\nSolution\nFind the measurement for which is one-half of the perimeter.\nApply Herons formula.\nThe developer has about 711.4 square meters.\nTRY IT #4 Find the area of a triangle given and\nMEDIA\nAccess these online resources for additional instruction and practice with the Law of Cosines.\nLaw of Cosines(http://openstax.org/l/lawcosines)\nLaw of Cosines: Applications(http://openstax.org/l/cosineapp)"
	},
	{
		"chunk_id" : "00002780",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Law of Cosines: Applications 2(http://openstax.org/l/cosineapp2)\n10.2 SECTION EXERCISES\nVerbal\n1. If you are looking for a 2. If you are looking for a 3. Explain what represents in\nmissing side of a triangle, missing angle of a triangle, Herons formula.\nwhat do you need to know what do you need to know\nwhen using the Law of when using the Law of\nCosines? Cosines?\n4. Explain the relationship 5. When must you use the Law\nbetween the Pythagorean of Cosines instead of the"
	},
	{
		"chunk_id" : "00002781",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "between the Pythagorean of Cosines instead of the\nTheorem and the Law of Pythagorean Theorem?\nCosines.\nAlgebraic\nFor the following exercises, assume is opposite side is opposite side and is opposite side If possible, solve\neach triangle for the unknown side. Round to the nearest tenth.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15.\n920 10  Further Applications of Trigonometry\nFor the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the\nnearest tenth."
	},
	{
		"chunk_id" : "00002782",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nearest tenth.\n16. find 17. 18. find\nangle find angle angle\n19. 20.\nfind angle find angle\nFor the following exercises, solve the triangle. Round to the nearest tenth.\n21. 22. 23.\n24. 25. 26.\nFor the following exercises, use Herons formula to find the area of the triangle. Round to the nearest hundredth.\n27. Find the area of a triangle 28. Find the area of a triangle 29.\nwith sides of length 18 in, with sides of length 20 cm,\n21 in, and 32 in. Round to 26 cm, and 37 cm. Round to"
	},
	{
		"chunk_id" : "00002783",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the nearest tenth. the nearest tenth.\n30. 31.\nGraphical\nFor the following exercises, find the length of side Round to the nearest tenth.\n32. 33. 34.\n35. 36. 37.\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 921\nFor the following exercises, find the measurement of angle\n38. 39. 40.\n41. 42. Find the measure of each angle in\nthe triangle shown inFigure 11.\nRound to the nearest tenth.\nFigure11\nFor the following exercises, solve for the unknown side. Round to the nearest tenth."
	},
	{
		"chunk_id" : "00002784",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "43. 44. 45.\n46.\n922 10  Further Applications of Trigonometry\nFor the following exercises, find the area of the triangle. Round to the nearest hundredth.\n47. 48. 49.\n50. 51.\nExtensions\n52. A parallelogram has sides 53. The sides of a 54. The sides of a\nof length 16 units and 10 parallelogram are 11 feet parallelogram are 28\nunits. The shorter diagonal and 17 feet. The longer centimeters and 40\nis 12 units. Find the diagonal is 22 feet. Find the centimeters. The measure"
	},
	{
		"chunk_id" : "00002785",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "measure of the longer length of the shorter of the larger angle is 100.\ndiagonal. diagonal. Find the length of the\nshorter diagonal.\n55. A regular octagon is 56. A regular pentagon is\ninscribed in a circle with a inscribed in a circle of\nradius of 8 inches. (See radius 12 cm. (SeeFigure\nFigure 12.) Find the 13.) Find the perimeter of\nperimeter of the octagon. the pentagon. Round to\nthe nearest tenth of a\ncentimeter.\nFigure12\nFigure13\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002786",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12\nFigure13\nAccess for free at openstax.org\n10.2  Non-right Triangles: Law of Cosines 923\nFor the following exercises, suppose that represents the relationship of three sides of a\ntriangle and the cosine of an angle.\n57. Draw the triangle. 58. Find the length of the third\nside.\nFor the following exercises, find the area of the triangle.\n59. 60. 61.\nReal-World Applications\n62. A surveyor has taken the 63. A satellite calculates the distances 64. An airplane flies 220 miles"
	},
	{
		"chunk_id" : "00002787",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "measurements shown inFigure 14. and angle shown inFigure 15(not with a heading of 40, and\nFind the distance across the lake. to scale). Find the distance between then flies 180 miles with a\nRound answers to the nearest tenth. the two cities. Round answers to the heading of 170. How far is\nnearest tenth. the plane from its starting\npoint, and at what\nheading? Round answers\nto the nearest tenth.\nFigure14\nFigure15\n924 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002788",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "924 10  Further Applications of Trigonometry\n65. A 113-foot tower is located on a hill66. Two ships left a port at the 67. The graph inFigure 17represents\nthat is inclined 34 to the horizontal, same time. One ship two boats departing at the same\nas shown inFigure 16. A guy-wire is traveled at a speed of 18 time from the same dock. The first\nto be attached to the top of the miles per hour at a boat is traveling at 18 miles per"
	},
	{
		"chunk_id" : "00002789",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tower and anchored at a point 98 heading of 320. The other hour at a heading of 327 and the\nfeet uphill from the base of the ship traveled at a speed of second boat is traveling at 4 miles\ntower. Find the length of wire 22 miles per hour at a per hour at a heading of 60. Find\nneeded. heading of 194. Find the the distance between the two boats\ndistance between the two after 2 hours.\nships after 10 hours of\ntravel.\nFigure16\nFigure17"
	},
	{
		"chunk_id" : "00002790",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ships after 10 hours of\ntravel.\nFigure16\nFigure17\n68. A triangular swimming 69. A pilot flies in a straight 70. Los Angeles is 1,744 miles\npool measures 40 feet on path for 1 hour 30 min. She from Chicago, Chicago is\none side and 65 feet on then makes a course 714 miles from New York,\nanother side. These sides correction, heading 10 to and New York is 2,451\nform an angle that the right of her original miles from Los Angeles.\nmeasures 50. How long is course, and flies 2 hours in Draw a triangle connecting"
	},
	{
		"chunk_id" : "00002791",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the third side (to the the new direction. If she these three cities, and find\nnearest tenth)? maintains a constant speed the angles in the triangle.\nof 680 miles per hour, how\nfar is she from her starting\nposition?\n71. Philadelphia is 140 miles 72. Two planes leave the same 73. Two airplanes take off in\nfrom Washington, D.C., airport at the same time. different directions. One\nWashington, D.C. is 442 One flies at 20 east of travels 300 mph due west"
	},
	{
		"chunk_id" : "00002792",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "miles from Boston, and north at 500 miles per and the other travels 25\nBoston is 315 miles from hour. The second flies at north of west at 420 mph.\nPhiladelphia. Draw a 30 east of south at 600 After 90 minutes, how far\ntriangle connecting these miles per hour. How far apart are they, assuming\nthree cities and find the apart are the planes after 2 they are flying at the same\nangles in the triangle. hours? altitude?\n74. A parallelogram has sides 75. The four sequential sides 76. The four sequential sides"
	},
	{
		"chunk_id" : "00002793",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of length 15.4 units and 9.8 of a quadrilateral have of a quadrilateral have\nunits. Its area is 72.9 lengths 4.5 cm, 7.9 cm, 9.4 lengths 5.7 cm, 7.2 cm, 9.4\nsquare units. Find the cm, and 12.9 cm. The angle cm, and 12.8 cm. The angle\nmeasure of the longer between the two smallest between the two smallest\ndiagonal. sides is 117. What is the sides is 106. What is the\narea of this quadrilateral? area of this quadrilateral?\nAccess for free at openstax.org\n10.3  Polar Coordinates 925"
	},
	{
		"chunk_id" : "00002794",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.3  Polar Coordinates 925\n77. Find the area of a 78. Find the area of a\ntriangular piece of land triangular piece of land\nthat measures 30 feet on that measures 110 feet on\none side and 42 feet on one side and 250 feet on\nanother; the included another; the included\nangle measures 132. angle measures 85. Round\nRound to the nearest to the nearest whole\nwhole square foot. square foot.\n10.3 Polar Coordinates\nLearning Objectives\nIn this section, you will:\nPlot points using polar coordinates."
	},
	{
		"chunk_id" : "00002795",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Plot points using polar coordinates.\nConvert from polar coordinates to rectangular coordinates.\nConvert from rectangular coordinates to polar coordinates.\nTransform equations between polar and rectangular forms.\nIdentify and graph polar equations by converting to rectangular equations.\nOver 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (seeFigure"
	},
	{
		"chunk_id" : "00002796",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of\nrepresenting location that is different from a standard coordinate grid.\nFigure1\nPlotting Points Using Polar Coordinates\nWhen we think about plotting points in the plane, we usually think ofrectangular coordinates in the Cartesian\ncoordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this"
	},
	{
		"chunk_id" : "00002797",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "section, we introduce topolar coordinates, which are points labeled and plotted on a polar grid. The polar grid is\nrepresented as a series of concentric circles radiating out from thepole, or the origin of the coordinate plane.\nThepolar gridis scaled as the unit circle with the positivex-axis now viewed as thepolar axisand the origin as the pole.\nThe first coordinate is the radius or length of the directed line segment from the pole. The angle measured in"
	},
	{
		"chunk_id" : "00002798",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "radians, indicates the direction of We move counterclockwise from the polar axis by an angle of and measure a\ndirected line segment the length of in the direction of Even though we measure first and then the polar point is\nwritten with ther-coordinate first. For example, to plot the point we would move units in the counterclockwise\ndirection and then a length of 2 from the pole. This point is plotted on the grid inFigure 2.\n926 10  Further Applications of Trigonometry\nFigure2\nEXAMPLE1"
	},
	{
		"chunk_id" : "00002799",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nEXAMPLE1\nPlotting a Point on the Polar Grid\nPlot the point on the polar grid.\nSolution\nThe angle is found by sweeping in a counterclockwise direction 90 from the polar axis. The point is located at a length\nof 3 units from the pole in the direction, as shown inFigure 3.\nFigure3\nTRY IT #1 Plot the point in thepolar grid.\nEXAMPLE2\nPlotting a Point in the Polar Coordinate System with a Negative Component\nPlot the point on the polar grid.\nSolution"
	},
	{
		"chunk_id" : "00002800",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Plot the point on the polar grid.\nSolution\nWe know that is located in the first quadrant. However, We can approach plotting a point with a negative in\ntwo ways:\nAccess for free at openstax.org\n10.3  Polar Coordinates 927\n1. Plot the point by moving in the counterclockwise direction and extending a directed line segment 2 units\ninto the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the\nthird quadrant;"
	},
	{
		"chunk_id" : "00002801",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "third quadrant;\n2. Move in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative\ndirection, into the third quadrant.\nSeeFigure 4(a). Compare this to the graph of the polar coordinate shown inFigure 4(b).\nFigure4\nTRY IT #2 Plot the points and on the same polar grid.\nConverting from Polar Coordinates to Rectangular Coordinates\nWhen given a set ofpolar coordinates, we may need to convert them torectangular coordinates. To do so, we can recall"
	},
	{
		"chunk_id" : "00002802",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the relationships that exist among the variables and\nDropping a perpendicular from the point in the plane to thex-axis forms a right triangle, as illustrated inFigure 5. An\neasy way to remember the equations above is to think of as the adjacent side over the hypotenuse and as\nthe opposite side over the hypotenuse.\nFigure5\nConverting from Polar Coordinates to Rectangular Coordinates\nTo convert polar coordinates to rectangular coordinates let\n928 10  Further Applications of Trigonometry\n...\nHOW TO"
	},
	{
		"chunk_id" : "00002803",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven polar coordinates, convert to rectangular coordinates.\n1. Given the polar coordinate write and\n2. Evaluate and\n3. Multiply by to find thex-coordinate of the rectangular form.\n4. Multiply by to find they-coordinate of the rectangular form.\nEXAMPLE3\nWriting Polar Coordinates as Rectangular Coordinates\nWrite the polar coordinates as rectangular coordinates.\nSolution\nUse the equivalent relationships.\nThe rectangular coordinates are SeeFigure 6.\nFigure6\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002804",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure6\nEXAMPLE4\nWriting Polar Coordinates as Rectangular Coordinates\nWrite the polar coordinates as rectangular coordinates.\nAccess for free at openstax.org\n10.3  Polar Coordinates 929\nSolution\nSeeFigure 7. Writing the polar coordinates as rectangular, we have\nThe rectangular coordinates are also\nFigure7\nTRY IT #3 Write the polar coordinates as rectangular coordinates.\nConverting from Rectangular Coordinates to Polar Coordinates"
	},
	{
		"chunk_id" : "00002805",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To convertrectangular coordinatestopolar coordinates, we will use two other familiar relationships. With this\nconversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.\nConverting from Rectangular Coordinates to Polar Coordinates\nConverting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships\nillustrated inFigure 8.\n930 10  Further Applications of Trigonometry\nFigure8\nEXAMPLE5"
	},
	{
		"chunk_id" : "00002806",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nEXAMPLE5\nWriting Rectangular Coordinates as Polar Coordinates\nConvert the rectangular coordinates to polar coordinates.\nSolution\nWe see that the original point is in the first quadrant. To find use the formula This gives\nTo find we substitute the values for and into the formula We know that must be positive, as is\nin the first quadrant. Thus\nSo, and giving us the polar point SeeFigure 9.\nFigure9\nAnalysis"
	},
	{
		"chunk_id" : "00002807",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure9\nAnalysis\nThere are other sets of polar coordinates that will be the same as our first solution. For example, the points\nAccess for free at openstax.org\n10.3  Polar Coordinates 931\nand will coincide with the original solution of The point\nindicates a move further counterclockwise by which is directly opposite The radius is expressed as\nHowever, the angle is located in the third quadrant and, as is negative, we extend the directed line segment in the"
	},
	{
		"chunk_id" : "00002808",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opposite direction, into the first quadrant. This is the same point as The point is a move\nfurther clockwise by from The radius, is the same.\nTransforming Equations between Polar and Rectangular Forms\nWe can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it\ncan be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot"
	},
	{
		"chunk_id" : "00002809",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points\nbetween the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the\npolar form of the equation.\n...\nHOW TO\nGiven an equation in polar form, graph it using a graphing calculator.\n1. Change theMODEtoPOL, representing polar form.\n2. Press theY=button to bring up a screen allowing the input of six equations:\n3. Enter the polar equation, set equal to"
	},
	{
		"chunk_id" : "00002810",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Enter the polar equation, set equal to\n4. PressGRAPH.\nEXAMPLE6\nWriting a Cartesian Equation in Polar Form\nWrite the Cartesian equation in polar form.\nSolution\nThe goal is to eliminate and from the equation and introduce and Ideally, we would write the equation as a\nfunction of To obtain the polar form, we will use the relationships between and Since and\nwe can substitute and solve for\nThus, and should generate the same graph. SeeFigure 10.\n932 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002811",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "932 10  Further Applications of Trigonometry\nFigure10 (a) Cartesian form (b) Polar form\nTo graph a circle in rectangular form, we must first solve for\nNote that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive\nand negative square roots into the calculator separately, as two equations in the form and\nPressGRAPH.\nEXAMPLE7\nRewriting a Cartesian Equation as a Polar Equation\nRewrite theCartesian equation as a polar equation.\nSolution"
	},
	{
		"chunk_id" : "00002812",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThis equation appears similar to the previous example, but it requires different steps to convert the equation.\nWe can still follow the same procedures we have already learned and make the following substitutions:\nTherefore, the equations and should give us the same graph. SeeFigure 11.\nAccess for free at openstax.org\n10.3  Polar Coordinates 933\nFigure11 (a) Cartesian form (b) polar form"
	},
	{
		"chunk_id" : "00002813",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure11 (a) Cartesian form (b) polar form\nThe Cartesian orrectangular equationis plotted on the rectangular grid, and thepolar equationis plotted on the polar\ngrid. Clearly, the graphs are identical.\nEXAMPLE8\nRewriting a Cartesian Equation in Polar Form\nRewrite the Cartesian equation as a polar equation.\nSolution\nWe will use the relationships and\nTRY IT #4 Rewrite the Cartesian equation in polar form.\nIdentify and Graph Polar Equations by Converting to Rectangular Equations"
	},
	{
		"chunk_id" : "00002814",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are\nindeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will\ndemonstrate that their graphs, while drawn on different grids, are identical.\nEXAMPLE9\nGraphing a Polar Equation by Converting to a Rectangular Equation\nCovert the polar equation to a rectangular equation, and draw its corresponding graph.\nSolution\nThe conversion is"
	},
	{
		"chunk_id" : "00002815",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe conversion is\nNotice that the equation drawn on the polar grid is clearly the same as the vertical line drawn on the\n934 10  Further Applications of Trigonometry\nrectangular grid (seeFigure 12). Just as is the standard form for a vertical line in rectangular form, is\nthe standard form for a vertical line in polar form.\nFigure12 (a) Polar grid (b) Rectangular coordinate system\nA similar discussion would demonstrate that the graph of the function will be the horizontal line In"
	},
	{
		"chunk_id" : "00002816",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fact, is the standard form for a horizontal line in polar form, corresponding to the rectangular form\nEXAMPLE10\nRewriting a Polar Equation in Cartesian Form\nRewrite the polar equation as a Cartesian equation.\nSolution\nThe goal is to eliminate and and introduce and We clear the fraction, and then use substitution. In order to\nreplace with and we must use the expression\nThe Cartesian equation is However, to graph it, especially using a graphing calculator or computer\nprogram, we want to isolate"
	},
	{
		"chunk_id" : "00002817",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "program, we want to isolate\nWhen our entire equation has been changed from and to and we can stop, unless asked to solve for or simplify.\nSeeFigure 13.\nAccess for free at openstax.org\n10.3  Polar Coordinates 935\nFigure13\nThe hour-glass shape of the graph is called ahyperbola. Hyperbolas have many interesting geometric features and\napplications, which we will investigate further inAnalytic Geometry.\nAnalysis"
	},
	{
		"chunk_id" : "00002818",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nIn this example, the right side of the equation can be expanded and the equation simplified further, as shown above.\nHowever, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular\nequation in the hyperbolas standard form. To do this, we can start with the initial equation.\nTRY IT #5 Rewrite the polar equation in Cartesian form.\nEXAMPLE11\nRewriting a Polar Equation in Cartesian Form\nRewrite the polar equation in Cartesian form.\nSolution"
	},
	{
		"chunk_id" : "00002819",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThis equation can also be written as\n936 10  Further Applications of Trigonometry\nMEDIA\nAccess these online resources for additional instruction and practice with polar coordinates.\nIntroduction to Polar Coordinates(http://openstax.org/l/intropolar)\nComparing Polar and Rectangular Coordinates(http://openstax.org/l/polarrect)\n10.3 SECTION EXERCISES\nVerbal\n1. How are polar coordinates 2. How are the polar axes 3. Explain how polar"
	},
	{
		"chunk_id" : "00002820",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "different from rectangular different from thex- and coordinates are graphed.\ncoordinates? y-axes of the Cartesian\nplane?\n4. How are the points 5. Explain why the points\nand related? and are\nthe same.\nAlgebraic\nFor the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the\nquadrant in which the given point is located when determining for the point.\n6. 7. 8.\n9. 10."
	},
	{
		"chunk_id" : "00002821",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7. 8.\n9. 10.\nFor the following exercises, convert the given Cartesian coordinates to polar coordinates with\nRemember to consider the quadrant in which the given point is located.\n11. 12. 13.\n14. 15.\nFor the following exercises, convert the given Cartesian equation to a polar equation.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\nAccess for free at openstax.org\n10.3  Polar Coordinates 937"
	},
	{
		"chunk_id" : "00002822",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.3  Polar Coordinates 937\nFor the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a\nconic if possible, and identify the conic section represented.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\nGraphical\nFor the following exercises, find the polar coordinates of the point.\n40. 41. 42.\n43. 44.\nFor the following exercises, plot the points.\n45. 46. 47.\n48. 49. 50.\n938 10  Further Applications of Trigonometry\n51. 52. 53.\n54."
	},
	{
		"chunk_id" : "00002823",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "51. 52. 53.\n54.\nFor the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.\n55. 56. 57.\n58. 59. 60.\n61.\nFor the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.\n62. 63. 64.\n65. 66. 67.\n68.\nTechnology\n69. Use a graphing calculator 70. Use a graphing calculator 71. Use a graphing calculator\nto find the rectangular to find the rectangular to find the polar"
	},
	{
		"chunk_id" : "00002824",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coordinates of coordinates of coordinates of in\nRound to the nearest Round to the nearest degrees. Round to the\nthousandth. thousandth. nearest thousandth.\n72. Use a graphing calculator 73. Use a graphing calculator\nto find the polar to find the polar\ncoordinates of in coordinates of in\ndegrees. Round to the radians. Round to the\nnearest hundredth. nearest hundredth.\nExtensions\n74. Describe the graph of 75. Describe the graph of 76. Describe the graph of"
	},
	{
		"chunk_id" : "00002825",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "77. Describe the graph of 78. What polar equations will\ngive an oblique line?\nFor the following exercise, graph the polar inequality.\n79. 80. 81.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 939\n82. 83. 84.\n10.4 Polar Coordinates: Graphs\nLearning Objectives\nIn this section you will:\nTest polar equations for symmetry.\nGraph polar equations by plotting points.\nThe planets move through space in elliptical, periodic orbits about the sun, as shown inFigure 1. They are in constant"
	},
	{
		"chunk_id" : "00002826",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planets\ninstantaneousposition. This is one application ofpolar coordinates, represented as We interpret as the distance\nfrom the sun and as the planets angular bearing, or its direction from a fixed point on the sun. In this section, we will\nfocus on the polar system and the graphs that are generated directly from polar coordinates."
	},
	{
		"chunk_id" : "00002827",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1 Planets follow elliptical paths as they orbit around the Sun. (credit: modification of work by NASA/JPL-Caltech)\nTesting Polar Equations for Symmetry\nJust as a rectangular equation such as describes the relationship between and on a Cartesian grid, apolar\nequationdescribes a relationship between and on a polar grid. Recall that the coordinate pair indicates that we\nmove counterclockwise from the polar axis (positivex-axis) by an angle of and extend a ray from the pole (origin)"
	},
	{
		"chunk_id" : "00002828",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "units in the direction of All points that satisfy the polar equation are on the graph.\nSymmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is\nsymmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on\none side would coincide with the portion on the other side. By performing three tests, we will see how to apply the"
	},
	{
		"chunk_id" : "00002829",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and\nmaximums of to determine the graph of a polar equation.\nIn the first test, we consider symmetry with respect to the line (y-axis). We replace with to determine\nif the new equation is equivalent to the original equation. For example, suppose we are given the equation\nThis equation exhibits symmetry with respect to the line"
	},
	{
		"chunk_id" : "00002830",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the second test, we consider symmetry with respect to the polar axis ( -axis). We replace with or\nto determine equivalency between the tested equation and the original. For example, suppose we are given\nthe equation\n940 10  Further Applications of Trigonometry\nThe graph of this equation exhibits symmetry with respect to the polar axis.\nIn the third test, we consider symmetry with respect to the pole (origin). We replace with to determine if the"
	},
	{
		"chunk_id" : "00002831",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tested equation is equivalent to the original equation. For example, suppose we are given the equation\nThe equation has failed thesymmetry test, but that does not mean that it is not symmetric with respect to the pole.\nPassing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the\nsymmetry tests does not necessarily indicate that a graph will not be symmetric about the line the polar axis, or"
	},
	{
		"chunk_id" : "00002832",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis\nof symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its\napplication is not perfect.\nSymmetry Tests\nApolar equationdescribes a curve on the polar grid. The graph of a polar equation can be evaluated for three types\nof symmetry, as shown inFigure 2."
	},
	{
		"chunk_id" : "00002833",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of symmetry, as shown inFigure 2.\nFigure2 (a) A graph is symmetric with respect to the line (y-axis) if replacing with yields an\nequivalent equation. (b) A graph is symmetric with respect to the polar axis (x-axis) if replacing with or\nyields an equivalent equation. (c) A graph is symmetric with respect to the pole (origin) if replacing\nwith yields an equivalent equation.\n...\nHOW TO\nGiven a polar equation, test for symmetry."
	},
	{
		"chunk_id" : "00002834",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "HOW TO\nGiven a polar equation, test for symmetry.\n1. Substitute the appropriate combination of components for for symmetry; for polar\naxis symmetry; and for symmetry with respect to the pole.\n2. If the resulting equations are equivalent in one or more of the tests, the graph produces the expected symmetry.\nEXAMPLE1\nTesting a Polar Equation for Symmetry\nTest the equation for symmetry.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 941\nSolution\nTest for each of the three types of symmetry."
	},
	{
		"chunk_id" : "00002835",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Test for each of the three types of symmetry.\n1) Replacing with yields the same result. Thus, the graph\nis symmetric with respect to the line\n2) Replacing with does not yield the same equation. Therefore,\nthe graph fails the test and may or may not be symmetric with respect\nto the polar axis.\n3) Replacing with changes the equation and fails the test. The\ngraph may or may not be symmetric with respect to the pole.\nTable1\nAnalysis"
	},
	{
		"chunk_id" : "00002836",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nAnalysis\nUsing a graphing calculator, we can see that the equation is a circle centered at with radius and is\nindeed symmetric to the line We can also see that the graph is not symmetric with the polar axis or the pole. See\nFigure 3.\nFigure3\nTRY IT #1 Test the equation for symmetry:\nGraphing Polar Equations by Plotting Points\nTo graph in the rectangular coordinate system we construct a table of and values. To graph in the polar coordinate"
	},
	{
		"chunk_id" : "00002837",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "system we construct a table of and values. We enter values of into apolar equationand calculate However, using\nthe properties of symmetry and finding key values of and means fewer calculations will be needed.\nFinding Zeros and Maxima\nTo find the zeros of a polar equation, we solve for the values of that result in Recall that, to find the zeros of\npolynomial functions, we set the equation equal to zero and then solve for We use the same process for polar\nequations. Set and solve for"
	},
	{
		"chunk_id" : "00002838",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations. Set and solve for\nFor many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values\n942 10  Further Applications of Trigonometry\nof into the equation that result in the maximum value of the trigonometric functions. Consider the\nmaximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when\nso our polar equation is and the value will yield the maximum"
	},
	{
		"chunk_id" : "00002839",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Similarly, the maximum value of the sine function is 1 when and if our polar equation is the value\nwill yield the maximum We may find additional information by calculating values of when These\npoints would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar\nequation.\nEXAMPLE2\nFinding Zeros and Maximum Values for a Polar Equation\nUsing the equation inExample 1, find the zeros and maximum and, if necessary, the polar axis intercepts of\nSolution"
	},
	{
		"chunk_id" : "00002840",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nTo find the zeros, set equal to zero and solve for\nSubstitute any one of the values into the equation. We will use\nThe points and are the zeros of the equation. They all coincide, so only one point is visible on the graph.\nThis point is also the only polar axis intercept.\nTo find the maximum value of the equation, look at the maximum value of the trigonometric function which\noccurs when resulting in Substitute for\nAnalysis"
	},
	{
		"chunk_id" : "00002841",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "occurs when resulting in Substitute for\nAnalysis\nThe point will be the maximum value on the graph. Lets plot a few more points to verify the graph of a circle. See\nTable 2andFigure 4.\n0\nTable2\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 943\nTable2\nFigure4\nTRY IT #2 Without converting to Cartesian coordinates, test the given equation for symmetry and find the\nzeros and maximum values of\nInvestigating Circles"
	},
	{
		"chunk_id" : "00002842",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "zeros and maximum values of\nInvestigating Circles\nNow we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation\nwas used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted\npoints that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.\nThere are five classic polar curves: cardioids,limaons, lemniscates, rose curves, andArchimedes spirals. We will"
	},
	{
		"chunk_id" : "00002843",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.\nFormulas for the Equation of a Circle\nSome of the formulas that produce the graph of a circle in polar coordinates are given by and\nwhere is the diameter of the circle or the distance from the pole to the farthest point on the circumference. The\nradius is or one-half the diameter. For the center is For the center is\nFigure 5shows the graphs of these four circles.\nFigure5"
	},
	{
		"chunk_id" : "00002844",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure5\n944 10  Further Applications of Trigonometry\nEXAMPLE3\nSketching the Graph of a Polar Equation for a Circle\nSketch the graph of\nSolution\nFirst, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find thezeros\nand maximum for First, set and solve for . Thus, a zero occurs at A key point to plot\nis\nTo find the maximum value of note that the maximum value of the cosine function is 1 when Substitute\ninto the equation:"
	},
	{
		"chunk_id" : "00002845",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "into the equation:\nThe maximum value of the equation is 4. A key point to plot is\nAs is symmetric with respect to the polar axis, we only need to calculater-values for over the interval\nPoints in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar toTable 3. The\ngraph is shown inFigure 6.\n0\n4 3.46 2.83 2 0 2 2.83 3.46 4\nTable3\nFigure6\nInvestigating Cardioids"
	},
	{
		"chunk_id" : "00002846",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table3\nFigure6\nInvestigating Cardioids\nWhile translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the\nclassic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart.\nThis shape is often included with the family of curves called limaons, but here we will discuss the cardioid on its own.\nFormulas for a Cardioid\nThe formulas that produce the graphs of acardioidare given by and where"
	},
	{
		"chunk_id" : "00002847",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and The cardioid graph passes through the pole, as we can see inFigure 7.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 945\nFigure7\n...\nHOW TO\nGiven the polar equation of a cardioid, sketch its graph.\n1. Check equation for the three types of symmetry.\n2. Find the zeros. Set\n3. Find the maximum value of the equation according to the maximum value of the trigonometric expression.\n4. Make a table of values for and\n5. Plot the points and sketch the graph.\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002848",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. Plot the points and sketch the graph.\nEXAMPLE4\nSketching the Graph of a Cardioid\nSketch the graph of\nSolution\nFirst, testing the equation for symmetry, we find that the graph of this equation will be symmetric about the polar axis.\nNext, we find the zeros and maximums. Setting we have The zero of the equation is located at\nThe graph passes through this point.\nThe maximum value of occurs when is a maximum, which is when or when\nSubstitute into the equation, and solve for"
	},
	{
		"chunk_id" : "00002849",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substitute into the equation, and solve for\nThe point is the maximum value on the graph.\nWe found that the polar equation is symmetric with respect to the polar axis, but as it extends to all four quadrants, we\nneed to plot values over the interval The upper portion of the graph is then reflected over the polar axis. Next, we\nmake a table of values, as inTable 4, and then we plot the points and draw the graph. SeeFigure 8.\n4 3.41 2 1 0\nTable4\n946 10  Further Applications of Trigonometry\nFigure8"
	},
	{
		"chunk_id" : "00002850",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure8\nInvestigating Limaons\nThe wordlimaonis Old French for snail, a name that describes the shape of the graph. As mentioned earlier, the\ncardioid is a member of the limaon family, and we can see the similarities in the graphs. The other images in this\ncategory include the one-loop limaon and the two-loop (or inner-loop) limaon.One-loop limaonsare sometimes\nreferred to asdimpled limaonswhen andconvex limaonswhen\nFormulas for One-Loop Limaons"
	},
	{
		"chunk_id" : "00002851",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Formulas for One-Loop Limaons\nThe formulas that produce the graph of a dimpledone-loop limaonare given by and\nwhere All four graphs are shown inFigure 9.\nFigure9 Dimpled limaons\n...\nHOW TO\nGiven a polar equation for a one-loop limaon, sketch the graph.\n1. Test the equation for symmetry. Remember that failing a symmetry test does not mean that the shape will not\nexhibit symmetry. Often the symmetry may reveal itself when the points are plotted.\n2. Find the zeros."
	},
	{
		"chunk_id" : "00002852",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Find the zeros.\n3. Find the maximum values according to the trigonometric expression.\n4. Make a table.\n5. Plot the points and sketch the graph.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 947\nEXAMPLE5\nSketching the Graph of a One-Loop Limaon\nGraph the equation\nSolution\nFirst, testing the equation for symmetry, we find that it fails all three symmetry tests, meaning that the graph may or"
	},
	{
		"chunk_id" : "00002853",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "may not exhibit symmetry, so we cannot use the symmetry to help us graph it. However, this equation has a graph that\nclearly displays symmetry with respect to the line yet it fails all the three symmetry tests. A graphing calculator\nwill immediately illustrate the graphs reflective quality.\nNext, we find the zeros and maximum, and plot the reflecting points to verify any symmetry. Setting results in\nbeing undefined. What does this mean? How could be undefined? The angle is undefined for any value of"
	},
	{
		"chunk_id" : "00002854",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Therefore, is undefined because there is no value of for which Consequently, the graph does not pass\nthrough the pole. Perhaps the graph does cross the polar axis, but not at the pole. We can investigate other intercepts\nby calculating when\nSo, there is at least one polar axis intercept at\nNext, as the maximum value of the sine function is 1 when we will substitute into the equation and solve\nfor Thus,\nMake a table of the coordinates similar toTable 5.\n4 2.5 1.4 1 1.4 2.5 4 5.5 6.6 7 6.6 5.5 4\nTable5"
	},
	{
		"chunk_id" : "00002855",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4 2.5 1.4 1 1.4 2.5 4 5.5 6.6 7 6.6 5.5 4\nTable5\nThe graph is shown inFigure 10.\nFigure10 One-loop limaon\nAnalysis\nThis is an example of a curve for which making a table of values is critical to producing an accurate graph. The symmetry\ntests fail; the zero is undefined. While it may be apparent that an equation involving is likely symmetric with\nrespect to the line evaluating more points helps to verify that the graph is correct.\n948 10  Further Applications of Trigonometry\nTRY IT #3 Sketch the graph of"
	},
	{
		"chunk_id" : "00002856",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #3 Sketch the graph of\nAnother type of limaon, theinner-loop limaon, is named for the loop formed inside the general limaon shape. It was\ndiscovered by the German artist AlbrechtDrer(1471-1528), who revealed a method for drawing the inner-loop limaon\nin his 1525 bookUnderweysung der Messing. A century later, the father of mathematician BlaisePascal, tienne\nPascal(1588-1651), rediscovered it.\nFormulas for Inner-Loop Limaons"
	},
	{
		"chunk_id" : "00002857",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Formulas for Inner-Loop Limaons\nThe formulas that generate theinner-loop limaonsare given by and where\nand The graph of the inner-loop limaon passes through the pole twice: once for the outer loop, and\nonce for the inner loop. SeeFigure 11for the graphs.\nFigure11\nEXAMPLE6\nSketching the Graph of an Inner-Loop Limaon\nSketch the graph of\nSolution\nTesting for symmetry, we find that the graph of the equation is symmetric about the polar axis. Next, finding the zeros"
	},
	{
		"chunk_id" : "00002858",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "reveals that when The maximum is found when or when Thus, the maximum is\nfound at the point (7, 0).\nEven though we have found symmetry, the zero, and the maximum, plotting more points will help to define the shape,\nand then a pattern will emerge.\nSeeTable 6.\n7 6.3 4.5 2 0.5 2.3 3 2.3 0.5 2 4.5 6.3 7\nTable6\nAs expected, the values begin to repeat after The graph is shown inFigure 12.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 949\nFigure12 Inner-loop limaon"
	},
	{
		"chunk_id" : "00002859",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12 Inner-loop limaon\nInvestigating Lemniscates\nThe lemniscate is a polar curve resembling the infinity symbolor a figure 8. Centered at the pole, a lemniscate is\nsymmetrical by definition.\nFormulas for Lemniscates\nThe formulas that generate the graph of alemniscateare given by and where\nThe formula is symmetric with respect to the pole. The formula is symmetric with\nrespect to the pole, the line and the polar axis. SeeFigure 13for the graphs.\nFigure13\nEXAMPLE7\nSketching the Graph of a Lemniscate"
	},
	{
		"chunk_id" : "00002860",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE7\nSketching the Graph of a Lemniscate\nSketch the graph of\nSolution\nThe equation exhibits symmetry with respect to the line the polar axis, and the pole.\nLets find the zeros. It should be routine by now, but we will approach this equation a little differently by making the\nsubstitution\n950 10  Further Applications of Trigonometry\nSo, the point is a zero of the equation.\nNow lets find the maximum value. Since the maximum of when the maximum when\nThus,"
	},
	{
		"chunk_id" : "00002861",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thus,\nWe have a maximum at (2, 0). Since this graph is symmetric with respect to the pole, the line and the polar axis,\nwe only need to plot points in the first quadrant.\nMake a table similar toTable 7.\n0\n0\nTable7\nPlot the points on the graph, such as the one shown inFigure 14.\nFigure14 Lemniscate\nAnalysis\nMaking a substitution such as is a common practice in mathematics because it can make calculations simpler."
	},
	{
		"chunk_id" : "00002862",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "However, we must not forget to replace the substitution term with the original term at the end, and then solve for the\nunknown.\nSome of the points on this graph may not show up using the Trace function on the TI-84 graphing calculator, and the\ncalculator table may show an error for these same points of This is because there are no real square roots for these\nvalues of In other words, the correspondingr-values of are complex numbers because there is a negative\nnumber under the radical."
	},
	{
		"chunk_id" : "00002863",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number under the radical.\nInvestigating Rose Curves\nThe next type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a\nsimple polar equation generates the pattern.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 951\nRose Curves\nThe formulas that generate the graph of arose curveare given by and where If is\neven, the curve has petals. If is odd, the curve has petals. SeeFigure 15.\nFigure15\nEXAMPLE8\nSketching the Graph of a Rose Curve (nEven)"
	},
	{
		"chunk_id" : "00002864",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketching the Graph of a Rose Curve (nEven)\nSketch the graph of\nSolution\nTesting for symmetry, we find again that the symmetry tests do not tell the whole story. The graph is not only symmetric\nwith respect to the polar axis, but also with respect to the line and the pole.\nNow we will find the zeros. First make the substitution\nThe zero is The point is on the curve.\nNext, we find the maximum We know that the maximum value of when Thus,\nThe point is on the curve."
	},
	{
		"chunk_id" : "00002865",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The point is on the curve.\nThe graph of the rose curve has unique properties, which are revealed inTable 8.\n0\n2 0 2 0 2 0 2\nTable8\n952 10  Further Applications of Trigonometry\nAs when it makes sense to divide values in the table by units. A definite pattern emerges. Look at the\nrange ofr-values: 2, 0, 2, 0, 2, 0, 2, and so on. This represents the development of the curve one petal at a time.\nStarting at each petal extends out a distance of and then turns back to zero times for a total of eight"
	},
	{
		"chunk_id" : "00002866",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "petals. See the graph inFigure 16.\nFigure16 Rose curve, even\nAnalysis\nWhen these curves are drawn, it is best to plot the points in order, as in theTable 8. This allows us to see how the graph\nhits a maximum (the tip of a petal), loops back crossing the pole, hits the opposite maximum, and loops back to the pole.\nThe action is continuous until all the petals are drawn.\nTRY IT #4 Sketch the graph of\nEXAMPLE9\nSketching the Graph of a Rose Curve (nOdd)\nSketch the graph of\nSolution"
	},
	{
		"chunk_id" : "00002867",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketch the graph of\nSolution\nThe graph of the equation shows symmetry with respect to the line Next, find the zeros and maximum. We will\nwant to make the substitution\nThe maximum value is calculated at the angle where is a maximum. Therefore,\nThus, the maximum value of the polar equation is 2. This is the length of each petal. As the curve for odd yields the\nsame number of petals as there will be five petals on the graph. SeeFigure 17.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 953"
	},
	{
		"chunk_id" : "00002868",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.4  Polar Coordinates: Graphs 953\nFigure17 Rose curve, odd\nCreate a table of values similar toTable 9.\n0\n0 1 1.73 2 1.73 1 0\nTable9\nTRY IT #5 Sketch the graph of\nInvestigating the Archimedes Spiral\nThe final polar equation we will discuss is the Archimedes spiral, named for its discoverer, the Greek mathematician\nArchimedes (c. 287 BCE-c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.\nArchimedes Spiral"
	},
	{
		"chunk_id" : "00002869",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Archimedes Spiral\nThe formula that generates the graph of theArchimedes spiralis given by for As increases,\nincreases at a constant rate in an ever-widening, never-ending, spiraling path. SeeFigure 18.\nFigure18\n...\nHOW TO\nGiven an Archimedes spiral over sketch the graph.\n954 10  Further Applications of Trigonometry\n1. Make a table of values for and over the given domain.\n2. Plot the points and sketch the graph.\nEXAMPLE10\nSketching the Graph of an Archimedes Spiral\nSketch the graph of over\nSolution"
	},
	{
		"chunk_id" : "00002870",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sketch the graph of over\nSolution\nAs is equal to the plot of the Archimedes spiral begins at the pole at the point (0, 0). While the graph hints of\nsymmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value,\nunless the domain is restricted.\nCreate a table such asTable 10.\n0.785 1.57 3.14 4.71 5.50 6.28\nTable10\nNotice that ther-values are just the decimal form of the angle measured in radians. We can see them on a graph in\nFigure 19."
	},
	{
		"chunk_id" : "00002871",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 19.\nFigure19 Archimedes spiral\nAnalysis\nThe domain of this polar curve is In general, however, the domain of this function is   Graphing the\nequation of the Archimedes spiral is rather simple, although the image makes it seem like it would be complex.\nTRY IT #6 Sketch the graph of over the interval\nSummary of Curves\nWe have explored a number of seemingly complex polar curves in this section.Figure 20andFigure 21summarize the\ngraphs and equations for each of these curves."
	},
	{
		"chunk_id" : "00002872",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graphs and equations for each of these curves.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 955\nFigure20\nFigure21\nMEDIA\nAccess these online resources for additional instruction and practice with graphs of polar coordinates.\nGraphing Polar Equations Part 1(http://openstax.org/l/polargraph1)\nGraphing Polar Equations Part 2(http://openstax.org/l/polargraph2)\nAnimation: The Graphs of Polar Equations(http://openstax.org/l/polaranim)"
	},
	{
		"chunk_id" : "00002873",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing Polar Equations on the TI-84(http://openstax.org/l/polarTI84)\n10.4 SECTION EXERCISES\nVerbal\n1. Describe the three types of 2. Which of the three types of 3. What are the steps to follow\nsymmetry in polar graphs, symmetries for polar graphs when graphing polar\nand compare them to the correspond to the equations?\nsymmetry of the Cartesian symmetries with respect to\nplane. thex-axis,y-axis, and\norigin?\n4. Describe the shapes of the 5. What part of the equation"
	},
	{
		"chunk_id" : "00002874",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graphs of cardioids, determines the shape of the\nlimaons, and lemniscates. graph of a polar equation?\n956 10  Further Applications of Trigonometry\nGraphical\nFor the following exercises, test the equation for symmetry.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15.\nFor the following exercises, graph the polar equation. Identify the name of the shape.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n43.\nTechnology"
	},
	{
		"chunk_id" : "00002875",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "37. 38. 39.\n40. 41. 42.\n43.\nTechnology\nFor the following exercises, use a graphing calculator to sketch the graph of the polar equation.\n44. 45. 46. a cissoid\n47. , a 48. 49.\nhippopede\n50. 51. 52.\n53.\nAccess for free at openstax.org\n10.4  Polar Coordinates: Graphs 957\nFor the following exercises, use a graphing utility to graph each pair of polar equations on a domain of and then\nexplain the differences shown in the graphs.\n54. 55. 56.\n57. 58. 59. On a graphing utility,\ngraph on ,\n, , , , , and"
	},
	{
		"chunk_id" : "00002876",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "graph on ,\n, , , , , and\n, Describe the\neffect of increasing the\nwidth of the domain.\n60. On a graphing utility, 61. On a graphing utility, 62. On a graphing utility,\ngraph and sketch graph each polar equation. graph each polar equation.\nExplain the similarities and Explain the similarities and\ndifferences you observe in differences you observe in\non\nthe graphs. the graphs.\n63. On a graphing utility,\ngraph each polar equation.\nExplain the similarities and\ndifferences you observe in\nthe graphs.\nExtensions"
	},
	{
		"chunk_id" : "00002877",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "differences you observe in\nthe graphs.\nExtensions\nFor the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.\n64. 65. 66.\n67. 68. 69. ,\n70. 71. 72. ,\n958 10  Further Applications of Trigonometry\n10.5 Polar Form of Complex Numbers\nLearning Objectives\nIn this section, you will:\nPlot complex numbers in the complex plane.\nFind the absolute value of a complex number.\nWrite complex numbers in polar form."
	},
	{
		"chunk_id" : "00002878",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write complex numbers in polar form.\nConvert a complex number from polar to rectangular form.\nFind products of complex numbers in polar form.\nFind quotients of complex numbers in polar form.\nFind powers of complex numbers in polar form.\nFind roots of complex numbers in polar form.\nGod made the integers; all else is the work of man. This rather famous quote by nineteenth-century German\nmathematician LeopoldKroneckersets the stage for this section on the polar form of a complex number. Complex"
	},
	{
		"chunk_id" : "00002879",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers were invented by people and represent over a thousand years of continuous investigation and struggle by\nmathematicians such asPythagoras,Descartes, De Moivre,Euler,Gauss, and others. Complex numbers answered\nquestions that for centuries had puzzled the greatest minds in science.\nWe first encountered complex numbers inComplex Numbers. In this section, we will focus on the mechanics of working\nwith complex numbers: translation of complex numbers from polar form to rectangular form and vice versa,"
	},
	{
		"chunk_id" : "00002880",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interpretation of complex numbers in the scheme of applications, and application of De Moivres Theorem.\nPlotting Complex Numbers in the Complex Plane\nPlotting acomplex number is similar to plotting a real number, except that the horizontal axis represents the real\npart of the number, and the vertical axis represents the imaginary part of the number,\n...\nHOW TO\nGiven a complex number plot it in the complex plane.\n1. Label the horizontal axis as therealaxis and the vertical axis as theimaginary axis."
	},
	{
		"chunk_id" : "00002881",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Plot the point in the complex plane by moving units in the horizontal direction and units in the vertical\ndirection.\nEXAMPLE1\nPlotting a Complex Number in the Complex Plane\nPlot the complex number in thecomplex plane.\nSolution\nFrom the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See\nFigure 1.\nAccess for free at openstax.org\n10.5  Polar Form of Complex Numbers 959\nFigure1\nTRY IT #1 Plot the point in the complex plane."
	},
	{
		"chunk_id" : "00002882",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #1 Plot the point in the complex plane.\nFinding the Absolute Value of a Complex Number\nThe first step toward working with a complex number inpolar formis to find the absolute value. The absolute value of a\ncomplex number is the same as itsmagnitude, or It measures the distance from the origin to a point in the plane.\nFor example, the graph of inFigure 2, shows\nFigure2\nAbsolute Value of a Complex Number\nGiven a complex number, the absolute value of is defined as"
	},
	{
		"chunk_id" : "00002883",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "It is the distance from the origin to the point\nNotice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a\ncomplex number gives the distance of the number from the origin,\n960 10  Further Applications of Trigonometry\nEXAMPLE2\nFinding the Absolute Value of a Complex Number with a Radical\nFind the absolute value of\nSolution\nUsing the formula, we have\nSeeFigure 3.\nFigure3\nTRY IT #2 Find the absolute value of the complex number\nEXAMPLE3"
	},
	{
		"chunk_id" : "00002884",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE3\nFinding the Absolute Value of a Complex Number\nGiven find\nSolution\nUsing the formula, we have\nThe absolute value is 5. SeeFigure 4.\nAccess for free at openstax.org\n10.5  Polar Form of Complex Numbers 961\nFigure4\nTRY IT #3 Given find\nWriting Complex Numbers in Polar Form\nThepolar form of a complex numberexpresses a number in terms of an angle and its distance from the origin\nGiven a complex number inrectangular formexpressed as we use the same conversion formulas as we do to"
	},
	{
		"chunk_id" : "00002885",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "write the number in trigonometric form:\nWe review these relationships inFigure 5.\nFigure5\nWe use the termmodulusto represent the absolute value of a complex number, or the distance from the origin to the\npoint The modulus, then, is the same as the radius in polar form. We use to indicate the angle of direction\n(just as with polar coordinates). Substituting, we have\nPolar Form of a Complex Number\nWriting a complex number in polar form involves the following conversion formulas:"
	},
	{
		"chunk_id" : "00002886",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "962 10  Further Applications of Trigonometry\nMaking a direct substitution, we have\nwhere is themodulusand is theargument. We often use the abbreviation to represent\nEXAMPLE4\nExpressing a Complex Number Using Polar Coordinates\nExpress the complex number using polar coordinates.\nSolution\nOn the complex plane, the number is the same as Writing it in polar form, we have to calculate first.\nNext, we look at If and then In polar coordinates, the complex number can be\nwritten as or SeeFigure 6.\nFigure6"
	},
	{
		"chunk_id" : "00002887",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "written as or SeeFigure 6.\nFigure6\nTRY IT #4 Express as in polar form.\nEXAMPLE5\nFinding the Polar Form of a Complex Number\nFind the polar form of\nSolution\nFirst, find the value of\nAccess for free at openstax.org\n10.5  Polar Form of Complex Numbers 963\nFind the angle using the formula:\nThus, the solution is\nTRY IT #5 Write in polar form.\nConverting a Complex Number from Polar to Rectangular Form"
	},
	{
		"chunk_id" : "00002888",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the\ndistributive property. In other words, given first evaluate the trigonometric functions and\nThen, multiply through by\nEXAMPLE6\nConverting from Polar to Rectangular Form\nConvert the polar form of the given complex number to rectangular form:\nSolution\nWe begin by evaluating the trigonometric expressions.\nAfter substitution, the complex number is\nWe apply the distributive property:"
	},
	{
		"chunk_id" : "00002889",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We apply the distributive property:\nThe rectangular form of the given point in complex form is\n964 10  Further Applications of Trigonometry\nEXAMPLE7\nFinding the Rectangular Form of a Complex Number\nFind the rectangular form of the complex number given and\nSolution\nIf and we first determine We then find and\nThe rectangular form of the given number in complex form is\nTRY IT #6 Convert the complex number to rectangular form:\nFinding Products of Complex Numbers in Polar Form"
	},
	{
		"chunk_id" : "00002890",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Products of Complex Numbers in Polar Form\nNow that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers\nin polar form. For the rest of this section, we will work with formulas developed by French mathematician AbrahamDe\nMoivre(1667-1754). These formulas have made working with products, quotients, powers, and roots of complex\nnumbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments."
	},
	{
		"chunk_id" : "00002891",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Products of Complex Numbers in Polar Form\nIf and then the product of these numbers is given as:\nNotice that the product calls for multiplying the moduli and adding the angles.\nEXAMPLE8\nFinding the Product of Two Complex Numbers in Polar Form\nFind the product of given and\nSolution\nFollow the formula\nFinding Quotients of Complex Numbers in Polar Form\nThe quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two\narguments.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002892",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "arguments.\nAccess for free at openstax.org\n10.5  Polar Form of Complex Numbers 965\nQuotients of Complex Numbers in Polar Form\nIf and then the quotient of these numbers is\nNotice that the moduli are divided, and the angles are subtracted.\n...\nHOW TO\nGiven two complex numbers in polar form, find the quotient.\n1. Divide\n2. Find\n3. Substitute the results into the formula: Replace with and replace with\n4. Calculate the new trigonometric expressions and multiply through by\nEXAMPLE9"
	},
	{
		"chunk_id" : "00002893",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nFinding the Quotient of Two Complex Numbers\nFind the quotient of and\nSolution\nUsing the formula, we have\nTRY IT #7 Find the product and the quotient of and\nFinding Powers of Complex Numbers in Polar Form\nFinding powers of complex numbers is greatly simplified usingDe Moivres Theorem. It states that, for a positive\ninteger is found by raising the modulus to the power and multiplying the argument by It is the standard\nmethod used in modern mathematics.\nDe Moivres Theorem"
	},
	{
		"chunk_id" : "00002894",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "De Moivres Theorem\nIf is a complex number, then\nwhere is a positive integer.\n966 10  Further Applications of Trigonometry\nEXAMPLE10\nEvaluating an Expression Using De Moivres Theorem\nEvaluate the expression using De Moivres Theorem.\nSolution\nSince De Moivres Theorem applies to complex numbers written in polar form, we must first write in polar form.\nLet us find\nThen we find Using the formula gives\nUse De Moivres Theorem to evaluate the expression.\nFinding Roots of Complex Numbers in Polar Form"
	},
	{
		"chunk_id" : "00002895",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Roots of Complex Numbers in Polar Form\nTo find thenth root of a complex numberin polar form, we use the Root Theorem orDe Moivres Theoremand raise\nthe complex number to a power with a rational exponent. There are several ways to represent a formula for finding\nroots of complex numbers in polar form.\nThenth Root Theorem\nTo find the root of a complex number in polar form, use the formula given as\nwhere We add to in order to obtain the periodic roots.\nEXAMPLE11\nFinding thenth Root of a Complex Number"
	},
	{
		"chunk_id" : "00002896",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE11\nFinding thenth Root of a Complex Number\nEvaluate the cube roots of\nSolution\nWe have\nAccess for free at openstax.org\n10.5  Polar Form of Complex Numbers 967\nThere will be three roots: When we have\nWhen we have\nWhen we have\nRemember to find the common denominator to simplify fractions in situations like this one. For the angle\nsimplification is\nTRY IT #8 Find the four fourth roots of\nMEDIA\nAccess these online resources for additional instruction and practice with polar forms of complex numbers."
	},
	{
		"chunk_id" : "00002897",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Product and Quotient of Complex Numbers in Trigonometric Form(http://openstax.org/l/prodquocomplex)\nDe Moivres Theorem(http://openstax.org/l/demoivre)\n10.5 SECTION EXERCISES\nVerbal\n1. A complex number is 2. What does the absolute 3. How is a complex number\nExplain each part. value of a complex number converted to polar form?\nrepresent?\n4. How do we find the product 5. What is De Moivres\nof two complex numbers? Theorem and what is it used\nfor?\n968 10  Further Applications of Trigonometry\nAlgebraic"
	},
	{
		"chunk_id" : "00002898",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Algebraic\nFor the following exercises, find the absolute value of the given complex number.\n6. 7. 8.\n9. 10. 11.\nFor the following exercises, write the complex number in polar form.\n12. 13. 14.\n15. 16.\nFor the following exercises, convert the complex number from polar to rectangular form.\n17. 18. 19.\n20. 21. 22.\nFor the following exercises, find in polar form.\n23. 24.\n25. 26.\n27. 28.\nFor the following exercises, find in polar form.\n29. 30.\n31. 32.\n33. 34."
	},
	{
		"chunk_id" : "00002899",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "29. 30.\n31. 32.\n33. 34.\nFor the following exercises, find the powers of each complex number in polar form.\n35. Find when 36. Find when 37. Find when\n38. Find when 39. Find when 40. Find when\nAccess for free at openstax.org\n10.6  Parametric Equations 969\nFor the following exercises, evaluate each root.\n41. Evaluate the cube root of 42. Evaluate the square root of 43. Evaluate the cube root of\nwhen when when\n44. Evaluate the square root of 45. Evaluate the square root of\nwhen when\nGraphical"
	},
	{
		"chunk_id" : "00002900",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when when\nGraphical\nFor the following exercises, plot the complex number in the complex plane.\n46. 47. 48.\n49. 50. 51.\n52. 53. 54.\n55.\nTechnology\nFor the following exercises, find all answers rounded to the nearest hundredth.\n56. Use the rectangular to 57. Use the rectangular to 58. Use the rectangular to\npolar feature on the polar feature on the polar feature on the\ngraphing calculator to graphing calculator to graphing calculator to\nchange to polar change to polar change to polar\nform. form. form."
	},
	{
		"chunk_id" : "00002901",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form. form. form.\n59. Use the polar to 60. Use the polar to 61. Use the polar to\nrectangular feature on the rectangular feature on the rectangular feature on the\ngraphing calculator to graphing calculator to graphing calculator to\nchange to change to change to\nrectangular form. rectangular form. rectangular form.\n10.6 Parametric Equations\nLearning Objectives\nIn this section, you will:\nParameterize a curve.\nEliminate the parameter.\nFind a rectangular equation for a curve defined parametrically."
	},
	{
		"chunk_id" : "00002902",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find parametric equations for curves defined by rectangular equations.\nConsider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen inFigure 1.\nAt any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the\nequation for the position of the moon when the distance from the planet, the speed of the moons orbit around the"
	},
	{
		"chunk_id" : "00002903",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.\n970 10  Further Applications of Trigonometry\nFigure1\nIn this section, we will consider sets of equations given by and where is the independent variable of time. We\ncan use these parametric equations in a number of applications when we are looking for not only a particular position\nbut also the direction of the movement. As we trace out successive values of the orientation of the curve becomes"
	},
	{
		"chunk_id" : "00002904",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "clear. This is one of the primary advantages of usingparametric equations: we are able to trace the movement of an\nobject along a path according to time. We begin this section with a look at the basic components of parametric\nequations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the\nequations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves\ndefined by rectangular equations."
	},
	{
		"chunk_id" : "00002905",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "defined by rectangular equations.\nParameterizing a Curve\nWhen an object moves along a curveorcurvilinear pathin a given direction and in a given amount of time, the\nposition of the object in the plane is given by thex-coordinate and they-coordinate. However, both and vary over\ntime and so are functions of time. For this reason, we add another variable, theparameter, upon which both and are\ndependent functions. In the example in the section opener, the parameter is time, The position of the moon at time,"
	},
	{
		"chunk_id" : "00002906",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is represented as the function and the position of the moon at time, is represented as the function\nTogether, and are called parametric equations, and generate an ordered pair Parametric equations\nprimarily describe motion and direction.\nWhen we parameterize a curve, we are translating a single equation in two variables, such as and into an\nequivalent pair of equations in three variables, and One of the reasons we parameterize a curve is because the"
	},
	{
		"chunk_id" : "00002907",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parametric equations yield more information: specifically, the direction of the objects motion over time.\nWhen we graph parametric equations, we can observe the individual behaviors of and of There are a number of\nshapes that cannot be represented in the form meaning that they are not functions. For example, consider the\ngraph of a circle, given as Solving for gives or two equations: and\nIf we graph and together, the graph will not pass the vertical line test, as shown inFigure 2."
	},
	{
		"chunk_id" : "00002908",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thus, the equation for the graph of a circle is not a function.\nFigure2\nAccess for free at openstax.org\n10.6  Parametric Equations 971\nHowever, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would\nrepresent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar\nto the concept of creating parametric equations, as we use two functions to produce a non-function. This will become"
	},
	{
		"chunk_id" : "00002909",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "clearer as we move forward.\nParametric Equations\nSuppose is a number on an interval, The set of ordered pairs, where and forms a\nplane curve based on the parameter The equations and are the parametric equations.\nEXAMPLE1\nParameterizing a Curve\nParameterize the curve letting Graph both equations.\nSolution\nIf then to find we replace the variable with the expression given in In other words,\nMake a table of values similar toTable 1, and sketch the graph.\nTable1"
	},
	{
		"chunk_id" : "00002910",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nSee the graphs inFigure 3. It may be helpful to use theTRACEfeature of a graphing calculator to see how the points are\ngenerated as increases.\n972 10  Further Applications of Trigonometry\nFigure3 (a) Parametric (b) Rectangular\nAnalysis\nThe arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of\nTRY IT #1 Construct a table of values and plot the parametric equations:\nEXAMPLE2\nFinding a Pair of Parametric Equations"
	},
	{
		"chunk_id" : "00002911",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nFinding a Pair of Parametric Equations\nFind a pair of parametric equations that models the graph of using the parameter Plot some points\nand sketch the graph.\nSolution\nIf and we substitute for into the equation, then Our pair of parametric equations is\nTo graph the equations, first we construct a table of values like that inTable 2. We can choose values around from\nto The values in the column will be the same as those in the column because Calculate\nvalues for the column\nTable2"
	},
	{
		"chunk_id" : "00002912",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values for the column\nTable2\nAccess for free at openstax.org\n10.6  Parametric Equations 973\nThe graph of is a parabola facing downward, as shown inFigure 4. We have mapped the curve over the\ninterval shown as a solid line with arrows indicating the orientation of the curve according to Orientation\nrefers to the path traced along the curve in terms of increasing values of As this parabola is symmetric with respect to\nthe line the values of are reflected across they-axis.\nFigure4"
	},
	{
		"chunk_id" : "00002913",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\nTRY IT #2 Parameterize the curve given by\nEXAMPLE3\nFinding Parametric Equations That Model Given Criteria\nAn object travels at a steady rate along a straight path to in the same plane in four seconds. The\ncoordinates are measured in meters. Find parametric equations for the position of the object.\nSolution\nThe parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The"
	},
	{
		"chunk_id" : "00002914",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x-value of the object starts at meters and goes to 3 meters. This means the distancexhas changed by 8 meters in 4\nseconds, which is a rate of or We can write thex-coordinate as a linear function with respect to time as\nIn the linear function template and\nSimilarly, they-value of the object starts at 3 and goes to which is a change in the distanceyof 4 meters in 4\nseconds, which is a rate of or We can also write they-coordinate as the linear function"
	},
	{
		"chunk_id" : "00002915",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Together, these are the parametric equations for the position of the object, where and are expressed in meters and\nrepresents time:\nUsing these equations, we can build a table of values for and (seeTable 3). In this example, we limited values of\nto non-negative numbers. In general, any value of can be used.\nTable3\n974 10  Further Applications of Trigonometry\nTable3\nFrom this table, we can create three graphs, as shown inFigure 5."
	},
	{
		"chunk_id" : "00002916",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure5 (a) A graph of vs. representing the horizontal position over time. (b) A graph of vs. representing the\nvertical position over time. (c) A graph of vs. representing the position of the object in the plane at time\nAnalysis\nAgain, we see that, inFigure 5(c), when the parameter represents time, we can indicate the movement of the object\nalong the path with arrows.\nEliminating the Parameter"
	},
	{
		"chunk_id" : "00002917",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Eliminating the Parameter\nIn many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation\ninvolves only two variables, such as and Eliminating the parameter is a method that may make graphing some\ncurves easier. However, if we are concerned with the mapping of the equation according to time, then it will be\nnecessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter"
	},
	{
		"chunk_id" : "00002918",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from a set of parametric equations; not every method works for every type of equation. Here we will review the methods\nfor the most common types of equations.\nEliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations\nFor polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation\nthat is most easily manipulated and solve for We substitute the resulting expression for into the second equation.\nThis gives one equation in and\nEXAMPLE4"
	},
	{
		"chunk_id" : "00002919",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This gives one equation in and\nEXAMPLE4\nEliminating the Parameter in Polynomials\nGiven and eliminate the parameter, and write the parametric equations as a Cartesian\nequation.\nSolution\nWe will begin with the equation for because the linear equation is easier to solve for\nNext, substitute for in\nAccess for free at openstax.org\n10.6  Parametric Equations 975\nThe Cartesian form is\nAnalysis\nThis is an equation for a parabola in which, in rectangular terms, is dependent on From the curves vertex at"
	},
	{
		"chunk_id" : "00002920",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the graph sweeps out to the right. SeeFigure 6. In this section, we consider sets of equations given by the functions\nand where is the independent variable of time. Notice, both and are functions of time; so in general is not a\nfunction of\nFigure6\nTRY IT #3 Given the equations below, eliminate the parameter and write as a rectangular equation for as a\nfunction of\nEXAMPLE5\nEliminating the Parameter in Exponential Equations\nEliminate the parameter and write as a Cartesian equation: and\nSolution\nIsolate"
	},
	{
		"chunk_id" : "00002921",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nIsolate\nSubstitute the expression into\nThe Cartesian form is\n976 10  Further Applications of Trigonometry\nAnalysis\nThe graph of the parametric equation is shown inFigure 7(a). The domain is restricted to The Cartesian equation,\nis shown inFigure 7(b)and has only one restriction on the domain,\nFigure7\nEXAMPLE6\nEliminating the Parameter in Logarithmic Equations\nEliminate the parameter and write as a Cartesian equation: and\nSolution\nSolve the first equation for"
	},
	{
		"chunk_id" : "00002922",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSolve the first equation for\nThen, substitute the expression for into the equation.\nThe Cartesian form is\nAnalysis\nTo be sure that the parametric equations are equivalent to the Cartesian equation, check the domains. The parametric\nequations restrict the domain on to we restrict the domain on to The domain for the\nparametric equation is restricted to we limit the domain on to\nTRY IT #4 Eliminate the parameter and write as arectangular equation."
	},
	{
		"chunk_id" : "00002923",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Eliminating the Parameter from Trigonometric Equations\nEliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the\nfamiliar trigonometric identities and the Pythagorean Theorem.\nFirst, we use the identities:\nSolving for and we have\nAccess for free at openstax.org\n10.6  Parametric Equations 977\nThen, use the Pythagorean Theorem:\nSubstituting gives\nEXAMPLE7\nEliminating the Parameter from a Pair of Trigonometric Parametric Equations"
	},
	{
		"chunk_id" : "00002924",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Eliminate the parameter from the given pair oftrigonometric equationswhere and sketch the graph.\nSolution\nSolving for and we have\nNext, use the Pythagorean identity and make the substitutions.\nThe graph for the equation is shown inFigure 8.\nFigure8\nAnalysis\nApplying the general equations forconic sections(introduced inAnalytic Geometry, we can identify as an\nellipse centered at Notice that when the coordinates are and when the coordinates are"
	},
	{
		"chunk_id" : "00002925",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This shows the orientation of the curve with increasing values of\nTRY IT #5 Eliminate the parameter from the given pair of parametric equations and write as a Cartesian\n978 10  Further Applications of Trigonometry\nequation: and\nFinding Cartesian Equations from Curves Defined Parametrically\nWhen we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially"
	},
	{
		"chunk_id" : "00002926",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "eliminating the parameter. However, there are various methods we can use to rewrite a set of parametric equations as\na Cartesian equation. The simplest method is to set one equation equal to the parameter, such as In this case,\ncan be any expression. For example, consider the following pair of equations.\nRewriting this set of parametric equations is a matter of substituting for Thus, the Cartesian equation is\nEXAMPLE8\nFinding a Cartesian Equation Using Alternate Methods"
	},
	{
		"chunk_id" : "00002927",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use two different methods to find the Cartesian equation equivalent to the given set of parametric equations.\nSolution\nMethod 1. First, lets solve the equation for Then we can substitute the result into the equation.\nNow substitute the expression for into the equation.\nMethod 2. Solve the equation for and substitute this expression in the equation.\nMake the substitution and then solve for\nTRY IT #6 Write the given parametric equations as a Cartesian equation: and"
	},
	{
		"chunk_id" : "00002928",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Parametric Equations for Curves Defined by Rectangular Equations\nAlthough we have just shown that there is only one way to interpret a set of parametric equations as a rectangular\nequation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we\nmay use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression"
	},
	{
		"chunk_id" : "00002929",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to represent and then substitute it into the equation, and it produces the same graph over the same domain as the\nAccess for free at openstax.org\n10.6  Parametric Equations 979\nrectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of\nparametric equations, and the function does not allow the same values for as the domain of the rectangular equation,\nthen the graphs will be different.\nEXAMPLE9"
	},
	{
		"chunk_id" : "00002930",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "then the graphs will be different.\nEXAMPLE9\nFinding a Set of Parametric Equations for Curves Defined by Rectangular Equations\nFind a set of equivalent parametric equations for\nSolution\nAn obvious choice would be to let Then But lets try something more interesting. What if we\nlet Then we have\nThe set of parametric equations is\nSeeFigure 9.\nFigure9\nMEDIA\nAccess these online resources for additional instruction and practice with parametric equations."
	},
	{
		"chunk_id" : "00002931",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Introduction to Parametric Equations(http://openstax.org/l/introparametric)\nConverting Parametric Equations to Rectangular Form(http://openstax.org/l/convertpara)\n980 10  Further Applications of Trigonometry\n10.6 SECTION EXERCISES\nVerbal\n1. What is a system of 2. Some examples of a third 3. Explain how to eliminate a\nparametric equations? parameter are time, length, parameter given a set of\nspeed, and scale. Explain parametric equations.\nwhen time is used as a\nparameter."
	},
	{
		"chunk_id" : "00002932",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "when time is used as a\nparameter.\n4. What is a benefit of writing a 5. What is a benefit of using 6. Why are there many sets of\nsystem of parametric parametric equations? parametric equations to\nequations as a Cartesian represent on Cartesian\nequation? function?\nAlgebraic\nFor the following exercises, eliminate the parameter to rewrite the parametric equation as a Cartesian equation.\n7. 8. 9.\n10. 11. 12.\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25."
	},
	{
		"chunk_id" : "00002933",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25.\nFor the following exercises, rewrite the parametric equation as a Cartesian equation by building an table.\n26. 27. 28.\n29.\nAccess for free at openstax.org\n10.6  Parametric Equations 981\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting or\nby setting\n30. 31. 32.\n33.\nFor the following exercises, parameterize (write parametric equations for) each Cartesian equation by using\nand Identify the curve.\n34. 35. 36."
	},
	{
		"chunk_id" : "00002934",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and Identify the curve.\n34. 35. 36.\n37. 38. Parameterize the line from 39. Parameterize the line from\nto so that to so that\nthe line is at at the line is at at\nand at at and at at\n40. Parameterize the line from 41. Parameterize the line from\nto so that the to so that the\nline is at at line is at at and\nand at at at at\nTechnology\nFor the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.\n42. 43."
	},
	{
		"chunk_id" : "00002935",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "42. 43.\nFor the following exercises, use a graphing calculator to complete the table of values for each set of parametric\nequations.\n44. 45. 46.\n1 1 -1\n0 2 0\n1 3 1\n2\n982 10  Further Applications of Trigonometry\nExtensions\n47. Find two different sets of 48. Find two different sets of 49. Find two different sets of\nparametric equations for parametric equations for parametric equations for\n10.7 Parametric Equations: Graphs\nLearning Objectives\nIn this section you will:"
	},
	{
		"chunk_id" : "00002936",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section you will:\nGraph plane curves described by parametric equations by plotting points.\nGraph parametric equations.\nWhile not every fan (or team manager) appreciates it, baseball and many other sports have become dependent on\nanalytics, which involve complex data recording and quantitative evaluation used to understand and predict behavior.\nThe earliest influence of analytics was mostly statistical; more recently, physics and other sciences have come into play."
	},
	{
		"chunk_id" : "00002937",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Foremost among these is the focus on launch angle and exit velocity, which when at certain values can almost guarantee\na home run. On the other hand, emphasis on launch angle and focusing on home runs rather than overall hitting results\nin far more outs. Consider the following situation: it is the bottom of the ninth inning, with two outs and two players on\nbase. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an"
	},
	{
		"chunk_id" : "00002938",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle of approximately to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home\nrun? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path\nof a projectile and predict approximately how far it will travel usingparametric equations. In this section, well discuss\nparametric equations and some common applications, such as projectile motion problems."
	},
	{
		"chunk_id" : "00002939",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1 Parametric equations can model the path of a projectile. (credit: Paul Kreher, Flickr)\nGraphing Parametric Equations by Plotting Points\nIn lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is\nthe standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.\n...\nHOW TO\nGiven a pair of parametric equations, sketch a graph by plotting points.\n1. Construct a table with three columns:"
	},
	{
		"chunk_id" : "00002940",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Construct a table with three columns:\n2. Evaluate and for values of over the interval for which the functions are defined.\n3. Plot the resulting pairs\nAccess for free at openstax.org\n10.7  Parametric Equations: Graphs 983\nEXAMPLE1\nSketching the Graph of a Pair of Parametric Equations by Plotting Points\nSketch the graph of theparametric equations\nSolution\nConstruct a table of values for and as inTable 1, and plot the points in a plane.\nTable1"
	},
	{
		"chunk_id" : "00002941",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Table1\nThe graph is aparabolawith vertex at the point opening to the right. SeeFigure 2.\nFigure2\nAnalysis\nAs values for progress in a positive direction from 0 to 5, the plotted points trace out the top half of the parabola. As\nvalues of become negative, they trace out the lower half of the parabola. There are no restrictions on the domain. The\n984 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002942",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "984 10  Further Applications of Trigonometry\narrows indicate direction according to increasing values of The graph does not represent a function, as it will fail the\nvertical line test. The graph is drawn in two parts: the positive values for and the negative values for\nTRY IT #1 Sketch the graph of the parametric equations\nEXAMPLE2\nSketching the Graph of Trigonometric Parametric Equations\nConstruct a table of values for the given parametric equations and sketch the graph:\nSolution"
	},
	{
		"chunk_id" : "00002943",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nConstruct a table like that inTable 2using angle measure in radians as inputs for and evaluating and Using angles\nwith known sine and cosine values for makes calculations easier.\n0\nTable2\nFigure 3shows the graph.\nAccess for free at openstax.org\n10.7  Parametric Equations: Graphs 985\nFigure3\nBy the symmetry shown in the values of and we see that the parametric equations represent anellipse. Theellipseis\nmapped in a counterclockwise direction as shown by the arrows indicating increasing values."
	},
	{
		"chunk_id" : "00002944",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe have seen that parametric equations can be graphed by plotting points. However, a graphing calculator will save\nsome time and reveal nuances in a graph that may be too tedious to discover using only hand calculations.\nMake sure to change the mode on the calculator to parametric (PAR). To confirm, the window should show\ninstead of\nTRY IT #2 Graph the parametric equations:\nEXAMPLE3\nGraphing Parametric Equations and Rectangular Form Together"
	},
	{
		"chunk_id" : "00002945",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph the parametric equations and First, construct the graph using data points generated from\ntheparametric form. Then graph therectangular formof the equation. Compare the two graphs.\nSolution\nConstruct a table of values like that inTable 3.\nTable3\n986 10  Further Applications of Trigonometry\nTable3\nPlot the values from the table. SeeFigure 4.\nFigure4\nNext, translate the parametric equations to rectangular form. To do this, we solve for in either or and then"
	},
	{
		"chunk_id" : "00002946",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitute the expression for in the other equation. The result will be a function if solving for as a function of\nor if solving for as a function of\nThen, use thePythagorean Theorem.\nAnalysis\nInFigure 5, the data from the parametric equations and the rectangular equation are plotted together. The parametric\nequations are plotted in blue; the graph for the rectangular equation is drawn on top of the parametric in a dashed style\ncolored red. Clearly, both forms produce the same graph."
	},
	{
		"chunk_id" : "00002947",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n10.7  Parametric Equations: Graphs 987\nFigure5\nEXAMPLE4\nGraphing Parametric Equations and Rectangular Equations on the Coordinate System\nGraph the parametric equations and and the rectangular equivalent on the same\ncoordinate system.\nSolution\nConstruct a table of values for the parametric equations, as we did in the previous example, and graph on\nthe same grid, as inFigure 6.\nFigure6\nAnalysis"
	},
	{
		"chunk_id" : "00002948",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the same grid, as inFigure 6.\nFigure6\nAnalysis\nWith the domain on restricted, we only plot positive values of The parametric data is graphed in blue and the graph\nof the rectangular equation is dashed in red. Once again, we see that the two forms overlap.\nTRY IT #3 Sketch the graph of the parametric equations along with the\nrectangular equation on the same grid.\nApplications of Parametric Equations\nMany of the advantages of parametric equations become obvious when applied to solving real-world problems."
	},
	{
		"chunk_id" : "00002949",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Although rectangular equations inxandygive an overall picture of an object's path, they do not reveal the position of\nan object at a specific time. Parametric equations, however, illustrate how the values ofxandychange depending ont,\nas the location of a moving object at a particular time.\nA common application of parametric equations is solving problems involving projectile motion. In this type of motion, an"
	},
	{
		"chunk_id" : "00002950",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "object is propelled forward in an upward direction forming an angle of to the horizontal, with an initial speed of\nand at a height above the horizontal.\n988 10  Further Applications of Trigonometry\nThe path of an object propelled at an inclination of to the horizontal, with initial speed and at a height above the\nhorizontal, is given by\nwhere accounts for the effects ofgravityand is the initial height of the object. Depending on the units involved in the"
	},
	{
		"chunk_id" : "00002951",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "problem, use or The equation for gives horizontal distance, and the equation for gives the\nvertical distance.\n...\nHOW TO\nGiven a projectile motion problem, use parametric equations to solve.\n1. The horizontal distance is given by Substitute the initial speed of the object for\n2. The expression indicates the angle at which the object is propelled. Substitute that angle in degrees for\n3. The vertical distance is given by the formula The term represents the effect"
	},
	{
		"chunk_id" : "00002952",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of gravity. Depending on units involved, use or Again, substitute the initial speed for\nand the height at which the object was propelled for\n4. Proceed by calculating each term to solve for\nEXAMPLE5\nFinding the Parametric Equations to Describe the Motion of a Baseball\nSolve the problem presented at the beginning of this section. Does the batter hit the game-winning home run? Assume\nthat the ball is hit with an initial velocity of 140 feet per second at an angle of to the horizontal, making contact 3 feet"
	},
	{
		"chunk_id" : "00002953",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "above the ground.\n Find the parametric equations to model the path of the baseball.  Where is the ball after 2 seconds?\n How long is the ball in the air?  Is it a home run?\nSolution\n\nUse the formulas to set up the equations. The horizontal position is found using the parametric equation for Thus,\nThe vertical position is found using the parametric equation for Thus,\n\nSubstitute 2 into the equations to find the horizontal and vertical positions of the ball."
	},
	{
		"chunk_id" : "00002954",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "After 2 seconds, the ball is 198 feet away from the batters box and 137 feet above the ground.\nAccess for free at openstax.org\n10.7  Parametric Equations: Graphs 989\n\nTo calculate how long the ball is in the air, we have to find out when it will hit ground, or when Thus,\nWhen seconds, the ball has hit the ground. (The quadratic equation can be solved in various ways, but this\nproblem was solved using a computer math program.)\n"
	},
	{
		"chunk_id" : "00002955",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "\nWe cannot confirm that the hit was a home run without considering the size of the outfield, which varies from field to\nfield. However, for simplicitys sake, lets assume that the outfield wall is 400 feet from home plate in the deepest part\nof the park. Lets also assume that the wall is 10 feet high. In order to determine whether the ball clears the wall, we\nneed to calculate how high the ball is whenx= 400 feet. So we will setx= 400, solve for and input into"
	},
	{
		"chunk_id" : "00002956",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The ball is 141.8 feet in the air when it soars out of the ballpark. It was indeed a home run. SeeFigure 7.\nFigure7\nMEDIA\nAccess the following online resource for additional instruction and practice with graphs of parametric equations.\nGraphing Parametric Equations on the TI-84(http://openstax.org/l/graphpara84)\n10.7 SECTION EXERCISES\nVerbal\n1. What are two methods used 2. What is one difference in 3. Why are some graphs drawn\nto graph parametric point-plotting parametric with arrows?"
	},
	{
		"chunk_id" : "00002957",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations? equations compared to\nCartesian equations?\n4. Name a few common types 5. Why are parametric graphs\nof graphs of parametric important in understanding\nequations. projectile motion?\n990 10  Further Applications of Trigonometry\nGraphical\nFor the following exercises, graph each set of parametric equations by making a table of values. Include the orientation\non the graph.\n6. 7.\n8. 9. 10.\n11.\nFor the following exercises, sketch the curve and include the orientation.\n12. 13. 14.\n15. 16. 17."
	},
	{
		"chunk_id" : "00002958",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13. 14.\n15. 16. 17.\n18. 19. 20.\n21. 22.\nAccess for free at openstax.org\n10.7  Parametric Equations: Graphs 991\nFor the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.\n23. 24. 25.\n26. 27.\nFor the following exercises, graph the equation and include the orientation.\n28. 29. 30.\n31. 32. 33.\nFor the following exercises, use the parametric equations for integersaandb:\n34. Graph on the domain 35. Graph on the domain 36. Graph on the domain"
	},
	{
		"chunk_id" : "00002959",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where and where and where and\nand include the , and include the , and include the\norientation. orientation. orientation.\n37. Graph on the domain 38. If is 1 more than 39. Describe the graph if\nwhere and describe the effect the and\n, and include the values of and have on\norientation. the graph of the\nparametric equations.\n40. What happens if is 1 41. If the parametric equations\nmore than Describe the and\ngraph. have the graph of a\nhorizontal parabola\nopening to the right, what\nwould change the direction"
	},
	{
		"chunk_id" : "00002960",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would change the direction\nof the curve?\nFor the following exercises, describe the graph of the set of parametric equations.\n42. and is 43. and is linear 44. and is\nlinear linear\n45. Write the parametric 46. Write the parametric\nequations of a circle with equations of an ellipse with\ncenter radius 5, and center major axis of\na counterclockwise length 10, minor axis of\norientation. length 6, and a\ncounterclockwise\norientation.\n992 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002961",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "992 10  Further Applications of Trigonometry\nFor the following exercises, use a graphing utility to graph on the window by on the domain for the\nfollowing values of and , and include the orientation.\n47. 48. 49.\n50. 51. 52.\nTechnology\nFor the following exercises, look at the graphs that were created by parametric equations of the form\nUse the parametric mode on the graphing calculator to find the values of and to achieve each graph.\n53.\n54.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00002962",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "53.\n54.\nAccess for free at openstax.org\n10.7  Parametric Equations: Graphs 993\n55.\n56.\nFor the following exercises, use a graphing utility to graph the given parametric equations.\na.\nb.\nc.\n57. Graph all three sets of 58. Graph all three sets of 59. Graph all three sets of\nparametric equations on parametric equations on parametric equations on\nthe domain the domain the domain\n994 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002963",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "994 10  Further Applications of Trigonometry\n60. The graph of each set of 61. Explain the effect on the 62. Explain the effect on the\nparametric equations graph of the parametric graph of the parametric\nappears to creep along equation when we equation when we\none of the axes. What switched and . changed the domain.\ncontrols which axis the\ngraph creeps along?\nExtensions\n63. An object is thrown in the air with vertical velocity 64. A skateboarder riding on a level surface at a"
	},
	{
		"chunk_id" : "00002964",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 20 ft/s and horizontal velocity of 15 ft/s. The constant speed of 9 ft/s throws a ball in the air,\nobjects height can be described by the equation the height of which can be described by the\n, while the object moves equation Write\nhorizontally with constant velocity 15 ft/s. Write parametric equations for the balls position, and\nparametric equations for the objects position, then eliminate time to write height as a function\nand then eliminate time to write height as a of horizontal position."
	},
	{
		"chunk_id" : "00002965",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function of horizontal position.\nFor the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 65 ft/s at an angle of\nelevation of 52. Consider the position of the dart at any time Neglect air resistance.\n65. Find parametric equations 66. Find all possible values of 67. When will the dart hit the\nthat model the problem that represent the ground?\nsituation. situation.\n68. Find the maximum height 69. At what time will the dart\nof the dart. reach maximum height?"
	},
	{
		"chunk_id" : "00002966",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the dart. reach maximum height?\nFor the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and\nbeautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each\non the indicated domain.\n70. An epicycloid: on the 71. A hypocycloid: on the\ndomain . domain .\n72. A hypotrochoid: on 73. A rose: on the domain\nthe domain . .\n10.8 Vectors\nLearning Objectives\nIn this section you will:"
	},
	{
		"chunk_id" : "00002967",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section you will:\nView vectors geometrically.\nFind magnitude and direction.\nPerform vector addition and scalar multiplication.\nFind the component form of a vector.\nFind the unit vector in the direction of .\nPerform operations with vectors in terms of and .\nFind the dot product of two vectors.\nAn airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140. A north wind (from north to\nAccess for free at openstax.org\n10.8  Vectors 995"
	},
	{
		"chunk_id" : "00002968",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.8  Vectors 995\nsouth) is blowing at 16.2 miles per hour, as shown inFigure 1. What are the ground speed and actual bearing of the\nplane?\nFigure1\nGround speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel\nrelative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier\nsection, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find"
	},
	{
		"chunk_id" : "00002969",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the airplanes groundspeed and bearing, while investigating another approach to problems of this type. First, however,\nlets examine the basics of vectors.\nA Geometric View of Vectors\nAvectoris a specific quantity drawn as a line segment with an arrowhead at one end. It has aninitial point, where it\nbegins, and aterminal point, where it ends. A vector is defined by itsmagnitude, or the length of the line, and its"
	},
	{
		"chunk_id" : "00002970",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various\nsymbols that distinguish vectors from other quantities:\n Lower case, boldfaced type, with or without an arrow on top such as\n Given initial point and terminal point a vector can be represented as The arrowhead on top is what\nindicates that it is not just a line, but a directed line segment.\n Given an initial point of and terminal point a vector may be represented as"
	},
	{
		"chunk_id" : "00002971",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This last symbol has special significance. It is called thestandard position. Theposition vectorhas an initial point\nand a terminal point To change any vector into the position vector, we think about the change in the\nx-coordinates and the change in they-coordinates. Thus, if the initial point of a vector is and the terminal\npoint is then the position vector is found by calculating\nInFigure 2, we see the original vector and the position vector\nFigure2\n996 10  Further Applications of Trigonometry"
	},
	{
		"chunk_id" : "00002972",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "996 10  Further Applications of Trigonometry\nProperties of Vectors\nA vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or\nthe length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position\nvector has an initial point at and is identified by its terminal point\nEXAMPLE1\nFind the Position Vector\nConsider the vector whose initial point is and terminal point is Find the position vector.\nSolution"
	},
	{
		"chunk_id" : "00002973",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe position vector is found by subtracting onex-coordinate from the otherx-coordinate, and oney-coordinate from the\nothery-coordinate. Thus\nThe position vector begins at and terminates at The graphs of both vectors are shown inFigure 3.\nFigure3\nWe see that the position vector is\nEXAMPLE2\nDrawing a Vector with the Given Criteria and Its Equivalent Position Vector\nFind the position vector given that vector has an initial point at and a terminal point at then graph both\nvectors in the same plane."
	},
	{
		"chunk_id" : "00002974",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vectors in the same plane.\nSolution\nThe position vector is found using the following calculation:\nThus, the position vector begins at and terminates at SeeFigure 4.\nAccess for free at openstax.org\n10.8  Vectors 997\nFigure4\nTRY IT #1 Draw a vector that connects from the origin to the point\nFinding Magnitude and Direction\nTo work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the"
	},
	{
		"chunk_id" : "00002975",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.\nMagnitude and Direction of a Vector\nGiven a position vector the magnitude is found by The direction is equal to the angle\nformed with thex-axis, or with they-axis, depending on the application. For a position vector, the direction is found\nby as illustrated inFigure 5.\nFigure5\nTwo vectorsvanduare considered equal if they have the same magnitude and the same direction. Additionally, if"
	},
	{
		"chunk_id" : "00002976",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both vectors have the same position vector, they are equal.\nEXAMPLE3\nFinding the Magnitude and Direction of a Vector\nFind the magnitude and direction of the vector with initial point and terminal point Draw the\nvector.\nSolution\nFirst, find theposition vector.\n998 10  Further Applications of Trigonometry\nWe use the Pythagorean Theorem to find the magnitude.\nThe direction is given as\nHowever, the angle terminates in the fourth quadrant, so we add 360 to obtain a positive angle. Thus,\nSeeFigure 6.\nFigure6"
	},
	{
		"chunk_id" : "00002977",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeFigure 6.\nFigure6\nEXAMPLE4\nShowing That Two Vectors Are Equal\nShow that vectorvwithinitial pointat andterminal pointat is equal to vectoruwith initial point at\nand terminal point at Draw the position vector on the same grid asvandu. Next, find the magnitude\nand direction of each vector.\nSolution\nAs shown inFigure 7, draw the vector starting at initial and terminal point Draw the vector with initial\npoint and terminal point Find the standard position for each."
	},
	{
		"chunk_id" : "00002978",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Next, find and sketch the position vector forvandu. We have\nSince the position vectors are the same,vanduare the same.\nAn alternative way to check for vector equality is to show that the magnitude and direction are the same for both\nvectors. To show that the magnitudes are equal, use the Pythagorean Theorem.\nAccess for free at openstax.org\n10.8  Vectors 999\nAs the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector\ngives"
	},
	{
		"chunk_id" : "00002979",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "gives\nHowever, we can see that the position vector terminates in the second quadrant, so we add Thus, the direction is\nFigure7\nPerforming Vector Addition and Scalar Multiplication\nNow that we understand the properties of vectors, we can perform operations involving them. While it is convenient to\nthink of the vector as an arrow or directed line segment from the origin to the point vectors can be\nsituated anywhere in the plane. The sum of two vectorsuandv, orvector addition, produces a third vectoru+v, the"
	},
	{
		"chunk_id" : "00002980",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "resultantvector.\nTo findu+v, we first draw the vectoru, and from the terminal end ofu, we drawn the vectorv. In other words, we have\nthe initial point ofvmeet the terminal end ofu. This position corresponds to the notion that we move along the first\nvector and then, from its terminal point, we move along the second vector. The sumu+vis the resultant vector because\nit results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning ofuto the"
	},
	{
		"chunk_id" : "00002981",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "end ofvin a straight path, as shown inFigure 8.\nFigure8\nVector subtraction is similar to vector addition. To finduv, view it asu+ (v). Adding vis reversing direction ofvand\nadding it to the end ofu. The new vector begins at the start ofuand stops at the end point of v. SeeFigure 9for a\nvisual that compares vector addition and vector subtraction usingparallelograms.\n1000 10  Further Applications of Trigonometry\nFigure9\nEXAMPLE5\nAdding and Subtracting Vectors\nGiven and find two new vectorsu+v, anduv."
	},
	{
		"chunk_id" : "00002982",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given and find two new vectorsu+v, anduv.\nSolution\nTo find the sum of two vectors, we add the components. Thus,\nSeeFigure 10(a).\nTo find the difference of two vectors, add the negative components of to Thus,\nSeeFigure 10(b).\nFigure10 (a) Sum of two vectors (b) Difference of two vectors\nMultiplying By a Scalar\nWhile adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of"
	},
	{
		"chunk_id" : "00002983",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "multiplying a vector by ascalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar\nmultiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting\nvector is opposite the direction of the original vector.\nScalar Multiplication\nScalar multiplicationinvolves the product of a vector and a scalar. Each component of the vector is multiplied by the\nscalar. Thus, to multiply by , we have"
	},
	{
		"chunk_id" : "00002984",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "scalar. Thus, to multiply by , we have\nOnly the magnitude changes, unless is negative, and then the vector reverses direction.\nAccess for free at openstax.org\n10.8  Vectors 1001\nEXAMPLE6\nPerforming Scalar Multiplication\nGiven vector find 3v, and v.\nSolution\nSeeFigure 11for a geometric interpretation. If then\nFigure11\nAnalysis\nNotice that the vector 3vis three times the length ofv, is half the length ofv, and vis the same length ofv, but in\nthe opposite direction.\nTRY IT #2 Find thescalar multiple3 given"
	},
	{
		"chunk_id" : "00002985",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #2 Find thescalar multiple3 given\nEXAMPLE7\nUsing Vector Addition and Scalar Multiplication to Find a New Vector\nGiven and find a new vectorw= 3u+ 2v.\nSolution\nFirst, we must multiply each vector by the scalar.\nThen, add the two together.\n1002 10  Further Applications of Trigonometry\nSo,\nFinding Component Form\nIn some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors"
	},
	{
		"chunk_id" : "00002986",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "are comprised of two components: the horizontal component is the direction, and the vertical component is the\ndirection. For example, we can see in the graph inFigure 12that the position vector comes from adding the\nvectorsv andv. We havev with initial point and terminal point\n1 2 1\nWe also havev with initial point and terminal point\n2\nTherefore, the position vector is\nUsing the Pythagorean Theorem, the magnitude ofv is 2, and the magnitude ofv is 3. To find the magnitude ofv, use\n1 2"
	},
	{
		"chunk_id" : "00002987",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 2\nthe formula with the position vector.\nThe magnitude ofvis To find the direction, we use the tangent function\nFigure12\nThus, the magnitude of is and the direction is off the horizontal.\nAccess for free at openstax.org\n10.8  Vectors 1003\nEXAMPLE8\nFinding the Components of the Vector\nFind the components of the vector with initial point and terminal point\nSolution\nFirst find the standard position.\nSee the illustration inFigure 13.\nFigure13\nThe horizontal component is and the vertical component is"
	},
	{
		"chunk_id" : "00002988",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Unit Vector in the Direction ofv\nIn addition to finding a vectors components, it is also useful in solving problems to find a vector in the same direction as\nthe given vector, but of magnitude 1. We call a vector with a magnitude of 1 aunit vector. We can then preserve the\ndirection of the original vector while simplifying calculations.\nUnit vectors are defined in terms of components. The horizontal unit vector is written as and is directed along"
	},
	{
		"chunk_id" : "00002989",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the positive horizontal axis. The vertical unit vector is written as and is directed along the positive vertical axis.\nSeeFigure 14.\nFigure14\nThe Unit Vectors\nIf is a nonzero vector, then is a unit vector in the direction of Any vector divided by its magnitude is a unit\nvector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal\nof the scalar.\n1004 10  Further Applications of Trigonometry\nEXAMPLE9\nFinding the Unit Vector in the Direction ofv"
	},
	{
		"chunk_id" : "00002990",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Unit Vector in the Direction ofv\nFind a unit vector in the same direction as\nSolution\nFirst, we will find the magnitude.\nThen we divide each component by which gives a unit vector in the same direction asv:\nor, in component form\nSeeFigure 15.\nFigure15\nVerify that the magnitude of the unit vector equals 1. The magnitude of is given as\nThe vectoru i jis the unit vector in the same direction asv\nPerforming Operations with Vectors in Terms ofiandj"
	},
	{
		"chunk_id" : "00002991",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar\nmultiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar\nwith the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of\niandj.\nAccess for free at openstax.org\n10.8  Vectors 1005\nVectors in the Rectangular Plane"
	},
	{
		"chunk_id" : "00002992",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Vectors in the Rectangular Plane\nGiven a vector with initial point and terminal point vis written as\nThe position vector from to where and is written asv=ai+bj. This vector\nsum is called a linear combination of the vectorsiandj.\nThe magnitude ofv=ai+bjis given as SeeFigure 16.\nFigure16\nEXAMPLE10\nWriting a Vector in Terms ofiandj\nGiven a vector with initial point and terminal point write the vector in terms of and\nSolution"
	},
	{
		"chunk_id" : "00002993",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nBegin by writing the general form of the vector. Then replace the coordinates with the given values.\nEXAMPLE11\nWriting a Vector in Terms ofiandjUsing Initial and Terminal Points\nGiven initial point and terminal point write the vector in terms of and\nSolution\nBegin by writing the general form of the vector. Then replace the coordinates with the given values.\nTRY IT #3 Write the vector with initial point and terminal point in terms of and\nPerforming Operations on Vectors in Terms ofiandj"
	},
	{
		"chunk_id" : "00002994",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Performing Operations on Vectors in Terms ofiandj\nWhen vectors are written in terms of and we can carry out addition, subtraction, and scalar multiplication by\nperforming operations on corresponding components.\n1006 10  Further Applications of Trigonometry\nAdding and Subtracting Vectors in Rectangular Coordinates\nGivenv=ai+bjandu=ci+dj, then\nEXAMPLE12\nFinding the Sum of the Vectors\nFind the sum of and\nSolution\nAccording to the formula, we have\nCalculating the Component Form of a Vector: Direction"
	},
	{
		"chunk_id" : "00002995",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We\nhave also examined notation for vectors drawn specifically in the Cartesian coordinate plane using For any of\nthese vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of\nmagnitude and direction.\nCalculating direction follows the same straightforward process we used for polar coordinates. We find the direction of"
	},
	{
		"chunk_id" : "00002996",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with\nreplacing\nVector Components in Terms of Magnitude and Direction\nGiven a position vector and a direction angle\nThus, and magnitude is expressed as\nEXAMPLE13\nWriting a Vector in Terms of Magnitude and Direction\nWrite a vector with length 7 at an angle of 135 to the positivex-axis in terms of magnitude and direction.\nSolution\nUsing the conversion formulas and we find that"
	},
	{
		"chunk_id" : "00002997",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using the conversion formulas and we find that\nThis vector can be written as or simplified as\nAccess for free at openstax.org\n10.8  Vectors 1007\nTRY IT #4 A vector travels from the origin to the point Write the vector in terms of magnitude and\ndirection.\nFinding the Dot Product of Two Vectors\nAs we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a"
	},
	{
		"chunk_id" : "00002998",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a\nvector, there are two possibilities: thedot productand thecross product. We will only examine the dot product here; you\nmay encounter the cross product in more advanced mathematics courses.\nThe dot product of two vectors involves multiplying two vectors together, and the result is a scalar.\nDot Product"
	},
	{
		"chunk_id" : "00002999",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Dot Product\nThedot productof two vectors and is the sum of the product of the horizontal components and\nthe product of the vertical components.\nTo find the angle between the two vectors, use the formula below.\nEXAMPLE14\nFinding the Dot Product of Two Vectors\nFind the dot product of and\nSolution\nUsing the formula, we have\nEXAMPLE15\nFinding the Dot Product of Two Vectors and the Angle between Them\nFind the dot product ofv = 5i+ 2jandv = 3i+ 7j. Then, find the angle between the two vectors.\n1 2\nSolution"
	},
	{
		"chunk_id" : "00003000",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 2\nSolution\nFinding the dot product, we multiply corresponding components.\nTo find the angle between them, we use the formula\n1008 10  Further Applications of Trigonometry\nSeeFigure 17.\nFigure17\nEXAMPLE16\nFinding the Angle between Two Vectors\nFind the angle between and\nSolution\nUsing the formula, we have\nSeeFigure 18.\nAccess for free at openstax.org\n10.8  Vectors 1009\nFigure18\nEXAMPLE17\nFinding Ground Speed and Bearing Using Vectors"
	},
	{
		"chunk_id" : "00003001",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding Ground Speed and Bearing Using Vectors\nWe now have the tools to solve the problem we introduced in the opening of the section.\nAn airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140. A north wind (from north to\nsouth) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane? SeeFigure 19.\nFigure19\nSolution\nThe ground speed is represented by in the diagram, and we need to find the angle in order to calculate the adjusted"
	},
	{
		"chunk_id" : "00003002",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bearing, which will be\nNotice inFigure 19, that angle must be equal to angle by the rule of alternating interior angles, so angle\nis 140. We can find by the Law of Cosines:\nThe ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.\n1010 10  Further Applications of Trigonometry\nTherefore, the plane has a SE bearing of 140+2.8=142.8. The ground speed is 212.7 miles per hour.\nMEDIA"
	},
	{
		"chunk_id" : "00003003",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with vectors.\nIntroduction to Vectors(http://openstax.org/l/introvectors)\nVector Operations(http://openstax.org/l/vectoroperation)\nThe Unit Vector(http://openstax.org/l/unitvector)\n10.8 SECTION EXERCISES\nVerbal\n1. What are the characteristics 2. How is a vector more 3. What are and and what\nof the letters that are specific than a line do they represent?\ncommonly used to segment?\nrepresent vectors?"
	},
	{
		"chunk_id" : "00003004",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "commonly used to segment?\nrepresent vectors?\n4. What is component form? 5. When a unit vector is\nexpressed as which\nletter is the coefficient of the\nand which the\nAlgebraic\n6. Given a vector with initial 7. Given a vector with initial 8. Given a vector with initial\npoint and terminal point and terminal point and terminal\npoint find an point find an point find an\nequivalent vector whose equivalent vector whose equivalent vector whose\ninitial point is Write initial point is Write initial point is Write"
	},
	{
		"chunk_id" : "00003005",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the vector in component the vector in component the vector in component\nform form form\nFor the following exercises, determine whether the two vectors and are equal, where has an initial point and a\nterminal point and has an initial point and a terminal point .\n9. and 10. and\n11. and 12. and\n13. and 14. Given initial point and terminal point\nwrite the vector in terms of and\nAccess for free at openstax.org\n10.8  Vectors 1011\n15. Given initial point and terminal point\nwrite the vector in terms of and"
	},
	{
		"chunk_id" : "00003006",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "write the vector in terms of and\nFor the following exercises, use the vectorsu=i+ 5j,v= 2i 3j,andw= 4ij.\n16. Findu+ (vw) 17. Find 4v+ 2u\nFor the following exercises, use the given vectors to computeu+v,uv, and 2u 3v.\n18. 19. 20. Letv= 4i+ 3j. Find a\nvector that is half the\nlength and points in the\nsame direction as\n21. Letv= 5i+ 2j. Find a vector\nthat is twice the length and\npoints in the opposite\ndirection as\nFor the following exercises, find a unit vector in the same direction as the given vector."
	},
	{
		"chunk_id" : "00003007",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "22. a= 3i+ 4j 23. b= 2i+ 5j 24. c= 10ij\n25. 26. u= 100i+ 200j 27. u= 14i+ 2j\nFor the following exercises, find the magnitude and direction of the vector,\n28. 29. 30.\n31. 32. Givenu= 3i 4jandv= 2i 33. Givenu= ijandv=i+\n+ 3j, calculate 5j, calculate\n34. Given and 35. Givenu andv\ncalculate calculate\nGraphical\nFor the following exercises, given draw 3vand\n36. 37. 38.\n1012 10  Further Applications of Trigonometry\nFor the following exercises, use the vectors shown to sketchu+v,uv, and 2u.\n39. 40. 41."
	},
	{
		"chunk_id" : "00003008",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "39. 40. 41.\nFor the following exercises, use the vectors shown to sketch 2u+v.\n42. 43.\nFor the following exercises, use the vectors shown to sketchu 3v.\n44. 45.\nAccess for free at openstax.org\n10.8  Vectors 1013\nFor the following exercises, write the vector shown in component form.\n46. 47. 48. Given initial point\nand terminal\npoint write\nthe vector in terms of\nand then draw the vector\non the graph.\n49. Given initial point 50. Given initial point\nand terminal and terminal\npoint write point write"
	},
	{
		"chunk_id" : "00003009",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and terminal and terminal\npoint write point write\nthe vector in terms of the vector in terms of\nand Draw the points and and Draw the points and\nthe vector on the graph. the vector on the graph.\nExtensions\nFor the following exercises, use the given magnitude and direction in standard position, write the vector in component\nform.\n51. 52. 53.\n54. 55. A 60-pound box is resting 56. A 25-pound box is resting\non a ramp that is inclined on a ramp that is inclined\n12. Rounding to the 8. Rounding to the nearest"
	},
	{
		"chunk_id" : "00003010",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. Rounding to the 8. Rounding to the nearest\nnearest tenth, tenth,\n Find the magnitude of  Find the magnitude of\nthe normal (perpendicular) the normal (perpendicular)\ncomponent of the force. component of the force.\n Find the magnitude of  Find the magnitude of\nthe component of the force the component of the force\nthat is parallel to the ramp. that is parallel to the ramp.\n57. Find the magnitude of the 58. Find the magnitude of the 59. Find the magnitude of the"
	},
	{
		"chunk_id" : "00003011",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontal and vertical horizontal and vertical horizontal and vertical\ncomponents of a vector components of the vector components of a vector\nwith magnitude 8 pounds with magnitude 4 pounds with magnitude 5 pounds\npointed in a direction of pointed in a direction of pointed in a direction of\n27 above the horizontal. 127 above the horizontal. 55 above the horizontal.\nRound to the nearest Round to the nearest Round to the nearest\nhundredth. hundredth. hundredth.\n60. Find the magnitude of the"
	},
	{
		"chunk_id" : "00003012",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "60. Find the magnitude of the\nhorizontal and vertical\ncomponents of the vector\nwith magnitude 1 pound\npointed in a direction of 8\nabove the horizontal.\nRound to the nearest\nhundredth.\n1014 10  Further Applications of Trigonometry\nReal-World Applications\n61. A woman leaves home and 62. A boat leaves the marina 63. A man starts walking from\nwalks 3 miles west, then 2 and sails 6 miles north, home and walks 4 miles\nmiles southwest. How far then 2 miles northeast. east, 2 miles southeast, 5"
	},
	{
		"chunk_id" : "00003013",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from home is she, and in How far from the marina is miles south, 4 miles\nwhat direction must she the boat, and in what southwest, and 2 miles\nwalk to head directly direction must it sail to east. How far has he\nhome? head directly back to the walked? If he walked\nmarina? straight home, how far\nwould he have to walk?\n64. A woman starts walking 65. A man starts walking from 66. A woman starts walking\nfrom home and walks 4 home and walks 3 miles at from home and walks 6"
	},
	{
		"chunk_id" : "00003014",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "miles east, 7 miles 20 north of west, then 5 miles at 40 north of east,\nsoutheast, 6 miles south, 5 miles at 10 west of south, then 2 miles at 15 east of\nmiles southwest, and 3 then 4 miles at 15 north of south, then 5 miles at 30\nmiles east. How far has she east. If he walked straight south of west. If she\nwalked? If she walked home, how far would he walked straight home, how\nstraight home, how far have to the walk, and in far would she have to walk,"
	},
	{
		"chunk_id" : "00003015",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would she have to walk? what direction? and in what direction?\n67. An airplane is heading 68. An airplane is heading 69. An airplane needs to head\nnorth at an airspeed of 600 north at an airspeed of 500 due north, but there is a\nkm/hr, but there is a wind km/hr, but there is a wind wind blowing from the\nblowing from the blowing from the southwest at 60 km/hr. The\nsouthwest at 80 km/hr. northwest at 50 km/hr. plane flies with an airspeed\nHow many degrees off How many degrees off of 550 km/hr. To end up"
	},
	{
		"chunk_id" : "00003016",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "course will the plane end course will the plane end flying due north, how many\nup flying, and what is the up flying, and what is the degrees west of north will\nplanes speed relative to planes speed relative to the pilot need to fly the\nthe ground? the ground? plane?\n70. An airplane needs to head 71. As part of a video game, 72. As part of a video game,\ndue north, but there is a the point is rotated the point is rotated\nwind blowing from the counterclockwise about the counterclockwise about the"
	},
	{
		"chunk_id" : "00003017",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "northwest at 80 km/hr. The origin through an angle of origin through an angle of\nplane flies with an airspeed 35. Find the new 40. Find the new\nof 500 km/hr. To end up coordinates of this point. coordinates of this point.\nflying due north, how many\ndegrees west of north will\nthe pilot need to fly the\nplane?\nAccess for free at openstax.org\n10.8  Vectors 1015\n73. Two children are throwing 74. Two children are throwing 75. A 50-pound object rests on"
	},
	{
		"chunk_id" : "00003018",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a ball back and forth a ball back and forth a ramp that is inclined 19.\nstraight across the back straight across the back Find the magnitude of the\nseat of a car. The ball is seat of a car. The ball is components of the force\nbeing thrown 10 mph being thrown 8 mph parallel to and\nrelative to the car, and the relative to the car, and the perpendicular to (normal)\ncar is traveling 25 mph car is traveling 45 mph the ramp to the nearest\ndown the road. If one child down the road. If one child tenth of a pound."
	},
	{
		"chunk_id" : "00003019",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "doesn't catch the ball, and doesn't catch the ball, and\nit flies out the window, in it flies out the window, in\nwhat direction does the what direction does the\nball fly (ignoring wind ball fly (ignoring wind\nresistance)? resistance)?\n76. Suppose a body has a force 77. Suppose a body has a force 78. The condition of\nof 10 pounds acting on it to of 10 pounds acting on it to equilibrium is when the\nthe right, 25 pounds acting the right, 25 pounds acting sum of the forces acting on"
	},
	{
		"chunk_id" : "00003020",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on it upward, and 5 pounds on it 135 from the a body is the zero vector.\nacting on it 45 from the horizontal, and 5 pounds Suppose a body has a force\nhorizontal. What single acting on it directed 150 of 2 pounds acting on it to\nforce is the resultant force from the horizontal. What the right, 5 pounds acting\nacting on the body? single force is the resultant on it upward, and 3 pounds\nforce acting on the body? acting on it 45 from the\nhorizontal. What single\nforce is needed to produce"
	},
	{
		"chunk_id" : "00003021",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "force is needed to produce\na state of equilibrium on\nthe body?\n79. Suppose a body has a force\nof 3 pounds acting on it to\nthe left, 4 pounds acting on\nit upward, and 2 pounds\nacting on it 30 from the\nhorizontal. What single\nforce is needed to produce\na state of equilibrium on\nthe body? Draw the vector.\n1016 10  Chapter Review\nChapter Review\nKey Terms\naltitude a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to"
	},
	{
		"chunk_id" : "00003022",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the line containing the opposite side, forming two right triangles\nambiguous case a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle\nArchimedes spiral a polar curve given by When multiplied by a constant, the equation appears as As\nthe curve continues to widen in a spiral path over the domain.\nargument the angle associated with a complex number; the angle between the line from the origin to the point and\nthe positive real axis"
	},
	{
		"chunk_id" : "00003023",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the positive real axis\ncardioid a member of the limaon family of curves, named for its resemblance to a heart; its equation is given as\nand where\nconvex limaon a type of one-loop limaon represented by and such that\nDe Moivres Theorem formula used to find the power ornth roots of a complex number; states that, for a positive\ninteger is found by raising the modulus to the power and multiplying the angles by\ndimpled limaon a type of one-loop limaon represented by and such that"
	},
	{
		"chunk_id" : "00003024",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "dot product given two vectors, the sum of the product of the horizontal components and the product of the vertical\ncomponents\nGeneralized Pythagorean Theorem an extension of the Law of Cosines; relates the sides of an oblique triangle and is\nused for SAS and SSS triangles\ninitial point the origin of a vector\ninner-loop limaon a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice;\nrepresented by and where"
	},
	{
		"chunk_id" : "00003025",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represented by and where\nLaw of Cosines states that the square of any side of a triangle is equal to the sum of the squares of the other two sides\nminus twice the product of the other two sides and the cosine of the included angle\nLaw of Sines states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is\nequal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for\na missing angle or side"
	},
	{
		"chunk_id" : "00003026",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a missing angle or side\nlemniscate a polar curve resembling a figure 8 and given by the equation and\nmagnitude the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean\nTheorem\nmodulus the absolute value of a complex number, or the distance from the origin to the point also called the\namplitude\noblique triangle any triangle that is not a right triangle\none-loop limaon a polar curve represented by and such that and"
	},
	{
		"chunk_id" : "00003027",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "may be dimpled or convex; does not pass through the pole\nparameter a variable, often representing time, upon which and are both dependent\npolar axis on the polar grid, the equivalent of the positivex-axis on the rectangular grid\npolar coordinates on the polar grid, the coordinates of a point labeled where indicates the angle of rotation\nfrom the polar axis and represents the radius, or the distance of the point from the pole in the direction of"
	},
	{
		"chunk_id" : "00003028",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "polar equation an equation describing a curve on the polar grid.\npolar form of a complex number a complex number expressed in terms of an angle and its distance from the origin\ncan be found by using conversion formulas and\npole the origin of the polar grid\nresultant a vector that results from addition or subtraction of two vectors, or from scalar multiplication\nrose curve a polar equation resembling a flower, given by the equations and when is even"
	},
	{
		"chunk_id" : "00003029",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "there are petals, and the curve is highly symmetrical; when is odd there are petals.\nscalar a quantity associated with magnitude but not direction; a constant\nscalar multiplication the product of a constant and each component of a vector\nstandard position the placement of a vector with the initial point at and the terminal point represented by\nthe change in thex-coordinates and the change in they-coordinates of the original vector"
	},
	{
		"chunk_id" : "00003030",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terminal point the end point of a vector, usually represented by an arrow indicating its direction\nunit vector a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along thex-axis\nand is defined as the vertical unit vector runs along they-axis and is defined as\nvector a quantity associated with both magnitude and direction, represented as a directed line segment with a starting\npoint (initial point) and an end point (terminal point)"
	},
	{
		"chunk_id" : "00003031",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vector addition the sum of two vectors, found by adding corresponding components\nAccess for free at openstax.org\n10  Chapter Review 1017\nKey Equations\nLaw of Sines\nArea for oblique triangles\nLaw of Cosines\nHerons formula\nConversion formulas\nKey Concepts\n10.1Non-right Triangles: Law of Sines\n The Law of Sines can be used to solve oblique triangles, which are non-right triangles.\n According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side"
	},
	{
		"chunk_id" : "00003032",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equals the other two ratios of angle measure to opposite side.\n There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate\nequation to find the requested solution. SeeExample 1.\n The ambiguous case arises when an oblique triangle can have different outcomes.\n There are three possible cases that arise from SSA arrangementa single solution, two possible solutions, and no\nsolution. SeeExample 2andExample 3."
	},
	{
		"chunk_id" : "00003033",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution. SeeExample 2andExample 3.\n The Law of Sines can be used to solve triangles with given criteria. SeeExample 4.\n The general area formula for triangles translates to oblique triangles by first finding the appropriate height value.\nSeeExample 5.\n There are many trigonometric applications. They can often be solved by first drawing a diagram of the given\ninformation and then using the appropriate equation. SeeExample 6.\n10.2Non-right Triangles: Law of Cosines"
	},
	{
		"chunk_id" : "00003034",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.2Non-right Triangles: Law of Cosines\n The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.\n The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS.\nDropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle,\nwhich allows sides to be related and measurements to be calculated. SeeExample 1andExample 2."
	},
	{
		"chunk_id" : "00003035",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally\nto draw a sketch of the problem presented. If the information given fits one of the three models (the three\nequations), then apply the Law of Cosines to find a solution. SeeExample 3andExample 4.\n Herons formula allows the calculation of area in oblique triangles. All three sides must be known to apply Herons\nformula. SeeExample 5and SeeExample 6.\n1018 10  Chapter Review"
	},
	{
		"chunk_id" : "00003036",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1018 10  Chapter Review\n10.3Polar Coordinates\n The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.\n To plot a point in the form move in a counterclockwise direction from the polar axis by an angle of\nand then extend a directed line segment from the pole the length of in the direction of If is negative, move in a\nclockwise direction, and extend a directed line segment the length of in the direction of SeeExample 1."
	},
	{
		"chunk_id" : "00003037",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If is negative, extend the directed line segment in the opposite direction of SeeExample 2.\n To convert from polar coordinates to rectangular coordinates, use the formulas and See\nExample 3andExample 4.\n To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:\nand SeeExample 5.\n Transforming equations between polar and rectangular forms means making the appropriate substitutions based"
	},
	{
		"chunk_id" : "00003038",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on the available formulas, together with algebraic manipulations. SeeExample 6,Example 7, andExample 8.\n Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and\nthen graph it in the rectangular plane. SeeExample 9,Example 10, andExample 11.\n10.4Polar Coordinates: Graphs\n It is easier to graph polar equations if we can test the equations for symmetry with respect to the line the\npolar axis, or the pole."
	},
	{
		"chunk_id" : "00003039",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "polar axis, or the pole.\n There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an\nequation fails a symmetry test, the graph may or may not exhibit symmetry. SeeExample 1.\n Polar equations may be graphed by making a table of values for and\n The maximum value of a polar equation is found by substituting the value that leads to the maximum value of the\ntrigonometric expression."
	},
	{
		"chunk_id" : "00003040",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "trigonometric expression.\n The zeros of a polar equation are found by setting and solving for SeeExample 2.\n Some formulas that produce the graph of a circle in polar coordinates are given by and See\nExample 3.\n The formulas that produce the graphs of a cardioid are given by and for\nand SeeExample 4.\n The formulas that produce the graphs of a one-loop limaon are given by and for\nSeeExample 5.\n The formulas that produce the graphs of an inner-loop limaon are given by and for\nand SeeExample 6."
	},
	{
		"chunk_id" : "00003041",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and SeeExample 6.\n The formulas that produce the graphs of a lemniscates are given by and where\nSeeExample 7.\n The formulas that produce the graphs of rose curves are given by and where if is\neven, there are petals, and if is odd, there are petals. SeeExample 8andExample 9.\n The formula that produces the graph of an Archimedes spiral is given by SeeExample 10.\n10.5Polar Form of Complex Numbers\n Complex numbers in the form are plotted in the complex plane similar to the way rectangular coordinates are"
	},
	{
		"chunk_id" : "00003042",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "plotted in the rectangular plane. Label thex-axis as therealaxis and they-axis as theimaginaryaxis. SeeExample 1.\n The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:\nSeeExample 2andExample 3.\n To write complex numbers in polar form, we use the formulas and Then,\nSeeExample 4andExample 5.\n To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through\nby SeeExample 6andExample 7."
	},
	{
		"chunk_id" : "00003043",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by SeeExample 6andExample 7.\n To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the\ntrigonometric functions, and multiply using the distributive property. SeeExample 8.\n To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference\nof the two angles. SeeExample 9.\n To find the power of a complex number raise to the power and multiply by SeeExample 10."
	},
	{
		"chunk_id" : "00003044",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational\nexponent. SeeExample 11.\nAccess for free at openstax.org\n10  Chapter Review 1019\n10.6Parametric Equations\n Parameterizing a curve involves translating a rectangular equation in two variables, and into two equations in\nthree variables,x,y, andt. Often, more information is obtained from a set of parametric equations. SeeExample 1,\nExample 2, andExample 3."
	},
	{
		"chunk_id" : "00003045",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 2, andExample 3.\n Sometimes equations are simpler to graph when written in rectangular form. By eliminating an equation in and\nis the result.\n To eliminate solve one of the equations for and substitute the expression into the second equation. See\nExample 4,Example 5,Example 6, andExample 7.\n Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the"
	},
	{
		"chunk_id" : "00003046",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parameter. Solve for in one of the equations, and substitute the expression into the second equation. SeeExample\n8.\n There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular\nequation.\n Find an expression for such that the domain of the set of parametric equations remains the same as the original\nrectangular equation. SeeExample 9.\n10.7Parametric Equations: Graphs"
	},
	{
		"chunk_id" : "00003047",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "10.7Parametric Equations: Graphs\n When there is a third variable, a third parameter on which and depend, parametric equations can be used.\n To graph parametric equations by plotting points, make a table with three columns labeled and Choose\nvalues for in increasing order. Plot the last two columns for and SeeExample 1andExample 2.\n When graphing a parametric curve by plotting points, note the associatedt-values and show arrows on the graph"
	},
	{
		"chunk_id" : "00003048",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "indicating the orientation of the curve. SeeExample 3andExample 4.\n Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that\nare not functions can be graphed and used in many applications involving motion. SeeExample 5.\n Projectile motion depends on two parametric equations: and Initial\nvelocity is symbolized as represents the initial angle of the object when thrown, and represents the height at\nwhich the object is propelled.\n10.8Vectors"
	},
	{
		"chunk_id" : "00003049",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "which the object is propelled.\n10.8Vectors\n The position vector has its initial point at the origin. SeeExample 1.\n If the position vector is the same for two vectors, they are equal. SeeExample 2.\n Vectors are defined by their magnitude and direction. SeeExample 3.\n If two vectors have the same magnitude and direction, they are equal. SeeExample 4.\n Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See\nExample 5."
	},
	{
		"chunk_id" : "00003050",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 5.\n Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the\nsame. SeeExample 6andExample 7.\n Vectors are comprised of two components: the horizontal component along the positivex-axis, and the vertical\ncomponent along the positivey-axis. SeeExample 8.\n The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude."
	},
	{
		"chunk_id" : "00003051",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The magnitude of a vector in the rectangular coordinate system is SeeExample 9.\n In the rectangular coordinate system, unit vectors may be represented in terms of and where represents the\nhorizontal component and represents the vertical component. Then,v= ai+ bjis a scalar multiple of by real\nnumbers SeeExample 10andExample 11.\n Adding and subtracting vectors in terms ofiandjconsists of adding or subtracting corresponding coefficients ofi\nand corresponding coefficients ofj. SeeExample 12."
	},
	{
		"chunk_id" : "00003052",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A vectorv=ai+bjis written in terms of magnitude and direction as SeeExample 13.\n The dot product of two vectors is the product of the terms plus the product of the terms. SeeExample 14.\n We can use the dot product to find the angle between two vectors.Example 15andExample 16.\n Dot products are useful for many types of physics applications. SeeExample 17.\n1020 10  Exercises\nExercises\nReview Exercises\nNon-right Triangles: Law of Sines"
	},
	{
		"chunk_id" : "00003053",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Non-right Triangles: Law of Sines\nFor the following exercises, assume is opposite side is opposite side and is opposite side Solve each triangle,\nif possible. Round each answer to the nearest tenth.\n1. 2. 3. Solve the triangle.\n4. Find the area of the triangle. 5. A pilot is flying over a straight\nhighway. He determines the angles\nof depression to two mileposts, 2.1\nkm apart, to be 25 and 49, as\nshown inFigure 1. Find the\ndistance of the plane from point\nand the elevation of the plane.\nFigure1"
	},
	{
		"chunk_id" : "00003054",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and the elevation of the plane.\nFigure1\nNon-right Triangles: Law of Cosines\n6. Solve the triangle, rounding 7. Solve the triangle inFigure 8. Find the area of a triangle\nto the nearest tenth, 2, rounding to the nearest with sides of length 8.3, 6.6,\nassuming is opposite side tenth. and 9.1.\nis opposite side and\ns opposite side\nFigure2\nAccess for free at openstax.org\n10  Exercises 1021\n9. To find the distance between two\ncities, a satellite calculates the\ndistances and angle shown in"
	},
	{
		"chunk_id" : "00003055",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distances and angle shown in\nFigure 3(not to scale). Find the\ndistance between the cities. Round\nanswers to the nearest tenth.\nFigure3\nPolar Coordinates\n10. Plot the point with polar 11. Plot the point with polar 12. Convert to\ncoordinates coordinates rectangular coordinates.\n13. Convert to 14. Convert to polar 15. Convert to polar\nrectangular coordinates. coordinates. coordinates.\nFor the following exercises, convert the given Cartesian equation to a polar equation.\n16. 17. 18."
	},
	{
		"chunk_id" : "00003056",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "16. 17. 18.\nFor the following exercises, convert the given polar equation to a Cartesian equation.\n19. 20.\nFor the following exercises, convert to rectangular form and graph.\n21. 22.\nPolar Coordinates: Graphs\nFor the following exercises, test each equation for symmetry.\n23. 24. 25. Sketch a graph of the polar\nequation\nLabel the axis intercepts.\n26. Sketch a graph of the polar 27. Sketch a graph of the polar\nequation equation\n1022 10  Exercises\nPolar Form of Complex Numbers"
	},
	{
		"chunk_id" : "00003057",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1022 10  Exercises\nPolar Form of Complex Numbers\nFor the following exercises, find the absolute value of each complex number.\n28. 29.\nWrite the complex number in polar form.\n30. 31.\nFor the following exercises, convert the complex number from polar to rectangular form.\n32. 33.\nFor the following exercises, find the product in polar form.\n34. 35.\nFor the following exercises, find the quotient in polar form.\n36. 37.\nFor the following exercises, find the powers of each complex number in polar form."
	},
	{
		"chunk_id" : "00003058",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "38. Find when 39. Find when\nFor the following exercises, evaluate each root.\n40. Evaluate the cube root of 41. Evaluate the square root of\nwhen when\nFor the following exercises, plot the complex number in the complex plane.\n42. 43.\nAccess for free at openstax.org\n10  Exercises 1023\nParametric Equations\nFor the following exercises, eliminate the parameter to rewrite the parametric equation as a Cartesian equation.\n44. 45. 46. Parameterize (write a\nparametric equation for)\neach Cartesian equation by"
	},
	{
		"chunk_id" : "00003059",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each Cartesian equation by\nusing and\nfor\n47. Parameterize the line from\nto so that the\nline is at at\nand at\nParametric Equations: Graphs\nFor the following exercises, make a table of values for each set of parametric equations, graph the equations, and\ninclude an orientation; then write the Cartesian equation.\n48. 49. 50.\n51. A ball is launched with an initial velocity of 80 feet\nper second at an angle of 40 to the horizontal.\nThe ball is released at a height of 4 feet above the\nground."
	},
	{
		"chunk_id" : "00003060",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ground.\n Find the parametric equations to model the\npath of the ball.\n Where is the ball after 3 seconds?\n How long is the ball in the air?\nVectors\nFor the following exercises, determine whether the two vectors, and are equal, where has an initial point and a\nterminal point and has an initial point and a terminal point\n52. and 53. and\nFor the following exercises, use the vectors and to evaluate the expression.\n54. uv 55. 2vu+w"
	},
	{
		"chunk_id" : "00003061",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "54. uv 55. 2vu+w\nFor the following exercises, find a unit vector in the same direction as the given vector.\n56. a= 8i 6j 57. b= 3ij\n1024 10  Exercises\nFor the following exercises, find the magnitude and direction of the vector.\n58. 59.\nFor the following exercises, calculate\n60. u= 2i+jandv= 3i+ 7j 61. u=i+ 4jandv= 4i+ 3j 62. Givenv drawv,\n2v, and v.\n63. Given the vectors shown inFigure 46,4. Given initial point\nsketchu+v,uvand 3v. and terminal\npoint write\nthe vector in terms of"
	},
	{
		"chunk_id" : "00003062",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "point write\nthe vector in terms of\nand Draw the points and\nthe vector on the graph.\nFigure4\nPractice Test\n1. Assume is opposite side 2. Find the area of the triangle 3. A pilot flies in a straight path\nis opposite side and inFigure 1. Round each for 2 hours. He then makes\nis opposite side Solve the answer to the nearest tenth. a course correction, heading\ntriangle, if possible, and 15 to the right of his\nround each answer to the original course, and flies 1"
	},
	{
		"chunk_id" : "00003063",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nearest tenth, given hour in the new direction. If\nhe maintains a constant\nspeed of 575 miles per hour,\nhow far is he from his\nstarting position?\nFigure1\n4. Convert to polar 5. Convert to 6. Convert the polar equation\ncoordinates, and then plot rectangular coordinates. to a Cartesian equation:\nthe point.\n7. Convert to rectangular form 8. Test the equation for 9. Graph\nand graph: symmetry:\n10. Graph 11. Find the absolute value of 12. Write the complex number\nthe complex number in polar form:"
	},
	{
		"chunk_id" : "00003064",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the complex number in polar form:\nAccess for free at openstax.org\n10  Exercises 1025\n13. Convert the complex\nnumber from polar to\nrectangular form:\nGiven and evaluate each expression.\n14. 15. 16.\n17. 18. Plot the complex number 19. Eliminate the parameter\nin the complex to rewrite the following\nplane. parametric equations as a\nCartesian equation:\n20. Parameterize (write a 21. Graph the set of 22. A ball is launched with an\nparametric equation for) parametric equations and initial velocity of 95 feet per"
	},
	{
		"chunk_id" : "00003065",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the following Cartesian find the Cartesian second at an angle of 52\nequation by using equation: to the horizontal. The ball\nand is released at a height of\n3.5 feet above the ground.\n Find the parametric\nequations to model the\npath of the ball.\n Where is the ball after 2\nseconds?\n How long is the ball in\nthe air?\nFor the following exercises, use the vectorsu=i 3jandv= 2i+ 3j.\n23. Find 2u 3v. 24. Calculate 25. Find a unit vector in the\nsame direction as\n26. Given vector has an initial\npoint and"
	},
	{
		"chunk_id" : "00003066",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "26. Given vector has an initial\npoint and\nterminal point\nwrite the\nvector in terms of and\nOn the graph, draw and\n1026 10  Exercises\nAccess for free at openstax.org\n11  Introduction 1027\n11 SYSTEMS OF EQUATIONS AND INEQUALITIES\nEnigma machines like this one were used by government and military officials for enciphering and deciphering top-\nsecret communications during World War II. By varying the combinations of the plugboard and the settings of the"
	},
	{
		"chunk_id" : "00003067",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rotors, encoders could add complex encryption to their messages. Notice that the three rotors each contain 26 pins, one\nfor each letter of the alphabet; later versions had four and five rotors. (credit: modification of \"Enigma Machine\"\" by"
	},
	{
		"chunk_id" : "00003068",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.4Partial Fractions\n11.5Matrices and Matrix Operations\n11.6Solving Systems with Gaussian Elimination\n11.7Solving Systems with Inverses\n11.8Solving Systems with Cramer's Rule\nIntroduction to Systems of Equations and Inequalities\nAt the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany\nwould require the Allies to know what the enemy was planning. This task was complicated by the fact that the German"
	},
	{
		"chunk_id" : "00003069",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "military transmitted all of its communications through a presumably uncrackable code created by a machine called\nEnigma. The Germans had been encoding their messages with this machine since the early 1930s, and were so confident\nin its security that they used it for everyday military communications as well as highly important strategic messages.\nConcerned about the increasing military threat, other European nations began working to decipher the Enigma codes."
	},
	{
		"chunk_id" : "00003070",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Poland was the first country to make significant advances when it trained and recruited a new group of codebreakers:\nmath students from Pozna University. With the help of intelligence obtained by French spies, Polish mathematicians,\nled by Marian Rejewski, were able to decipher initial codes and later to understand the wiring of the machines;\neventually they create replicas. However, the German military eventually increased the complexity of the machines by"
	},
	{
		"chunk_id" : "00003071",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "adding additional rotors, requiring a new method of decryption.\nThe machine attached letters on a keyboard to three, four, or five rotors (depending on the version), each with 26\nstarting positions that could be set prior to encoding; a decryption code (called a cipher key) essentially conveyed these\nsettings to the message recipient, and allowed people to interpret the message using another Enigma machine. Even"
	},
	{
		"chunk_id" : "00003072",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with the simpler three-rotor scrambler, there were 17,576 different combinations of starting positions (26 x 26 x 26); plus\nthe machine had numerous other methods of introducing variation. Not long after the war started, the British recruited\na team of brilliant codebreakers to crack the Enigma code. The codebreakers, led by Alan Turing, used what they knew\n1028 11  Systems of Equations and Inequalities"
	},
	{
		"chunk_id" : "00003073",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1028 11  Systems of Equations and Inequalities\nabout the Enigma machine to build a mechanical computer that could crack the code. And that knowledge of what the\nGermans were planning proved to be a key part of the ultimate Allied victory of Nazi Germany in 1945.\nThe Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role\ncryptography has played in society. Now, technology has moved cryptanalysis to the digital world."
	},
	{
		"chunk_id" : "00003074",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a\nmatrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must\nalso know the key that, when used with the matrix inverse, will allow the message to be read.\nIn this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of"
	},
	{
		"chunk_id" : "00003075",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial\nfractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.\n11.1 Systems of Linear Equations: Two Variables\nLearning Objectives\nIn this section, you will:\nSolve systems of equations by graphing.\nSolve systems of equations by substitution.\nSolve systems of equations by addition.\nIdentify inconsistent systems of equations containing two variables."
	},
	{
		"chunk_id" : "00003076",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Express the solution of a system of dependent equations containing two variables.\nFigure1 (credit: Thomas Srenes)\nA skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it\nspends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the\ncompany determine if it is making a profit with its new line? How many skateboards must be produced and sold before a"
	},
	{
		"chunk_id" : "00003077",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "profit is possible? In this section, we will consider linear equations with two variables to answer these and similar\nquestions.\nIntroduction to Systems of Equations\nIn order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing\nwith more than one variable and likely more than one equation. Asystem of linear equationsconsists of two or more"
	},
	{
		"chunk_id" : "00003078",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "linear equations made up of two or more variables such that all equations in the system are considered simultaneously.\nTo find the unique solution to a system of linear equations, we must find a numerical value for each variable in the\nsystem that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and\nothers may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at"
	},
	{
		"chunk_id" : "00003079",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "least as many equations as there are variables. Even so, this does not guarantee a unique solution.\nIn this section, we will look at systems of linear equations in two variables, which consist of two equations that contain\ntwo different variables. For example, consider the following system of linear equations in two variables.\nThesolutionto a system of linear equations in two variables is any ordered pair that satisfies each equation"
	},
	{
		"chunk_id" : "00003080",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the\nsolution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will\ninvestigate methods of finding such a solution if it exists.\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1029"
	},
	{
		"chunk_id" : "00003081",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In addition to considering the number of equations and variables, we can categorize systems of linear equations by the\nnumber of solutions. Aconsistent systemof equations has at least one solution. A consistent system is considered to\nbe anindependent systemif it has a single solution, such as the example we just explored. The two lines have different\nslopes and intersect at one point in the plane. A consistent system is considered to be adependent systemif the"
	},
	{
		"chunk_id" : "00003082",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations have the same slope and the samey-intercepts. In other words, the lines coincide so the equations represent\nthe same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite\nnumber of solutions.\nAnother type of system of linear equations is aninconsistent system, which is one in which the equations represent\ntwo parallel lines. The lines have the same slope and differenty-intercepts. There are no points common to both lines;"
	},
	{
		"chunk_id" : "00003083",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hence, there is no solution to the system.\nTypes of Linear Systems\nThere are three types of systems of linear equations in two variables, and three types of solutions.\n Anindependent systemhas exactly one solution pair The point where the two lines intersect is the only\nsolution.\n Aninconsistent systemhas no solution. Notice that the two lines are parallel and will never intersect.\n Adependent systemhas infinitely many solutions. The lines are coincident. They are the same line, so every"
	},
	{
		"chunk_id" : "00003084",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coordinate pair on the line is a solution to both equations.\nFigure 2compares graphical representations of each type of system.\nFigure2\n...\nHOW TO\nGiven a system of linear equations and an ordered pair, determine whether the ordered pair is a solution.\n1. Substitute the ordered pair into each equation in the system.\n2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a\nsolution.\nEXAMPLE1"
	},
	{
		"chunk_id" : "00003085",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution.\nEXAMPLE1\nDetermining Whether an Ordered Pair Is a Solution to a System of Equations\nDetermine whether the ordered pair is a solution to the given system of equations.\n1030 11  Systems of Equations and Inequalities\nSolution\nSubstitute the ordered pair into both equations.\nThe ordered pair satisfies both equations, so it is the solution to the system.\nAnalysis\nWe can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies"
	},
	{
		"chunk_id" : "00003086",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both equations, it is a point on both of the lines and thus the point of intersection of the two lines. SeeFigure 3.\nFigure3\nTRY IT #1 Determine whether the ordered pair is a solution to the following system.\nSolving Systems of Equations by Graphing\nThere are multiple methods of solving systems of linear equations. For asystem of linear equationsin two variables, we\ncan determine both the type of system and the solution by graphing the system of equations on the same set of axes.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00003087",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nSolving a System of Equations in Two Variables by Graphing\nSolve the following system of equations by graphing. Identify the type of system.\nSolution\nSolve the first equation for\nSolve the second equation for\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1031\nGraph both equations on the same set of axes as inFigure 4.\nFigure4\nThe lines appear to intersect at the point We can check to make sure that this is the solution to the system by"
	},
	{
		"chunk_id" : "00003088",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substituting the ordered pair into both equations.\nThe solution to the system is the ordered pair so the system is independent.\nTRY IT #2 Solve the following system of equations by graphing.\nQ&A Can graphing be used if the system is inconsistent or dependent?\nYes, in both cases we can still graph the system to determine the type of system and solution. If the two\nlines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system"
	},
	{
		"chunk_id" : "00003089",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has infinite solutions and is a dependent system.\nSolving Systems of Equations by Substitution\nSolving a linear system in two variables by graphing works well when the solution consists of integer values, but if our\nsolution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving\nasystem of linear equationsthat are more precise than graphing. One such method is solving a system of equations by"
	},
	{
		"chunk_id" : "00003090",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thesubstitution method, in which we solve one of the equations for one variable and then substitute the result into the\nsecond equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the\nreason the substitution method is both valuable and practical.\n...\nHOW TO\nGiven a system of two equations in two variables, solve using the substitution method.\n1. Solve one of the two equations for one of the variables in terms of the other."
	},
	{
		"chunk_id" : "00003091",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Substitute the expression for this variable into the second equation, then solve for the remaining variable.\n3. Substitute that solution into either of the original equations to find the value of the first variable. If possible,\nwrite the solution as an ordered pair.\n4. Check the solution in both equations.\n1032 11  Systems of Equations and Inequalities\nEXAMPLE3\nSolving a System of Equations in Two Variables by Substitution\nSolve the following system of equations by substitution.\nSolution"
	},
	{
		"chunk_id" : "00003092",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nFirst, we will solve the first equation for\nNow we can substitute the expression for in the second equation.\nNow, we substitute into the first equation and solve for\nOur solution is\nCheck the solution by substituting into both equations.\nTRY IT #3 Solve the following system of equations by substitution.\nQ&A Can the substitution method be used to solve any linear system in two variables?\nYes, but the method works best if one of the equations contains a coefficient of 1 or 1 so that we do not"
	},
	{
		"chunk_id" : "00003093",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "have to deal with fractions.\nSolving Systems of Equations in Two Variables by the Addition Method\nA third method ofsolving systems of linear equationsis theaddition method. In this method, we add two terms with\nthe same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two\nterms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication\nso that one variable will be eliminated by addition.\n..."
	},
	{
		"chunk_id" : "00003094",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a system of equations, solve using the addition method.\n1. Write both equations withx- andy-variables on the left side of the equal sign and constants on the right.\n2. Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1033\nhas the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating"
	},
	{
		"chunk_id" : "00003095",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has\nthe opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the\nvariable.\n3. Solve the resulting equation for the remaining variable.\n4. Substitute that value into one of the original equations and solve for the second variable.\n5. Check the solution by substituting the values into the other equation.\nEXAMPLE4\nSolving a System by the Addition Method"
	},
	{
		"chunk_id" : "00003096",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE4\nSolving a System by the Addition Method\nSolve the given system of equations by addition.\nSolution\nBoth equations are already set equal to a constant. Notice that the coefficient of in the second equation, 1, is the\nopposite of the coefficient of in the first equation, 1. We can add the two equations to eliminate without needing to\nmultiply by a constant.\nNow that we have eliminated we can solve the resulting equation for"
	},
	{
		"chunk_id" : "00003097",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then, we substitute this value for into one of the original equations and solve for\nThe solution to this system is\nCheck the solution in the first equation.\nAnalysis\nWe gain an important perspective on systems of equations by looking at the graphical representation. SeeFigure 5to\nfind that the equations intersect at the solution. We do not need to ask whether there may be a second solution because\nobserving the graph confirms that the system has exactly one solution."
	},
	{
		"chunk_id" : "00003098",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1034 11  Systems of Equations and Inequalities\nFigure5\nEXAMPLE5\nUsing the Addition Method When Multiplication of One Equation Is Required\nSolve the given system of equations by theaddition method.\nSolution\nAdding these equations as presented will not eliminate a variable. However, we see that the first equation has in it\nand the second equation has So if we multiply the second equation by thex-terms will add to zero.\nNow, lets add them."
	},
	{
		"chunk_id" : "00003099",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now, lets add them.\nFor the last step, we substitute into one of the original equations and solve for\nOur solution is the ordered pair SeeFigure 6. Check the solution in the original second equation.\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1035\nFigure6\nTRY IT #4 Solve the system of equations by addition.\nEXAMPLE6\nUsing the Addition Method When Multiplication of Both Equations Is Required\nSolve the given system of equations in two variables by addition.\nSolution"
	},
	{
		"chunk_id" : "00003100",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nOne equation has and the other has The least common multiple is so we will have to multiply both equations\nby a constant in order to eliminate one variable. Lets eliminate by multiplying the first equation by and the second\nequation by\nThen, we add the two equations together.\nSubstitute into the original first equation.\nThe solution is Check it in the other equation.\n1036 11  Systems of Equations and Inequalities\nSeeFigure 7.\nFigure7\nEXAMPLE7"
	},
	{
		"chunk_id" : "00003101",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeFigure 7.\nFigure7\nEXAMPLE7\nUsing the Addition Method in Systems of Equations Containing Fractions\nSolve the given system of equations in two variables by addition.\nSolution\nFirst clear each equation of fractions by multiplying both sides of the equation by the least common denominator.\nNow multiply the second equation by so that we can eliminate thex-variable.\nAdd the two equations to eliminate thex-variable and solve the resulting equation.\nSubstitute into the first equation."
	},
	{
		"chunk_id" : "00003102",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substitute into the first equation.\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1037\nThe solution is Check it in the other equation.\nTRY IT #5 Solve the system of equations by addition.\nIdentifying Inconsistent Systems of Equations Containing Two Variables\nNow that we have several methods for solving systems of equations, we can use the methods to identify inconsistent"
	},
	{
		"chunk_id" : "00003103",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "systems. Recall that aninconsistent systemconsists of parallel lines that have the same slope but different -intercepts.\nThey will never intersect. When searching for a solution to an inconsistent system, we will come up with a false\nstatement, such as\nEXAMPLE8\nSolving an Inconsistent System of Equations\nSolve the following system of equations.\nSolution\nWe can approach this problem in two ways. Because one equation is already solved for the most obvious step is to use\nsubstitution."
	},
	{
		"chunk_id" : "00003104",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitution.\nClearly, this statement is a contradiction because Therefore, the system has no solution.\nThe second approach would be to first manipulate the equations so that they are both in slope-intercept form. We\nmanipulate the first equation as follows.\nWe then convert the second equation expressed to slope-intercept form.\nComparing the equations, we see that they have the same slope but differenty-intercepts. Therefore, the lines are\nparallel and do not intersect."
	},
	{
		"chunk_id" : "00003105",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "parallel and do not intersect.\n1038 11  Systems of Equations and Inequalities\nAnalysis\nWriting the equations in slope-intercept form confirms that the system is inconsistent because all lines will intersect\neventually unless they are parallel. Parallel lines will never intersect; thus, the two lines have no points in common. The\ngraphs of the equations in this example are shown inFigure 8.\nFigure8\nTRY IT #6 Solve the following system of equations in two variables."
	},
	{
		"chunk_id" : "00003106",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Expressing the Solution of a System of Dependent Equations Containing Two\nVariables\nRecall that adependent systemof equations in two variables is a system in which the two equations represent the same\nline. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other\nline. After using substitution or addition, the resulting equation will be an identity, such as\nEXAMPLE9\nFinding a Solution to a Dependent System of Linear Equations"
	},
	{
		"chunk_id" : "00003107",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find a solution to the system of equations using theaddition method.\nSolution\nWith the addition method, we want to eliminate one of the variables by adding the equations. In this case, lets focus on\neliminating If we multiply both sides of the first equation by then we will be able to eliminate the -variable.\nNow add the equations.\nWe can see that there will be an infinite number of solutions that satisfy both equations.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003108",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1039\nAnalysis\nIf we rewrote both equations in the slope-intercept form, we might know what the solution would look like before\nadding. Lets look at what happens when we convert the system to slope-intercept form.\nSeeFigure 9. Notice the results are the same. The general solution to the system is\nFigure9\nTRY IT #7 Solve the following system of equations in two variables.\nUsing Systems of Equations to Investigate Profits"
	},
	{
		"chunk_id" : "00003109",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Systems of Equations to Investigate Profits\nUsing what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the\nbeginning of the section. The skateboard manufacturersrevenue functionis the function used to calculate the amount\nof money that comes into the business. It can be represented by the equation where quantity and price.\nThe revenue function is shown in orange inFigure 10."
	},
	{
		"chunk_id" : "00003110",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thecost functionis the function used to calculate the costs of doing business. It includes fixed costs, such as rent and\nsalaries, and variable costs, such as utilities. The cost function is shown in blue inFigure 10. The -axis represents\nquantity in hundreds of units. They-axis represents either cost or revenue in hundreds of dollars.\n1040 11  Systems of Equations and Inequalities\nFigure10"
	},
	{
		"chunk_id" : "00003111",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure10\nThe point at which the two lines intersect is called thebreak-even point. We can see from the graph that if 700 units are\nproduced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce\nand sell 700 units. They neither make money nor lose money.\nThe shaded region to the right of the break-even point represents quantities for which the company makes a profit. The"
	},
	{
		"chunk_id" : "00003112",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shaded region to the left represents quantities for which the company suffers a loss. Theprofit functionis the revenue\nfunction minus the cost function, written as Clearly, knowing the quantity for which the cost equals\nthe revenue is of great importance to businesses.\nEXAMPLE10\nFinding the Break-Even Point and the Profit Function Using Substitution\nGiven the cost function and the revenue function find the break-even point and\nthe profit function.\nSolution"
	},
	{
		"chunk_id" : "00003113",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the profit function.\nSolution\nWrite the system of equations using to replace function notation.\nSubstitute the expression from the first equation into the second equation and solve for\nThen, we substitute into either the cost function or the revenue function.\nThe break-even point is\nThe profit function is found using the formula\nThe profit function is\nAnalysis\nThe cost to produce 50,000 units is $77,500, and the revenue from the sales of 50,000 units is also $77,500. To make a"
	},
	{
		"chunk_id" : "00003114",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "profit, the business must produce and sell more than 50,000 units. SeeFigure 11.\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1041\nFigure11\nWe see from the graph inFigure 12that the profit function has a negative value until when the graph\ncrosses thex-axis. Then, the graph emerges into positivey-values and continues on this path as the profit function is a"
	},
	{
		"chunk_id" : "00003115",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "straight line. This illustrates that the break-even point for businesses occurs when the profit function is 0. The area to the\nleft of the break-even point represents operating at a loss.\nFigure12\nEXAMPLE11\nWriting and Solving a System of Equations in Two Variables\nThe cost of a ticket to the circus is for children and for adults. On a certain day, attendance at the circus is\nand the total gate revenue is How many children and how many adults bought tickets?\nSolution"
	},
	{
		"chunk_id" : "00003116",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nLetc= the number of children anda= the number of adults in attendance.\nThe total number of people is We can use this to write an equation for the number of people at the circus that\nday.\nThe revenue from all children can be found by multiplying by the number of children, The revenue from all\nadults can be found by multiplying by the number of adults, The total revenue is We can use this to\nwrite an equation for the revenue.\nWe now have a system of linear equations in two variables."
	},
	{
		"chunk_id" : "00003117",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the first equation, the coefficient of both variables is 1. We can quickly solve the first equation for either or We will\n1042 11  Systems of Equations and Inequalities\nsolve for\nSubstitute the expression in the second equation for and solve for\nSubstitute into the first equation to solve for\nWe find that children and adults bought tickets to the circus that day.\nTRY IT #8 Meal tickets at the circus cost for children and for adults. If meal tickets were"
	},
	{
		"chunk_id" : "00003118",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bought for a total of how many children and how many adults bought meal tickets?\nMEDIA\nAccess these online resources for additional instruction and practice with systems of linear equations.\nSolving Systems of Equations Using Substitution(http://openstax.org/l/syssubst)\nSolving Systems of Equations Using Elimination(http://openstax.org/l/syselim)\nApplications of Systems of Equations(http://openstax.org/l/sysapp)\n11.1 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00003119",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.1 SECTION EXERCISES\nVerbal\n1. Can a system of linear 2. If you are performing a 3. If you are solving a break-\nequations have exactly two break-even analysis for a even analysis and get a\nsolutions? Explain why or business and their cost and negative break-even point,\nwhy not. revenue equations are explain what this signifies\ndependent, explain what for the company?\nthis means for the\ncompanys profit margins.\n4. If you are solving a break- 5. Given a system of equations,"
	},
	{
		"chunk_id" : "00003120",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "even analysis and there is explain at least two different\nno break-even point, explain methods of solving that\nwhat this means for the system.\ncompany. How should they\nensure there is a break-even\npoint?\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1043\nAlgebraic\nFor the following exercises, determine whether the given ordered pair is a solution to the system of equations.\n6. and 7. and 8. and\n9. and 10. and"
	},
	{
		"chunk_id" : "00003121",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. and 7. and 8. and\n9. and 10. and\nFor the following exercises, solve each system by substitution.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20.\nFor the following exercises, solve each system by addition.\n21. 22. 23.\n24. 25. 26.\n27. 28. 29.\n30.\nFor the following exercises, solve each system by any method.\n31. 32. 33.\n34. 35. 36.\n1044 11  Systems of Equations and Inequalities\n37. 38. 39.\n40.\nGraphical"
	},
	{
		"chunk_id" : "00003122",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "37. 38. 39.\n40.\nGraphical\nFor the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or\ndependent and whether the system has one solution, no solution, or infinite solutions.\n41. 42. 43.\n44. 45.\nTechnology\nFor the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to\nthe nearest hundredth.\n46. 47. 48.\n49. 50.\nExtensions"
	},
	{
		"chunk_id" : "00003123",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47. 48.\n49. 50.\nExtensions\nFor the following exercises, solve each system in terms of and where are nonzero numbers. Note\nthat and\n51. 52. 53.\n54. 55.\nAccess for free at openstax.org\n11.1  Systems of Linear Equations: Two Variables 1045\nReal-World Applications\nFor the following exercises, solve for the desired quantity.\n56. A stuffed animal business 57. An Ethiopian restaurant 58. A cell phone factory has a\nhas a total cost of has a cost of production cost of production\nproduction and a and"
	},
	{
		"chunk_id" : "00003124",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "production and a and\nand a revenue function revenue function a revenue function\nFind the break- When does the What is the\neven point. company start to turn a break-even point?\nprofit?\n59. A musician charges 60. A guitar factory has a cost\nof production\nwhere is the total If\nnumber of attendees at the the company needs to\nconcert. The venue charges break even after 150 units\n$80 per ticket. After how sold, at what price should\nmany people buy tickets they sell each guitar?"
	},
	{
		"chunk_id" : "00003125",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "many people buy tickets they sell each guitar?\ndoes the venue break even, Round up to the nearest\nand what is the value of dollar, and write the\nthe total tickets sold at that revenue function.\npoint?\nFor the following exercises, use a system of linear equations with two variables and two equations to solve.\n61. Find two numbers whose 62. A number is 9 more than 63. The startup cost for a\nsum is 28 and difference is another number. Twice the restaurant is $120,000, and"
	},
	{
		"chunk_id" : "00003126",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13. sum of the two numbers is each meal costs $10 for the\n10. Find the two numbers. restaurant to make. If each\nmeal is then sold for $15,\nafter how many meals\ndoes the restaurant break\neven?\n64. A moving company 65. A total of 1,595 first- and 66. 276 students enrolled in an\ncharges a flat rate of $150, second-year college introductory chemistry\nand an additional $5 for students gathered at a pep class. By the end of the\neach box. If a taxi service rally. The number of first- semester, 5 times the"
	},
	{
		"chunk_id" : "00003127",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "would charge $20 for each years exceeded the number of students passed\nbox, how many boxes number of second-years by as failed. Find the number\nwould you need for it to be 15. How many students of students who passed,\ncheaper to use the moving from each year group were and the number of\ncompany, and what would in attendance? students who failed.\nbe the total cost?\n1046 11  Systems of Equations and Inequalities\n67. There were 130 faculty at a 68. A jeep and a pickup truck 69. If a scientist mixed 10%"
	},
	{
		"chunk_id" : "00003128",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "conference. If there were enter a highway running saline solution with 60%\n18 more women than men east-west at the same exit saline solution to get 25\nattending, how many of heading in opposite gallons of 40% saline\neach gender attended the directions. The jeep solution, how many gallons\nconference? entered the highway 30 of 10% and 60% solutions\nminutes before the pickup were mixed?\ndid, and traveled 7 mph\nslower than the pickup.\nAfter 2 hours from the time\nthe pickup entered the\nhighway, the cars were"
	},
	{
		"chunk_id" : "00003129",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the pickup entered the\nhighway, the cars were\n306.5 miles apart. Find the\nspeed of each car,\nassuming they were driven\non cruise control and\nretained the same speed.\n70. An investor earned triple 71. An investor invested 1.1 72. If an investor invests a total\nthe profits of what they million dollars into two of $25,000 into two bonds,\nearned last year. If they land investments. On the one that pays 3% simple\nmade $500,000.48 total for first investment, Swan interest, and the other that"
	},
	{
		"chunk_id" : "00003130",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "both years, how much did Peak, her return was a pays interest, and the\nthe investor earn in profits 110% increase on the investor earns $737.50\neach year? money she invested. On annual interest, how much\nthe second investment, was invested in each\nRiverside Community, she account?\nearned 50% over what she\ninvested. If she earned $1\nmillion in profits, how\nmuch did she invest in\neach of the land deals?\n73. If an investor invests 74. Blu-rays cost $5.96 more 75. A store clerk sold 60 pairs"
	},
	{
		"chunk_id" : "00003131",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$23,000 into two bonds, than regular DVDs at All of sneakers. The high-tops\none that pays 4% in simple Bets Are Off Electronics. sold for $98.99 and the\ninterest, and the other How much would 6 Blu- low-tops sold for $129.99. If\npaying 2% simple interest, rays and 2 DVDs cost if 5 the receipts for the two\nand the investor earns Blu-rays and 2 DVDs cost types of sales totaled\n$710.00 annual interest, $127.73? $6,404.40, how many of\nhow much was invested in each type of sneaker were\neach account? sold?"
	},
	{
		"chunk_id" : "00003132",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each account? sold?\n76. A concert manager 77. Admission into an\ncounted 350 ticket receipts amusement park for 4\nthe day after a concert. The children and 2 adults is\nprice for a student ticket $116.90. For 6 children and\nwas $12.50, and the price 3 adults, the admission is\nfor an adult ticket was $175.35. Assuming a\n$16.00. The register different price for children\nconfirms that $5,075 was and adults, what is the\ntaken in. How many price of the childs ticket"
	},
	{
		"chunk_id" : "00003133",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "taken in. How many price of the childs ticket\nstudent tickets and adult and the price of the adult\ntickets were sold? ticket?\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1047\n11.2 Systems of Linear Equations: Three Variables\nLearning Objectives\nIn this section, you will:\nSolve systems of three equations in three variables.\nIdentify inconsistent systems of equations containing three variables."
	},
	{
		"chunk_id" : "00003134",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Express the solution of a system of dependent equations containing three variables.\nFigure1 (credit: Elembis, Wikimedia Commons)\nJordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market\nfund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual\ninterest. Jordi invested $4,000 more in municipal funds than in municipal bonds. He earned $670 in interest the first year."
	},
	{
		"chunk_id" : "00003135",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "How much did Jordi invest in each type of fund?\nUnderstanding the correct approach to setting up problems such as this one makes finding a solution a matter of\nfollowing a pattern. We will solve this and similar problems involving three equations and three variables in this section.\nDoing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding\nsolutions to systems of three equations requires a bit more organization and a touch of visualization."
	},
	{
		"chunk_id" : "00003136",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Systems of Three Equations in Three Variables\nIn order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be\nusing is calledGaussian elimination, named after the prolific German mathematician Karl FriedrichGauss. While there is\nno definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can"
	},
	{
		"chunk_id" : "00003137",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a\ntime to achieveupper triangular form, the ideal form for a three-by-three system because it allows for straightforward\nback-substitution to find a solution which we call anordered triple. A system in upper triangular form looks like\nthe following:\nThe third equation can be solved for and then we back-substitute to find and To write the system in upper"
	},
	{
		"chunk_id" : "00003138",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "triangular form, we can perform the following operations:\n1. Interchange the order of any two equations.\n2. Multiply both sides of an equation by a nonzero constant.\n3. Add a nonzero multiple of one equation to another equation.\nThesolution setto a three-by-three system is an ordered triple Graphically, the ordered triple defines the\npoint that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a"
	},
	{
		"chunk_id" : "00003139",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rectangular room. A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two\nwalls and the floor meet represents the intersection of three planes.\n1048 11  Systems of Equations and Inequalities\nNumber of Possible Solutions\nFigure 2andFigure 3illustrate possible solution scenarios for three-by-three systems.\n Systems that have a single solution are those which, after elimination, result in asolution setconsisting of an"
	},
	{
		"chunk_id" : "00003140",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ordered triple Graphically, the ordered triple defines a point that is the intersection of three planes in\nspace.\n Systems that have an infinite number of solutions are those which, after elimination, result in an expression that\nis always true, such as Graphically, an infinite number of solutions represents a line or coincident plane\nthat serves as the intersection of three planes in space."
	},
	{
		"chunk_id" : "00003141",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Systems that have no solution are those that, after elimination, result in a statement that is a contradiction, such\nas Graphically, a system with no solution is represented by three planes with no point in common.\nFigure2 (a)Three planes intersect at a single point, representing a three-by-three system with a single solution. (b)\nThree planes intersect in a line, representing a three-by-three system with infinite solutions."
	},
	{
		"chunk_id" : "00003142",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure3 All three figures represent three-by-three systems with no solution. (a) The three planes intersect with each\nother, but not at a common point. (b) Two of the planes are parallel and intersect with the third plane, but not with\neach other. (c) All three planes are parallel, so there is no point of intersection.\nEXAMPLE1\nDetermining Whether an Ordered Triple Is a Solution to a System\nDetermine whether the ordered triple is a solution to the system.\nSolution"
	},
	{
		"chunk_id" : "00003143",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe will check each equation by substituting in the values of the ordered triple for and\nThe ordered triple is indeed a solution to the system.\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1049\n...\nHOW TO\nGiven a linear system of three equations, solve for three unknowns.\n1. Pick any pair of equations and solve for one variable.\n2. Pick another pair of equations and solve for the same variable."
	},
	{
		"chunk_id" : "00003144",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.\n4. Back-substitute known variables into any one of the original equations and solve for the missing variable.\nEXAMPLE2\nSolving a System of Three Equations in Three Variables by Elimination\nFind a solution to the following system:\nSolution\nThere will always be several choices as to where to begin, but the most obvious first step here is to eliminate by adding\nequations (1) and (2)."
	},
	{
		"chunk_id" : "00003145",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equations (1) and (2).\nThe second step is multiplying equation (1) by and adding the result to equation (3). These two steps will eliminate\nthe variable\nIn equations (4) and (5), we have created a new two-by-two system. We can solve for by adding the two equations.\nChoosing one equation from each new system, we obtain the upper triangular form:\nNext, we back-substitute into equation (4) and solve for\nFinally, we can back-substitute and into equation (1). This will yield the solution for"
	},
	{
		"chunk_id" : "00003146",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1050 11  Systems of Equations and Inequalities\nThe solution is the ordered triple SeeFigure 4.\nFigure4\nEXAMPLE3\nSolving a Real-World Problem Using a System of Three Equations in Three Variables\nIn the problem posed at the beginning of the section, Jordi invested his inheritance of $12,000 in three different funds:\npart in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in"
	},
	{
		"chunk_id" : "00003147",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "mutual funds paying 7% annually. Jordi invested $4,000 more in mutual funds than he invested in municipal bonds. The\ntotal interest earned in one year was $670. How much did he invest in each type of fund?\nSolution\nTo solve this problem, we use all of the information given and set up three equations. First, we assign a variable to each\nof the three investment amounts:\nThe first equation indicates that the sum of the three principal amounts is $12,000."
	},
	{
		"chunk_id" : "00003148",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We form the second equation according to the information that Jordi invested $4,000 more in mutual funds than he\ninvested in municipal bonds.\nThe third equation shows that the total amount of interest earned from each fund equals $670.\nThen, we write the three equations as a system.\nTo make the calculations simpler, we can multiply the third equation by 100. Thus,\nStep 1. Interchange equation (2) and equation (3) so that the two equations with three variables will line up."
	},
	{
		"chunk_id" : "00003149",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Step 2. Multiply equation (1) by and add to equation (2). Write the result as row 2.\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1051\nStep 3. Add equation (2) to equation (3) and write the result as equation (3).\nStep 4. Solve for in equation (3). Back-substitute that value in equation (2) and solve for Then, back-substitute the\nvalues for and into equation (1) and solve for"
	},
	{
		"chunk_id" : "00003150",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "values for and into equation (1) and solve for\nJordi invested $2,000 in a money-market fund, $3,000 in municipal bonds, and $7,000 in mutual funds.\nTRY IT #1 Solve the system of equations in three variables.\nIdentifying Inconsistent Systems of Equations Containing Three Variables\nJust as with systems of equations in two variables, we may come across aninconsistent systemof equations in three"
	},
	{
		"chunk_id" : "00003151",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variables, which means that it does not have a solution that satisfies all three equations. The equations could represent\nthree parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not\nat the same location. The process of elimination will result in a false statement, such as or some other\ncontradiction.\nEXAMPLE4\nSolving an Inconsistent System of Three Equations in Three Variables\nSolve the following system.\nSolution"
	},
	{
		"chunk_id" : "00003152",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the following system.\nSolution\nLooking at the coefficients of we can see that we can eliminate by adding equation (1) to equation (2).\nNext, we multiply equation (1) by and add it to equation (3).\n1052 11  Systems of Equations and Inequalities\nThen, we multiply equation (4) by 2 and add it to equation (5).\nThe final equation is a contradiction, so we conclude that the system of equations in inconsistent and, therefore,\nhas no solution.\nAnalysis"
	},
	{
		"chunk_id" : "00003153",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has no solution.\nAnalysis\nIn this system, each plane intersects the other two, but not at the same location. Therefore, the system is inconsistent.\nTRY IT #2 Solve the system of three equations in three variables.\nExpressing the Solution of a System of Dependent Equations Containing Three\nVariables\nWe know from working with systems of equations in two variables that adependent systemof equations has an infinite"
	},
	{
		"chunk_id" : "00003154",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of\nsolutions can result from several situations. The three planes could be the same, so that a solution to one equation will\nbe the solution to the other two equations. All three equations could be different but they intersect on a line, which has\ninfinite solutions. Or two of the equations could be the same and intersect the third on a line.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00003155",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE5\nFinding the Solution to a Dependent System of Equations\nFind the solution to the given system of three equations in three variables.\nSolution\nFirst, we can multiply equation (1) by and add it to equation (2).\nWe do not need to proceed any further. The result we get is an identity, which tells us that this system has an\ninfinite number of solutions. There are other ways to begin to solve this system, such as multiplying equation (3) by"
	},
	{
		"chunk_id" : "00003156",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and adding it to equation (1). We then perform the same steps as above and find the same result,\nWhen a system is dependent, we can find general expressions for the solutions. Adding equations (1) and (3), we have\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1053\nWe then solve the resulting equation for\nWe back-substitute the expression for into one of the equations and solve for"
	},
	{
		"chunk_id" : "00003157",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So the general solution is In this solution, can be any real number. The values of and are dependent\non the value selected for\nAnalysis\nAs shown inFigure 5, two of the planes are the same and they intersect the third plane on a line. The solution set is\ninfinite, as all points along the intersection line will satisfy all three equations.\nFigure5\nQ&A Does the generic solution to a dependent system always have to be written in terms of"
	},
	{
		"chunk_id" : "00003158",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "No, you can write the generic solution in terms of any of the variables, but it is common to write it in\nterms of x and if needed and\nTRY IT #3 Solve the following system.\nMEDIA\nAccess these online resources for additional instruction and practice with systems of equations in three variables.\nEx 1: System of Three Equations with Three Unknowns Using Elimination(http://openstax.org/l/systhree)\nEx. 2: System of Three Equations with Three Unknowns Using Elimination(http://openstax.org/l/systhelim)"
	},
	{
		"chunk_id" : "00003159",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1054 11  Systems of Equations and Inequalities\n11.2 SECTION EXERCISES\nVerbal\n1. Can a linear system of three 2. If a given ordered triple 3. If a given ordered triple\nequations have exactly two solves the system of does not solve the system of\nsolutions? Explain why or equations, is that solution equations, is there no\nwhy not unique? If so, explain why. If solution? If so, explain why.\nnot, give an example where If not, give an example.\nit is not unique.\n4. Using the method of 5. Can you explain whether"
	},
	{
		"chunk_id" : "00003160",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Using the method of 5. Can you explain whether\naddition, is there only one there can be only one\nway to solve the system? method to solve a linear\nsystem of equations? If yes,\ngive an example of such a\nsystem of equations. If not,\nexplain why not.\nAlgebraic\nFor the following exercises, determine whether the ordered triple given is the solution to the system of equations.\n6. and 7. and\n8. and 9. and\n10. and\nFor the following exercises, solve each system by elimination.\n11. 12. 13.\n14. 15. 16."
	},
	{
		"chunk_id" : "00003161",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11. 12. 13.\n14. 15. 16.\nFor the following exercises, solve each system by Gaussian elimination.\n17. 18. 19.\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1055\n20. 21. 22.\n23. 24. 25.\n26. 27. 28.\n29. 30. 31.\n32. 33. 34.\n35. 36. 37.\n38. 39. 40.\n41. 42. 43.\n44. 45.\n1056 11  Systems of Equations and Inequalities\nExtensions\nFor the following exercises, solve the system for and\n46. 47. 48.\n49. 50.\nReal-World Applications"
	},
	{
		"chunk_id" : "00003162",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47. 48.\n49. 50.\nReal-World Applications\n51. Three even numbers sum 52. Three numbers sum up to 53. At a family reunion, there\nup to 108. The smaller is 147. The smallest number were only blood relatives,\nhalf the larger and the is half the middle number, consisting of children,\nmiddle number is the which is half the largest parents, and grandparents,\nlarger. What are the three number. What are the in attendance. There were\nnumbers? three numbers? 400 people total. There\nwere twice as many"
	},
	{
		"chunk_id" : "00003163",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "were twice as many\nparents as grandparents,\nand 50 more children than\nparents. How many\nchildren, parents, and\ngrandparents were in\nattendance?\n54. An animal shelter has a 55. Your roommate, Shani, 56. Your roommate, John,\ntotal of 350 animals offered to buy groceries for offered to buy household\ncomprised of cats, dogs, you and your other supplies for you and your\nand rabbits. If the number roommate. The total bill other roommate. You live"
	},
	{
		"chunk_id" : "00003164",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of rabbits is 5 less than was $82. She forgot to save near the border of three\none-half the number of the individual receipts but states, each of which has a\ncats, and there are 20 more remembered that your different sales tax. The\ncats than dogs, how many groceries were $0.05 total amount of money\nof each animal are at the cheaper than half of her spent was $100.75. Your\nshelter? groceries, and that your supplies were bought with\nother roommates 5% tax, Johns with 8% tax,"
	},
	{
		"chunk_id" : "00003165",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "other roommates 5% tax, Johns with 8% tax,\ngroceries were $2.10 more and your third roommates\nthan your groceries. How with 9% sales tax. The total\nmuch was each of your amount of money spent\nshare of the groceries? without taxes is $93.50. If\nyour supplies before tax\nwere $1 more than half of\nwhat your third\nroommates supplies were\nbefore tax, how much did\neach of you spend? Give\nyour answer both with and\nwithout taxes.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003166",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "without taxes.\nAccess for free at openstax.org\n11.2  Systems of Linear Equations: Three Variables 1057\n57. Three coworkers work for 58. At a carnival, $2,914.25 in 59. A local band sells out for\nthe same employer. Their receipts were taken at the their concert. They sell all\njobs are warehouse end of the day. The cost of 1,175 tickets for a total\nmanager, office manager, a childs ticket was $20.50, purse of $28,112.50. The\nand truck driver. The sum an adult ticket was $29.75, tickets were priced at $20"
	},
	{
		"chunk_id" : "00003167",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the annual salaries of and a senior citizen ticket for student tickets, $22.50\nthe warehouse manager was $15.25. There were for children, and $29 for\nand office manager is twice as many senior adult tickets. If the band\n$82,000. The office citizens as adults in sold twice as many adult as\nmanager makes $4,000 attendance, and 20 more children tickets, how many\nmore than the truck driver children than senior of each type was sold?\nannually. The annual citizens. How many"
	},
	{
		"chunk_id" : "00003168",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "annually. The annual citizens. How many\nsalaries of the warehouse children, adult, and senior\nmanager and the truck citizen tickets were sold?\ndriver total $78,000. What\nis the annual salary of each\nof the co-workers?\n60. In a bag, a child has 325 61. Last year, at Havens Pond 62. When his youngest child\ncoins worth $19.50. There Car Dealership, for a moved out, Deandre sold\nwere three types of coins: particular model of BMW, his home and made three"
	},
	{
		"chunk_id" : "00003169",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "pennies, nickels, and Jeep, and Toyota, one could investments using gains\ndimes. If the bag contained purchase all three cars for from the sale. He invested\nthe same number of a total of $140,000. This $80,500 into three\nnickels as dimes, how year, due to inflation, the accounts, one that paid 4%\nmany of each type of coin same cars would cost simple interest, one that\nwas in the bag? $151,830. The cost of the paid simple interest,\nBMW increased by 8%, the\nand one that paid\nJeep by 5%, and the Toyota"
	},
	{
		"chunk_id" : "00003170",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and one that paid\nJeep by 5%, and the Toyota\nsimple interest. He earned\nby 12%. If the price of last\n$2,670 interest at the end\nyears Jeep was $7,000 less\nof one year. If the amount\nthan the price of last years\nof the money invested in\nBMW, what was the price\nthe second account was\nof each of the three cars\nfour times the amount\nlast year?\ninvested in the third\naccount, how much was\ninvested in each account?\n1058 11  Systems of Equations and Inequalities"
	},
	{
		"chunk_id" : "00003171",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1058 11  Systems of Equations and Inequalities\n63. You inherit one million 64. An entrepreneur sells a 65. The top three countries in\ndollars. You invest it all in portion of their business oil consumption in a\nthree accounts for one for one hundred thousand certain year are as follows:\nyear. The first account pays dollars and invests it all in the United States, Japan,\n3% compounded annually, three accounts for one and China. In millions of"
	},
	{
		"chunk_id" : "00003172",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the second account pays year. The first account pays barrels per day, the three\n4% compounded annually, 4% compounded annually, top countries consumed\nand the third account pays the second account pays 39.8% of the worlds\n2% compounded annually. 3% compounded annually, consumed oil. The United\nAfter one year, you earn and the third account pays States consumed 0.7%\n$34,000 in interest. If you 2% compounded annually. more than four times\ninvest four times the After one year, the Chinas consumption. The"
	},
	{
		"chunk_id" : "00003173",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "money into the account entrepreneur earns $3,650 United States consumed\nthat pays 3% compared to in interest. If they invested 5% more than triple\n2%, how much did you five times the money in the Japans consumption. What\ninvest in each account? account that pays 4% percent of the world oil\ncompared to 3%, how consumption did the\nmuch did they invest in United States, Japan, and\neach account? China consume?1\n66. The top three countries in 67. The top three sources of oil 68. The top three oil producers"
	},
	{
		"chunk_id" : "00003174",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "oil production in the same imports for the United in the United States in a\nyear are Saudi Arabia, the States in the same year certain year are the Gulf of\nUnited States, and Russia. were Saudi Arabia, Mexico, Mexico, Texas, and Alaska.\nIn millions of barrels per and Canada. The three top The three regions were\nday, the top three countries accounted for responsible for 64% of the\ncountries produced 31.4% 47% of oil imports. The United States oil"
	},
	{
		"chunk_id" : "00003175",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the worlds produced oil. United States imported production. The Gulf of\nSaudi Arabia and the 1.8% more from Saudi Mexico and Texas\nUnited States combined for Arabia than they did from combined for 47% of oil\n22.1% of the worlds Mexico, and 1.7% more production. Texas\nproduction, and Saudi from Saudi Arabia than produced 3% more than\nArabia produced 2% more they did from Canada. Alaska. What percent of\noil than Russia. What What percent of the United United States oil"
	},
	{
		"chunk_id" : "00003176",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "percent of the world oil States oil imports were production came from\nproduction did Saudi from these three these regions?4\nArabia, the United States, countries?3\nand Russia produce?2\n1 Oil reserves, production and consumption in 2001, accessed April 6, 2014, http://scaruffi.com/politics/oil.html.\n2 Oil reserves, production and consumption in 2001, accessed April 6, 2014, http://scaruffi.com/politics/oil.html."
	},
	{
		"chunk_id" : "00003177",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3 Oil reserves, production and consumption in 2001, accessed April 6, 2014, http://scaruffi.com/politics/oil.html.\n4 USA: The coming global oil crisis, accessed April 6, 2014, http://www.oilcrisis.com/us/.\nAccess for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1059\n69. At one time, in the United 70. Meat consumption in the\nStates, 398 species of United States can be\nanimals were on the broken into three\nendangered species list. categories: red meat,"
	},
	{
		"chunk_id" : "00003178",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "endangered species list. categories: red meat,\nThe top groups were poultry, and fish. If fish\nmammals, birds, and fish, makes up 4% less than\nwhich comprised 55% of one-quarter of poultry\nthe endangered species. consumption, and red\nBirds accounted for 0.7% meat consumption is 18.2%\nmore than fish, and fish higher than poultry\naccounted for 1.5% more consumption, what are the\nthan mammals. What percentages of meat\npercent of the endangered consumption?5\nspecies came from\nmammals, birds, and fish?"
	},
	{
		"chunk_id" : "00003179",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "species came from\nmammals, birds, and fish?\n11.3 Systems of Nonlinear Equations and Inequalities: Two Variables\nLearning Objectives\nIn this section, you will:\nSolve a system of nonlinear equations using substitution.\nSolve a system of nonlinear equations using elimination.\nGraph a nonlinear inequality.\nGraph a system of nonlinear inequalities.\nHalleys Comet (Figure 1) orbits the sun about once every 75 years. Its path can be considered to be a very elongated"
	},
	{
		"chunk_id" : "00003180",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These\nsystems, however, are different from the ones we considered in the previous section because the equations are not\nlinear.\nFigure1 Halleys Comet (credit: \"NASA Blueshift\"\"/Flickr)"
	},
	{
		"chunk_id" : "00003181",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The methods for solving systems of nonlinear equations are similar to those for linear equations.\nSolving a System of Nonlinear Equations Using Substitution\nAsystem of nonlinear equationsis a system of two or more equations in two or more variables containing at least one\nequation that is not linear. Recall that a linear equation can take the form Any equation that cannot\nbe written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use"
	},
	{
		"chunk_id" : "00003182",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to\nsolve for another variable, and so on. There is, however, a variation in the possible outcomes.\nIntersection of a Parabola and a Line\nThere are three possible types of solutions for a system of nonlinear equations involving aparabolaand a line.\n5 The United States Meat Industry at a Glance, accessed April 6, 2014, http://www.meatami.com/ht/d/sp/i/47465/pid/47465."
	},
	{
		"chunk_id" : "00003183",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1060 11  Systems of Equations and Inequalities\nPossible Types of Solutions for Points of Intersection of a Parabola and a Line\nFigure 2illustrates possible solution sets for a system of equations involving a parabola and a line.\n No solution. The line will never intersect the parabola.\n One solution. The line is tangent to the parabola and intersects the parabola at exactly one point.\n Two solutions. The line crosses on the inside of the parabola and intersects the parabola at two points.\nFigure2\n..."
	},
	{
		"chunk_id" : "00003184",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\n...\nHOW TO\nGiven a system of equations containing a line and a parabola, find the solution.\n1. Solve the linear equation for one of the variables.\n2. Substitute the expression obtained in step one into the parabola equation.\n3. Solve for the remaining variable.\n4. Check your solutions in both equations.\nEXAMPLE1\nSolving a System of Nonlinear Equations Representing a Parabola and a Line\nSolve the system of equations.\nSolution"
	},
	{
		"chunk_id" : "00003185",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve the system of equations.\nSolution\nSolve the first equation for and then substitute the resulting expression into the second equation.\nExpand the equation and set it equal to zero.\nAccess for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1061\nSolving for gives and Next, substitute each value for into the first equation to solve for Always\nsubstitute the value into the linear equation to check for extraneous solutions."
	},
	{
		"chunk_id" : "00003186",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solutions are and which can be verified by substituting these values into both of the original\nequations. SeeFigure 3.\nFigure3\nQ&A Could we have substituted values for into the second equation to solve for inExample 1?\nYes, but because is squared in the second equation this could give us extraneous solutions for\nFor\nThis gives us the same value as in the solution.\nFor\nNotice that is an extraneous solution.\nTRY IT #1 Solve the given system of equations by substitution."
	},
	{
		"chunk_id" : "00003187",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Intersection of a Circle and a Line\nJust as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a\ncircle and a line.\n1062 11  Systems of Equations and Inequalities\nPossible Types of Solutions for the Points of Intersection of a Circle and a Line\nFigure 4illustrates possible solution sets for a system of equations involving acircleand a line.\n No solution. The line does not intersect the circle."
	},
	{
		"chunk_id" : "00003188",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " One solution. The line is tangent to the circle and intersects the circle at exactly one point.\n Two solutions. The line crosses the circle and intersects it at two points.\nFigure4\n...\nHOW TO\nGiven a system of equations containing a line and a circle, find the solution.\n1. Solve the linear equation for one of the variables.\n2. Substitute the expression obtained in step one into the equation for the circle.\n3. Solve for the remaining variable.\n4. Check your solutions in both equations.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00003189",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nFinding the Intersection of a Circle and a Line by Substitution\nFind the intersection of the given circle and the given line by substitution.\nSolution\nOne of the equations has already been solved for We will substitute into the equation for the circle.\nNow, we factor and solve for\nSubstitute the twox-values into the original linear equation to solve for\nAccess for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1063"
	},
	{
		"chunk_id" : "00003190",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The line intersects the circle at and which can be verified by substituting these values into both of\nthe original equations. SeeFigure 5.\nFigure5\nTRY IT #2 Solve the system of nonlinear equations.\nSolving a System of Nonlinear Equations Using Elimination\nWe have seen that substitution is often the preferred method when a system of equations includes a linear equation and\na nonlinear equation. However, when both equations in the system have like variables of the second degree, solving"
	},
	{
		"chunk_id" : "00003191",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "them using elimination by addition is often easier than substitution. Generally,eliminationis a far simpler method when\nthe system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as\nthere are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of\nequations representing acircleand an ellipse.\nPossible Types of Solutions for the Points of Intersection of a Circle and an Ellipse"
	},
	{
		"chunk_id" : "00003192",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure 6illustrates possible solution sets for a system of equations involving a circle and anellipse.\n No solution. The circle and ellipse do not intersect. One shape is inside the other or the circle and the ellipse are\na distance away from the other.\n One solution. The circle and ellipse are tangent to each other, and intersect at exactly one point.\n Two solutions. The circle and the ellipse intersect at two points.\n Three solutions. The circle and the ellipse intersect at three points."
	},
	{
		"chunk_id" : "00003193",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Four solutions. The circle and the ellipse intersect at four points.\n1064 11  Systems of Equations and Inequalities\nFigure6\nEXAMPLE3\nSolving a System of Nonlinear Equations Representing a Circle and an Ellipse\nSolve the system of nonlinear equations.\nSolution\nLets begin by multiplying equation (1) by and adding it to equation (2).\nAfter we add the two equations together, we solve for\nSubstitute into one of the equations and solve for\nThere are four solutions: SeeFigure 7."
	},
	{
		"chunk_id" : "00003194",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "There are four solutions: SeeFigure 7.\nAccess for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1065\nFigure7\nTRY IT #3 Find the solution set for the given system of nonlinear equations.\nGraphing a Nonlinear Inequality\nAll of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter\nsystems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding"
	},
	{
		"chunk_id" : "00003195",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation, and then shading the region represented by theinequalitysymbol. Now, we will follow similar steps to graph a\nnonlinear inequality so that we can learn to solve systems of nonlinear inequalities. Anonlinear inequalityis an\ninequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.\nRecall that when the inequality is greater than, or less than, the graph is drawn with a dashed line. When"
	},
	{
		"chunk_id" : "00003196",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the inequality is greater than or equal to, or less than or equal to, the graph is drawn with a solid line. The\ngraphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the\nwhole region works. That is the region we shade. SeeFigure 8.\nFigure8 (a) an example of (b) an example of (c) an example of (d) an example of\n1066 11  Systems of Equations and Inequalities\n...\nHOW TO\nGiven an inequality bounded by a parabola, sketch a graph."
	},
	{
		"chunk_id" : "00003197",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Graph the parabola as if it were an equation. This is the boundary for the region that is the solution set.\n2. If the boundary is included in the region (the operator is or ), the parabola is graphed as a solid line.\n3. If the boundary is not included in the region (the operator is < or >), the parabola is graphed as a dashed line.\n4. Test a point in one of the regions to determine whether it satisfies the inequality statement. If the statement is"
	},
	{
		"chunk_id" : "00003198",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "true, the solution set is the region including the point. If the statement is false, the solution set is the region on\nthe other side of the boundary line.\n5. Shade the region representing the solution set.\nEXAMPLE4\nGraphing an Inequality for a Parabola\nGraph the inequality\nSolution\nFirst, graph the corresponding equation Since has a greater than symbol, we draw the graph\nwith a dashed line. Then we choose points to test both inside and outside the parabola. Lets test the points"
	},
	{
		"chunk_id" : "00003199",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and One point is clearly inside the parabola and the other point is clearly outside.\nThe graph is shown inFigure 9. We can see that the solution set consists of all points inside the parabola, but not on the\ngraph itself.\nFigure9\nGraphing a System of Nonlinear Inequalities\nNow that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities.\nAsystem of nonlinear inequalitiesis a system of two or more inequalities in two or more variables containing at least"
	},
	{
		"chunk_id" : "00003200",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear\ninequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in\na system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the\nshaded regions of the graph of each inequality overlap, or where the regions intersect, called thefeasible region.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003201",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1067\n...\nHOW TO\nGiven a system of nonlinear inequalities, sketch a graph.\n1. Find the intersection points by solving the corresponding system of nonlinear equations.\n2. Graph the nonlinear equations.\n3. Find the shaded regions of each inequality.\n4. Identify the feasible region as the intersection of the shaded regions of each inequality or the set of points\ncommon to each inequality.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00003202",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common to each inequality.\nEXAMPLE5\nGraphing a System of Inequalities\nGraph the given system of inequalities.\nSolution\nThese two equations are clearly parabolas. We can find the points of intersection by the elimination process: Add both\nequations and the variable will be eliminated. Then we solve for\nSubstitute thex-values into one of the equations and solve for\nThe two points of intersection are and Notice that the equations can be rewritten as follows."
	},
	{
		"chunk_id" : "00003203",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph each inequality. SeeFigure 10. The feasible region is the region between the two equations bounded by\non the top and on the bottom.\n1068 11  Systems of Equations and Inequalities\nFigure10\nTRY IT #4 Graph the given system of inequalities.\nMEDIA\nAccess these online resources for additional instruction and practice with nonlinear equations.\nSolve a System of Nonlinear Equations Using Substitution(http://openstax.org/l/nonlinsub)"
	},
	{
		"chunk_id" : "00003204",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solve a System of Nonlinear Equations Using Elimination(http://openstax.org/l/nonlinelim)\n11.3 SECTION EXERCISES\nVerbal\n1. Explain whether a system of 2. When graphing an 3. When you graph a system of\ntwo nonlinear equations can inequality, explain why we inequalities, will there\nhave exactly two solutions. only need to test one point always be a feasible region?\nWhat about exactly three? If to determine whether an If so, explain why. If not,"
	},
	{
		"chunk_id" : "00003205",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "not, explain why not. If so, entire region is the solution? give an example of a graph\ngive an example of such a of inequalities that does not\nsystem, in graph form, and have a feasible region. Why\nexplain why your choice does it not have a feasible\ngives two or three answers. region?\n4. If you graph a revenue and 5. If you perform your break-\ncost function, explain how to even analysis and there is\ndetermine in what regions more than one solution,\nthere is profit. explain how you would"
	},
	{
		"chunk_id" : "00003206",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "there is profit. explain how you would\ndetermine whichx-values\nare profit and which are not.\nAlgebraic\nFor the following exercises, solve the system of nonlinear equations using substitution.\n6. 7. 8.\nAccess for free at openstax.org\n11.3  Systems of Nonlinear Equations and Inequalities: Two Variables 1069\n9. 10.\nFor the following exercises, solve the system of nonlinear equations using elimination.\n11. 12. 13.\n14. 15.\nFor the following exercises, use any method to solve the system of nonlinear equations."
	},
	{
		"chunk_id" : "00003207",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "16. 17. 18.\n19. 20. 21.\n22. 23.\nFor the following exercises, use any method to solve the nonlinear system.\n24. 25. 26.\n27. 28. 29.\n30. 31. 32.\n33. 34. 35.\n36. 37. 38.\nGraphical\nFor the following exercises, graph the inequality.\n39. 40.\n1070 11  Systems of Equations and Inequalities\nFor the following exercises, graph the system of inequalities. Label all points of intersection.\n41. 42. 43.\n44. 45.\nExtensions\nFor the following exercises, graph the inequality.\n46. 47."
	},
	{
		"chunk_id" : "00003208",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47.\nFor the following exercises, find the solutions to the nonlinear equations with two variables.\n48. 49. 50.\n51. 52.\nTechnology\nFor the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.\n53. 54.\nReal-World Applications\nFor the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the\nrequested solutions.\n55. Two numbers add up to 56. The squares of two"
	},
	{
		"chunk_id" : "00003209",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "55. Two numbers add up to 56. The squares of two\n300. One number is twice numbers add to 360. The\nthe square of the other second number is half the\nnumber. What are the value of the first number\nnumbers? squared. What are the\nnumbers?\n57. A laptop company has discovered their cost and 58. A cell phone company has the following cost and\nrevenue functions for each day: revenue functions:\nand and What is the range of\nIf they want to make a cell phones they should produce each day so"
	},
	{
		"chunk_id" : "00003210",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "profit, what is the range of laptops per day that there is profit? Round to the nearest number that\nthey should produce? Round to the nearest generates profit.\nnumber which would generate profit.\nAccess for free at openstax.org\n11.4  Partial Fractions 1071\n11.4 Partial Fractions\nLearning Objectives\nIn this section, you will:\nDecompose ,where has only nonrepeated linear factors.\nDecompose ,where has repeated linear factors.\nDecompose ,where has a nonrepeated irreducible quadratic factor."
	},
	{
		"chunk_id" : "00003211",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Decompose ,where has a repeated irreducible quadratic factor.\nEarlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three\nvariables, and nonlinear systems. Here we introduce another way that systems of equations can be utilizedthe\ndecomposition of rational expressions.\nFractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in\nthis section will help simplify the concept of a rational expression."
	},
	{
		"chunk_id" : "00003212",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Decomposing WhereQ(x)Has Only Nonrepeated Linear Factors\nRecall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a\ncommon denominator so that we can write the sum or difference as a single, simplified rational expression. In this\nsection, we will look atpartial fraction decomposition, which is the undoing of the procedure to add or subtract rational"
	},
	{
		"chunk_id" : "00003213",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expressions. In other words, it is a return from the single simplifiedrational expressionto the original expressions, called\nthepartial fraction.\nFor example, suppose we add the following fractions:\nWe would first need to find a common denominator,\nNext, we would write each expression with this common denominator and find the sum of the terms.\nPartial fractiondecompositionis the reverse of this procedure. We would start with the solution and rewrite\n(decompose) it as the sum of two fractions."
	},
	{
		"chunk_id" : "00003214",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(decompose) it as the sum of two fractions.\nWe will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of\nthe numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing,\nthe first and most important thing to do is factor the denominator.\nWhen the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original"
	},
	{
		"chunk_id" : "00003215",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words,\nusing the example above, the factors of are the denominators of the decomposed rational\nexpression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each\nnumerator. Then, we will solve for each numerator using one of several methods available for partial fraction\ndecomposition.\nPartial Fraction Decomposition of Has Nonrepeated Linear Factors"
	},
	{
		"chunk_id" : "00003216",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thepartial fraction decompositionof when has nonrepeated linear factors and the degree of is less\nthan the degree of is\n1072 11  Systems of Equations and Inequalities\n...\nHOW TO\nGiven a rational expression with distinct linear factors in the denominator, decompose it.\n1. Use a variable for the original numerators, usually or depending on the number of factors, placing\neach variable over a single factor. For the purpose of this definition, we use for each numerator"
	},
	{
		"chunk_id" : "00003217",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Multiply both sides of the equation by the common denominator to eliminate fractions.\n3. Expand the right side of the equation and collect like terms.\n4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system\nof equations to solve for the numerators.\nEXAMPLE1\nDecomposing a Rational Function with Distinct Linear Factors\nDecompose the givenrational expressionwith distinct linear factors.\nSolution"
	},
	{
		"chunk_id" : "00003218",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe will separate the denominator factors and give each numerator a symbolic label, like or\nMultiply both sides of the equation by the common denominator to eliminate the fractions:\nThe resulting equation is\nExpand the right side of the equation and collect like terms.\nSet up a system of equations associating corresponding coefficients.\nAdd the two equations and solve for\nSubstitute into one of the original equations in the system.\nAccess for free at openstax.org\n11.4  Partial Fractions 1073"
	},
	{
		"chunk_id" : "00003219",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.4  Partial Fractions 1073\nThus, the partial fraction decomposition is\nAnother method to use to solve for or is by considering the equation that resulted from eliminating the fractions\nand substituting a value for that will make either theA- orB-term equal 0. If we let the\nterm becomes 0 and we can simply solve for\nNext, either substitute into the equation and solve for or make theB-term 0 by substituting into the\nequation."
	},
	{
		"chunk_id" : "00003220",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation.\nWe obtain the same values for and using either method, so the decompositions are the same using either method.\nAlthough this method is not seen very often in textbooks, we present it here as an alternative that may make some\npartial fraction decompositions easier. It is known as theHeaviside method, named after Charles Heaviside, a pioneer in\nthe study of electronics.\nTRY IT #1 Find the partial fraction decomposition of the following expression.\nDecomposing WhereQ(x)Has Repeated Linear Factors"
	},
	{
		"chunk_id" : "00003221",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Decomposing WhereQ(x)Has Repeated Linear Factors\nSome fractions we may come across are special cases that we can decompose into partial fractions with repeated linear\nfactors. We must remember that we account for repeated factors by writing each factor in increasing powers.\nPartial Fraction Decomposition of Has Repeated Linear Factors\nThe partial fraction decomposition of when has a repeated linear factor occurring times and the degree\nof is less than the degree of is"
	},
	{
		"chunk_id" : "00003222",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of is less than the degree of is\nWrite the denominator powers in increasing order.\n1074 11  Systems of Equations and Inequalities\n...\nHOW TO\nGiven a rational expression with repeated linear factors, decompose it.\n1. Use a variable like or for the numerators and account for increasing powers of the denominators.\n2. Multiply both sides of the equation by the common denominator to eliminate fractions.\n3. Expand the right side of the equation and collect like terms."
	},
	{
		"chunk_id" : "00003223",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system\nof equations to solve for the numerators.\nEXAMPLE2\nDecomposing with Repeated Linear Factors\nDecompose the given rational expression with repeated linear factors.\nSolution\nThe denominator factors are To allow for the repeated factor of the decomposition will include three\ndenominators: and Thus,\nNext, we multiply both sides by the common denominator."
	},
	{
		"chunk_id" : "00003224",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "On the right side of the equation, we expand and collect like terms.\nNext, we compare the coefficients of both sides. This will give the system of equations in three variables:\nSolving for , we have\nSubstitute into equation (1).\nThen, to solve for substitute the values for and into equation (2).\nAccess for free at openstax.org\n11.4  Partial Fractions 1075\nThus,\nTRY IT #2 Find the partial fraction decomposition of the expression with repeated linear factors."
	},
	{
		"chunk_id" : "00003225",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Decomposing WhereQ(x)Has a Nonrepeated Irreducible Quadratic Factor\nSo far, we have performed partial fraction decomposition with expressions that have had linear factors in the\ndenominator, and we applied numerators or representing constants. Now we will look at an example where\none of the factors in the denominator is aquadraticexpression that does not factor. This is referred to as an irreducible\nquadratic factor. In cases like this, we use a linear numerator such as etc."
	},
	{
		"chunk_id" : "00003226",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Decomposition of Has a Nonrepeated Irreducible Quadratic Factor\nThe partial fraction decomposition of such that has a nonrepeated irreducible quadratic factor and\nthe degree of is less than the degree of is written as\nThe decomposition may contain more rational expressions if there are linear factors. Each linear factor will have a\ndifferent constant numerator: and so on.\n...\nHOW TO\nGiven a rational expression where the factors of the denominator are distinct, irreducible quadratic factors,\ndecompose it."
	},
	{
		"chunk_id" : "00003227",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decompose it.\n1. Use variables such as or for the constant numerators over linear factors, and linear expressions\nsuch as etc., for the numerators of each quadratic factor in the denominator.\n2. Multiply both sides of the equation by the common denominator to eliminate fractions.\n3. Expand the right side of the equation and collect like terms.\n4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system\nof equations to solve for the numerators."
	},
	{
		"chunk_id" : "00003228",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of equations to solve for the numerators.\n1076 11  Systems of Equations and Inequalities\nEXAMPLE3\nDecomposing WhenQ(x)Contains a Nonrepeated Irreducible Quadratic Factor\nFind a partial fraction decomposition of the given expression.\nSolution\nWe have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant\nand the other numerator will be a linear expression. Thus,"
	},
	{
		"chunk_id" : "00003229",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by\nthe common denominator.\nNotice we could easily solve for by choosing a value for that will make the term equal 0. Let\nand substitute it into the equation.\nNow that we know the value of substitute it back into the equation. Then expand the right side and collect like\nterms."
	},
	{
		"chunk_id" : "00003230",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms.\nSetting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of\nequations.\nSolve for using equation (1) and solve for using equation (3).\nThus, the partial fraction decomposition of the expression is\nAccess for free at openstax.org\n11.4  Partial Fractions 1077\nQ&A Could we have just set up a system of equations to solveExample 3?\nYes, we could have solved it by setting up a system of equations without solving for first. The"
	},
	{
		"chunk_id" : "00003231",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expansion on the right would be:\nSo the system of equations would be:\nTRY IT #3 Find the partial fraction decomposition of the expression with a nonrepeating irreducible\nquadratic factor.\nDecomposing WhenQ(x)Has a Repeated Irreducible Quadratic Factor\nNow that we can decompose a simplifiedrational expressionwith an irreduciblequadraticfactor, we will learn how to do\npartial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The"
	},
	{
		"chunk_id" : "00003232",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented\nin increasing powers.\nDecomposition of WhenQ(x)Has a Repeated Irreducible Quadratic Factor\nThe partial fraction decomposition of when has a repeated irreducible quadratic factor and the\ndegree of is less than the degree of is\nWrite the denominators in increasing powers.\n...\nHOW TO\nGiven a rational expression that has a repeated irreducible factor, decompose it."
	},
	{
		"chunk_id" : "00003233",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Use variables like or for the constant numerators over linear factors, and linear expressions such as\netc., for the numerators of each quadratic factor in the denominator written in\nincreasing powers, such as\n2. Multiply both sides of the equation by the common denominator to eliminate fractions.\n3. Expand the right side of the equation and collect like terms.\n4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system"
	},
	{
		"chunk_id" : "00003234",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of equations to solve for the numerators.\n1078 11  Systems of Equations and Inequalities\nEXAMPLE4\nDecomposing a Rational Function with a Repeated Irreducible Quadratic Factor in the Denominator\nDecompose the given expression that has a repeated irreducible factor in the denominator.\nSolution\nThe factors of the denominator are and Recall that, when a factor in the denominator is a\nquadratic that includes at least two terms, the numerator must be of the linear form So, lets begin the\ndecomposition."
	},
	{
		"chunk_id" : "00003235",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "decomposition.\nWe eliminate the denominators by multiplying each term by Thus,\nExpand the right side.\nNow we will collect like terms.\nSet up the system of equations matching corresponding coefficients on each side of the equal sign.\nWe can use substitution from this point. Substitute into the first equation.\nSubstitute and into the third equation.\nSubstitute into the fourth equation.\nNow we have solved for all of the unknowns on the right side of the equal sign. We have"
	},
	{
		"chunk_id" : "00003236",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and We can write the decomposition as follows:\nTRY IT #4 Find the partial fraction decomposition of the expression with a repeated irreducible quadratic\nfactor.\nAccess for free at openstax.org\n11.4  Partial Fractions 1079\nMEDIA\nAccess these online resources for additional instruction and practice with partial fractions.\nPartial Fraction Decomposition(http://openstax.org/l/partdecomp)\nPartial Fraction Decomposition With Repeated Linear Factors(http://openstax.org/l/partdecomprlf)"
	},
	{
		"chunk_id" : "00003237",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Partial Fraction Decomposition With Linear and Quadratic Factors(http://openstax.org/l/partdecomlqu)\n11.4 SECTION EXERCISES\nVerbal\n1. Can any quotient of 2. Can you explain why a 3. Can you explain how to\npolynomials be decomposed partial fraction verify a partial fraction\ninto at least two partial decomposition is unique? decomposition graphically?\nfractions? If so, explain why, (Hint: Think about it as a\nand if not, give an example system of equations.)\nof such a fraction"
	},
	{
		"chunk_id" : "00003238",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of such a fraction\n4. You are unsure if you correctly decomposed the 5. Once you have a system of equations generated by\npartial fraction correctly. Explain how you could the partial fraction decomposition, can you explain\ndouble-check your answer. another method to solve it? For example if you had\n, we eventually simplify\nto Explain how\nyou could intelligently choose an -value that will\neliminate either or and solve for and\nAlgebraic"
	},
	{
		"chunk_id" : "00003239",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "eliminate either or and solve for and\nAlgebraic\nFor the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n18. 19.\nFor the following exercises, find the decomposition of the partial fraction for the repeating linear factors.\n20. 21. 22.\n1080 11  Systems of Equations and Inequalities\n23. 24. 25.\n26. 27. 28.\n29. 30."
	},
	{
		"chunk_id" : "00003240",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. 24. 25.\n26. 27. 28.\n29. 30.\nFor the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic\nfactor.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n43.\nFor the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.\n44. 45. 46.\n47. 48. 49.\n50. 51. 52.\n53. 54.\nExtensions\nFor the following exercises, find the partial fraction expansion.\n55. 56.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003241",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "55. 56.\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1081\nFor the following exercises, perform the operation and then find the partial fraction decomposition.\n57. 58. 59.\n11.5 Matrices and Matrix Operations\nLearning Objectives\nIn this section, you will:\nFind the sum and difference of two matrices.\nFind scalar multiples of a matrix.\nFind the product of two matrices.\nFigure1 (credit: SD Dirk, Flickr)"
	},
	{
		"chunk_id" : "00003242",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1 (credit: SD Dirk, Flickr)\nTwo club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.\nTable 1shows the needs of both teams.\nWildcats Mud Cats\nGoals 6 10\nBalls 30 24\nJerseys 14 20\nTable1\nA goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for\neach team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and"
	},
	{
		"chunk_id" : "00003243",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "used for calculating other information. Then, we will be able to calculate the cost of the equipment.\nFinding the Sum and Difference of Two Matrices\nTo solve a problem like the one described for the soccer teams, we can use amatrix, which is a rectangular array of\nnumbers. Arowin a matrix is a set of numbers that are aligned horizontally. Acolumnin a matrix is a set of numbers\nthat are aligned vertically. Each number is anentry, sometimes called an element, of the matrix. Matrices (plural) are"
	},
	{
		"chunk_id" : "00003244",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named and are\nshown below.\n1082 11  Systems of Equations and Inequalities\nDescribing Matrices\nA matrix is often referred to by its size or dimensions: indicating rows and columns. Matrix entries are defined\nfirst by row and then by column. For example, to locate the entry in matrix identified as we look for the entry in\nrow column In matrix shown below, the entry in row 2, column 3 is"
	},
	{
		"chunk_id" : "00003245",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Asquare matrixis a matrix with dimensions meaning that it has the same number of rows as columns. The\nmatrix above is an example of a square matrix.\nArow matrixis a matrix consisting of one row with dimensions\nAcolumn matrixis a matrix consisting of one column with dimensions\nA matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the"
	},
	{
		"chunk_id" : "00003246",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with\nvariables. We will investigate this idea further in the next section, but first we will look at basicmatrix operations.\nMatrices\nAmatrixis a rectangular array of numbers that is usually named by a capital letter: and so on. Each entry in\na matrix is referred to as such that represents the row and represents the column. Matrices are often referred"
	},
	{
		"chunk_id" : "00003247",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to by their dimensions: indicating rows and columns.\nEXAMPLE1\nFinding the Dimensions of the Given Matrix and Locating Entries\nGiven matrix\n What are the dimensions of matrix  What are the entries at and\nSolution\n The dimensions are because there are three rows and three columns.\n Entry is the number at row 3, column 1, which is 3. The entry is the number at row 2, column 2, which is 4.\nRemember, the row comes first, then the column.\nAdding and Subtracting Matrices"
	},
	{
		"chunk_id" : "00003248",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Adding and Subtracting Matrices\nWe use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on\nmatrices. We add or subtract matrices by adding or subtracting corresponding entries.\nIn order to do this, the entries must correspond. Therefore,addition and subtraction of matrices is only possible when\nthe matrices have the same dimensions. We can add or subtract a matrix and another matrix, but we cannot"
	},
	{
		"chunk_id" : "00003249",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "add or subtract a matrix and a matrix because some entries in one matrix will not have a corresponding\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1083\nentry in the other matrix.\nAdding and Subtracting Matrices\nGiven matrices and of like dimensions, addition and subtraction of and will produce matrix or\nmatrix of the same dimension.\nMatrix addition is commutative.\nIt is also associative.\nEXAMPLE2\nFinding the Sum of Matrices\nFind the sum of and given\nSolution"
	},
	{
		"chunk_id" : "00003250",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the sum of and given\nSolution\nAdd corresponding entries.\nEXAMPLE3\nAdding MatrixAand MatrixB\nFind the sum of and\nSolution\nAdd corresponding entries. Add the entry in row 1, column 1, of matrix to the entry in row 1, column 1, of\nContinue the pattern until all entries have been added.\n1084 11  Systems of Equations and Inequalities\nEXAMPLE4\nFinding the Difference of Two Matrices\nFind the difference of and\nSolution\nWe subtract the corresponding entries of each matrix.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00003251",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE5\nFinding the Sum and Difference of Two 3 x 3 Matrices\nGiven and\n Find the sum.  Find the difference.\nSolution\n Add the corresponding entries.  Subtract the corresponding entries.\nTRY IT #1 Add matrix and matrix\nFinding Scalar Multiples of a Matrix\nBesides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a\nconstant called a scalar. Recall that ascalaris a real number quantity that has magnitude, but not direction. For example,"
	},
	{
		"chunk_id" : "00003252",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "time, temperature, and distance are scalar quantities. The process ofscalar multiplicationinvolves multiplying each\nentry in a matrix by a scalar. Ascalar multipleis any entry of a matrix that results from scalar multiplication.\nConsider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1085"
	},
	{
		"chunk_id" : "00003253",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.5  Matrices and Matrix Operations 1085\nchairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both\nlabs. The schools current inventory is displayed inTable 2.\nLab A Lab B\nComputers 15 27\nComputer Tables 16 34\nChairs 16 34\nTable2\nConverting the data to a matrix, we have\nTo calculate how much computer equipment will be needed, we multiply all entries in matrix by 0.15.\nWe must round up to the next integer, so the amount of new equipment needed is"
	},
	{
		"chunk_id" : "00003254",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Adding the two matrices as shown below, we see the new inventory amounts.\nThis means\nThus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables,\nand 40 chairs.\nScalar Multiplication\nScalar multiplication involves finding the product of a constant by each entry in the matrix. Given\nthe scalar multiple is\nScalar multiplication is distributive. For the matrices and with scalars and\n1086 11  Systems of Equations and Inequalities\nEXAMPLE6"
	},
	{
		"chunk_id" : "00003255",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nMultiplying the Matrix by a Scalar\nMultiply matrix by the scalar 3.\nSolution\nMultiply each entry in by the scalar 3.\nTRY IT #2 Given matrix find where\nEXAMPLE7\nFinding the Sum of Scalar Multiples\nFind the sum\nSolution\nFirst, find then\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1087\nNow, add\nFinding the Product of Two Matrices\nIn addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding theproduct of two matricesis only"
	},
	{
		"chunk_id" : "00003256",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to\nthe number of rows of the second matrix. If is an matrix and is an matrix, then the product matrix is\nan matrix. For example, the product is possible because the number of columns in is the same as the\nnumber of rows in If the inner dimensions do not match, the product is not defined."
	},
	{
		"chunk_id" : "00003257",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We multiply entries of with entries of according to a specific pattern as outlined below. The process ofmatrix\nmultiplicationbecomes clearer when working a problem with real numbers.\nTo obtain the entries in row of we multiply the entries in row of by column in and add. For example, given\nmatrices and where the dimensions of are and the dimensions of are the product of will be a\nmatrix.\nMultiply and add as follows to obtain the first entry of the product matrix"
	},
	{
		"chunk_id" : "00003258",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. To obtain the entry in row 1, column 1 of multiply the first row in by the first column in and add.\n2. To obtain the entry in row 1, column 2 of multiply the first row of by the second column in and add.\n1088 11  Systems of Equations and Inequalities\n3. To obtain the entry in row 1, column 3 of multiply the first row of by the third column in and add.\nWe proceed the same way to obtain the second row of In other words, row 2 of times column 1 of row 2 of"
	},
	{
		"chunk_id" : "00003259",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "times column 2 of row 2 of times column 3 of When complete, the product matrix will be\nProperties of Matrix Multiplication\nFor the matrices and the following properties hold.\n Matrix multiplication is associative:\n Matrix multiplication is distributive:\nNote that matrix multiplication is not commutative.\nEXAMPLE8\nMultiplying Two Matrices\nMultiply matrix and matrix\nSolution\nFirst, we check the dimensions of the matrices. Matrix has dimensions and matrix has dimensions The"
	},
	{
		"chunk_id" : "00003260",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inner dimensions are the same so we can perform the multiplication. The product will have the dimensions\nWe perform the operations outlined previously.\nEXAMPLE9\nMultiplying Two Matrices\nGiven and\n Find  Find\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1089\nSolution\n As the dimensions of are and the dimensions of are these matrices can be multiplied together\nbecause the number of columns in matches the number of rows in The resulting product will be a matrix,"
	},
	{
		"chunk_id" : "00003261",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the number of rows in by the number of columns in\n The dimensions of are and the dimensions of are The inner dimensions match so the product is\ndefined and will be a matrix.\nAnalysis\nNotice that the products and are not equal.\nThis illustrates the fact that matrix multiplication is not commutative.\nQ&A Is it possible forABto be defined but notBA?\nYes, consider a matrix A with dimension and matrix B with dimension For the product AB the"
	},
	{
		"chunk_id" : "00003262",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and\n3 so the product is undefined.\nEXAMPLE10\nUsing Matrices in Real-World Problems\nLets return to the problem presented at the opening of this section. We haveTable 3, representing the equipment\nneeds of two soccer teams.\nWildcats Mud Cats\nGoals 6 10\nTable3\n1090 11  Systems of Equations and Inequalities\nWildcats Mud Cats\nBalls 30 24\nJerseys 14 20\nTable3"
	},
	{
		"chunk_id" : "00003263",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Balls 30 24\nJerseys 14 20\nTable3\nWe are also given the prices of the equipment, as shown inTable 4.\nGoal $300\nBall $10\nJersey $30\nTable4\nWe will convert the data to matrices. Thus, the equipment need matrix is written as\nThe cost matrix is written as\nWe perform matrix multiplication to obtain costs for the equipment.\nThe total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.\n...\nHOW TO\nGiven a matrix operation, evaluate using a calculator."
	},
	{
		"chunk_id" : "00003264",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Save each matrix as a matrix variable\n2. Enter the operation into the calculator, calling up each matrix variable as needed.\n3. If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will\ndisplay an error message.\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1091\nEXAMPLE11\nUsing a Calculator to Perform Matrix Operations\nFind given\nSolution"
	},
	{
		"chunk_id" : "00003265",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find given\nSolution\nOn the matrix page of the calculator, we enter matrix above as the matrix variable matrix above as the matrix\nvariable and matrix above as the matrix variable\nOn the home screen of the calculator, we type in the problem and call up each matrix variable as needed.\nThe calculator gives us the following matrix.\nMEDIA\nAccess these online resources for additional instruction and practice with matrices and matrix operations.\nDimensions of a Matrix(http://openstax.org/l/matrixdimen)"
	},
	{
		"chunk_id" : "00003266",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Matrix Addition and Subtraction(http://openstax.org/l/matrixaddsub)\nMatrix Operations(http://openstax.org/l/matrixoper)\nMatrix Multiplication(http://openstax.org/l/matrixmult)\n11.5 SECTION EXERCISES\nVerbal\n1. Can we add any two 2. Can we multiply any column 3. Can both the products\nmatrices together? If so, matrix by any row matrix? and be defined? If so,\nexplain why; if not, explain Explain why or why not. explain how; if not, explain\nwhy not and give an why.\nexample of two matrices\nthat cannot be added"
	},
	{
		"chunk_id" : "00003267",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "example of two matrices\nthat cannot be added\ntogether.\n4. Can any two matrices of the 5. Does matrix multiplication\nsame size be multiplied? If commute? That is, does\nso, explain why, and if not, If so, prove why\nexplain why not and give an it does. If not, explain why it\nexample of two matrices of does not.\nthe same size that cannot be\nmultiplied together.\n1092 11  Systems of Equations and Inequalities\nAlgebraic"
	},
	{
		"chunk_id" : "00003268",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Algebraic\nFor the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the\noperation is undefined.\n6. 7. 8.\n9. 10. 11.\nFor the following exercises, use the matrices below to perform scalar multiplication.\n12. 13. 14.\n15. 16. 17.\nFor the following exercises, use the matrices below to perform matrix multiplication.\n18. 19. 20.\n21. 22. 23."
	},
	{
		"chunk_id" : "00003269",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "18. 19. 20.\n21. 22. 23.\nFor the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain\nwhy the operation cannot be performed.\n24. 25. 26.\n27. 28. 29.\nFor the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain\nwhy the operation cannot be performed. (Hint: )\n30. 31. 32.\nAccess for free at openstax.org\n11.5  Matrices and Matrix Operations 1093\n33. 34. 35.\n36. 37. 38.\n39. 40."
	},
	{
		"chunk_id" : "00003270",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "33. 34. 35.\n36. 37. 38.\n39. 40.\nFor the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain\nwhy the operation cannot be performed. (Hint: )\n41. 42. 43.\n44. 45. 46.\n47. 48. 49.\nTechnology\nFor the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain\nwhy the operation cannot be performed. Use a calculator to verify your solution.\n50. 51. 52.\n53. 54.\nExtensions"
	},
	{
		"chunk_id" : "00003271",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "50. 51. 52.\n53. 54.\nExtensions\nFor the following exercises, use the matrix below to perform the indicated operation on the given matrix.\n55. 56. 57.\n58. 59. Using the above questions,\nfind a formula for Test\nthe formula for and\nusing a calculator.\n1094 11  Systems of Equations and Inequalities\n11.6 Solving Systems with Gaussian Elimination\nLearning Objectives\nIn this section, you will:\nWrite the augmented matrix of a system of equations.\nWrite the system of equations from an augmented matrix."
	},
	{
		"chunk_id" : "00003272",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Perform row operations on a matrix.\nSolve a system of linear equations using matrices.\nFigure1 German mathematician Carl Friedrich Gauss (17771855).\nCarl FriedrichGausslived during the late 18th century and early 19th century, but he is still considered one of the most\nprolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as\nalgebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding"
	},
	{
		"chunk_id" : "00003273",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "matrix theory changed the way mathematicians have worked for the last two centuries.\nWe first encountered Gaussian elimination inSystems of Linear Equations: Two Variables. In this section, we will revisit\nthis technique for solving systems, this time using matrices.\nWriting the Augmented Matrix of a System of Equations\nAmatrixcan serve as a device for representing and solving a system of equations. To express a system in matrix form,"
	},
	{
		"chunk_id" : "00003274",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a\nvertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is\nwritten in this form, we call it anaugmented matrix.\nFor example, consider the following system of equations.\nWe can write this system as an augmented matrix:\nWe can also write a matrix containing just the coefficients. This is called thecoefficient matrix."
	},
	{
		"chunk_id" : "00003275",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A three-by-threesystem of equationssuch as\nhas a coefficient matrix\nAccess for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1095\nand is represented by the augmented matrix\nNotice that the matrix is written so that the variables line up in their own columns:x-terms go in the first column,\ny-terms in the second column, andz-terms in the third column. It is very important that each equation is written in"
	},
	{
		"chunk_id" : "00003276",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "standard form so that the variables line up. When there is a missing variable term in an equation, the\ncoefficient is 0.\n...\nHOW TO\nGiven a system of equations, write an augmented matrix.\n1. Write the coefficients of thex-terms as the numbers down the first column.\n2. Write the coefficients of they-terms as the numbers down the second column.\n3. If there arez-terms, write the coefficients as the numbers down the third column.\n4. Draw a vertical line and write the constants to the right of the line."
	},
	{
		"chunk_id" : "00003277",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nWriting the Augmented Matrix for a System of Equations\nWrite the augmented matrix for the given system of equations.\nSolution\nThe augmented matrix displays the coefficients of the variables, and an additional column for the constants.\nTRY IT #1 Write the augmented matrix of the given system of equations.\nWriting a System of Equations from an Augmented Matrix\nWe can use augmented matrices to help us solve systems of equations because they simplify operations when the"
	},
	{
		"chunk_id" : "00003278",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "systems are not encumbered by the variables. However, it is important to understand how to move back and forth\nbetween formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an\naugmented matrix to write thesystem of equationsin standard form.\nEXAMPLE2\nWriting a System of Equations from an Augmented Matrix Form\nFind the system of equations from the augmented matrix.\n1096 11  Systems of Equations and Inequalities\nSolution"
	},
	{
		"chunk_id" : "00003279",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWhen the columns represent the variables and\nTRY IT #2 Write the system of equations from the augmented matrix.\nPerforming Row Operations on a Matrix\nNow that we can write systems of equations in augmented matrix form, we will examine the variousrow operations\nthat can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.\nPerforming row operations on a matrix is the method we use for solving a system of equations. In order to solve the"
	},
	{
		"chunk_id" : "00003280",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "system of equations, we want to convert the matrix torow-echelon form, in which there are ones down themain\ndiagonalfrom the upper left corner to the lower right corner, and zeros in every position below the main diagonal as\nshown.\nWe use row operations corresponding to equation operations to obtain a new matrix that isrow-equivalentin a simpler\nform. Here are the guidelines to obtaining row-echelon form.\n1. In any nonzero row, the first nonzero number is a 1. It is called aleading1."
	},
	{
		"chunk_id" : "00003281",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Any all-zero rows are placed at the bottom on the matrix.\n3. Any leading 1 is below and to the right of a previous leading 1.\n4. Any column containing a leading 1 has zeros in all other positions in the column.\nTo solve a system of equations we can perform the following row operations to convert thecoefficient matrixto row-\nechelon form and do back-substitution to find the solution.\n1. Interchange rows. (Notation: )\n2. Multiply a row by a constant. (Notation: )"
	},
	{
		"chunk_id" : "00003282",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Multiply a row by a constant. (Notation: )\n3. Add the product of a row multiplied by a constant to another row. (Notation:\nEach of the row operations corresponds to the operations we have already learned to solve systems of equations in\nthree variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in\nrow-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method"
	},
	{
		"chunk_id" : "00003283",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.\nGaussian Elimination\nTheGaussian eliminationmethod refers to a strategy used to obtain the row-echelon form of a matrix. The goal is\nto write matrix with the number 1 as the entry down the main diagonal and have all zeros below.\nAccess for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1097"
	},
	{
		"chunk_id" : "00003284",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first step of the Gaussian strategy includes obtaining a 1 as the first entry, so that row 1 may be used to alter the\nrows below.\n...\nHOW TO\nGiven an augmented matrix, perform row operations to achieve row-echelon form.\n1. The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary.\n2. Use row operations to obtain zeros down the first column below the first entry of 1.\n3. Use row operations to obtain a 1 in row 2, column 2."
	},
	{
		"chunk_id" : "00003285",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Use row operations to obtain zeros down column 2, below the entry of 1.\n5. Use row operations to obtain a 1 in row 3, column 3.\n6. Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only\nzeros below.\n7. If any rows contain all zeros, place them at the bottom.\nEXAMPLE3\nSolving a System by Gaussian Elimination\nSolve the given system by Gaussian elimination.\nSolution\nFirst, we write this as an augmented matrix."
	},
	{
		"chunk_id" : "00003286",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "First, we write this as an augmented matrix.\nWe want a 1 in row 1, column 1. This can be accomplished by interchanging row 1 and row 2.\nWe now have a 1 as the first entry in row 1, column 1. Now lets obtain a 0 in row 2, column 1. This can be accomplished\nby multiplying row 1 by and then adding the result to row 2.\nWe only have one more step, to multiply row 2 by\nUse back-substitution. The second row of the matrix represents Back-substitute into the first equation."
	},
	{
		"chunk_id" : "00003287",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1098 11  Systems of Equations and Inequalities\nThe solution is the point\nTRY IT #3 Solve the given system by Gaussian elimination.\nEXAMPLE4\nUsing Gaussian Elimination to Solve a System of Equations\nUseGaussian eliminationto solve the given system of equations.\nSolution\nWrite the system as anaugmented matrix.\nObtain a 1 in row 1, column 1. This can be accomplished by multiplying the first row by\nNext, we want a 0 in row 2, column 1. Multiply row 1 by and add row 1 to row 2."
	},
	{
		"chunk_id" : "00003288",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The second row represents the equation Therefore, the system is inconsistent and has no solution.\nEXAMPLE5\nSolving a Dependent System\nSolve the system of equations.\nSolution\nPerformrow operationson the augmented matrix to try and achieverow-echelon form.\nThe matrix ends up with all zeros in the last row: Thus, there are an infinite number of solutions and the system\nis classified as dependent. To find the generic solution, return to one of the original equations and solve for"
	},
	{
		"chunk_id" : "00003289",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1099\nSo the solution to this system is\nEXAMPLE6\nPerforming Row Operations on a 33 Augmented Matrix to Obtain Row-Echelon Form\nPerform row operations on the given matrix to obtain row-echelon form.\nSolution\nThe first row already has a 1 in row 1, column 1. The next step is to multiply row 1 by and add it to row 2. Then replace\nrow 2 with the result.\nNext, obtain a zero in row 3, column 1."
	},
	{
		"chunk_id" : "00003290",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Next, obtain a zero in row 3, column 1.\nNext, obtain a zero in row 3, column 2.\nThe last step is to obtain a 1 in row 3, column 3.\nTRY IT #4 Write the system of equations in row-echelon form.\nSolving a System of Linear Equations Using Matrices\nWe have seen how to write asystem of equationswith anaugmented matrix, and then how to use row operations and\nback-substitution to obtainrow-echelon form. Now, we will take row-echelon form a step farther to solve a 3 by 3 system"
	},
	{
		"chunk_id" : "00003291",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to\nsolve for the other variables.\n1100 11  Systems of Equations and Inequalities\nEXAMPLE7\nSolving a System of Linear Equations Using Matrices\nSolve the system of linear equations using matrices.\nSolution\nFirst, we write the augmented matrix.\nNext, we perform row operations to obtain row-echelon form.\nThe easiest way to obtain a 1 in row 2 of column 1 is to interchange and\nThen"
	},
	{
		"chunk_id" : "00003292",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then\nThe last matrix represents the equivalent system.\nUsing back-substitution, we obtain the solution as\nEXAMPLE8\nSolving a Dependent System of Linear Equations Using Matrices\nSolve the following system of linear equations using matrices.\nSolution\nWrite the augmented matrix.\nAccess for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1101\nFirst, multiply row 1 by to get a 1 in row 1, column 1. Then, performrow operationsto obtain row-echelon form."
	},
	{
		"chunk_id" : "00003293",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The last matrix represents the following system.\nWe see by the identity that this is a dependent system with an infinite number of solutions. We then find the\ngeneric solution. By solving the second equation for and substituting it into the first equation we can solve for in\nterms of\nNow we substitute the expression for into the second equation to solve for in terms of\nThe generic solution is\nTRY IT #5 Solve the system using matrices."
	},
	{
		"chunk_id" : "00003294",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 Solve the system using matrices.\nQ&A Can any system of linear equations be solved by Gaussian elimination?\nYes, a system of linear equations of any size can be solved by Gaussian elimination.\n1102 11  Systems of Equations and Inequalities\n...\nHOW TO\nGiven a system of equations, solve with matrices using a calculator.\n1. Save the augmented matrix as a matrix variable\n2. Use theref(function in the calculator, calling up each matrix variable as needed.\nEXAMPLE9"
	},
	{
		"chunk_id" : "00003295",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE9\nSolving Systems of Equations with Matrices Using a Calculator\nSolve the system of equations.\nSolution\nWrite the augmented matrix for the system of equations.\nOn the matrix page of the calculator, enter the augmented matrix above as the matrix variable\nUse theref(function in the calculator, calling up the matrix variable\nEvaluate.\nUsing back-substitution, the solution is\nEXAMPLE10\nApplying 2  2 Matrices to Finance"
	},
	{
		"chunk_id" : "00003296",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE10\nApplying 2  2 Matrices to Finance\nCarolyn invests a total of $12,000 in two municipal bonds, one paying 10.5% interest and the other paying 12% interest.\nThe annual interest earned on the two investments last year was $1,335. How much was invested at each rate?\nSolution\nWe have a system of two equations in two variables. Let the amount invested at 10.5% interest, and the amount\ninvested at 12% interest.\nAs a matrix, we have\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003297",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1103\nMultiply row 1 by and add the result to row 2.\nThen,\nSo\nThus, $5,000 was invested at 12% interest and $7,000 at 10.5% interest.\nEXAMPLE11\nApplying 3  3 Matrices to Finance\nAva invests a total of $10,000 in three accounts, one paying 5% interest, another paying 8% interest, and the third paying\n9% interest. The annual interest earned on the three investments last year was $770. The amount invested at 9% was"
	},
	{
		"chunk_id" : "00003298",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "twice the amount invested at 5%. How much was invested at each rate?\nSolution\nWe have a system of three equations in three variables. Let be the amount invested at 5% interest, let be the amount\ninvested at 8% interest, and let be the amount invested at 9% interest. Thus,\nAs a matrix, we have\nNow, we perform Gaussian elimination to achieve row-echelon form.\nThe third row tells us thus\nThe second row tells us Substituting we get\n1104 11  Systems of Equations and Inequalities"
	},
	{
		"chunk_id" : "00003299",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1104 11  Systems of Equations and Inequalities\nThe first row tells us Substituting and we get\nThe answer is $3,000 invested at 5% interest, $1,000 invested at 8%, and $6,000 invested at 9% interest.\nTRY IT #6 A small shoe company took out a loan of $1,500,000 to expand their inventory. Part of the money\nwas borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount\nborrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three"
	},
	{
		"chunk_id" : "00003300",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "loans was $130,500. Use matrices to find the amount borrowed at each rate.\nMEDIA\nAccess these online resources for additional instruction and practice with solving systems of linear equations using\nGaussian elimination.\nSolve a System of Two Equations Using an Augmented Matrix(http://openstax.org/l/system2augmat)\nSolve a System of Three Equations Using an Augmented Matrix(http://openstax.org/l/system3augmat)\nAugmented Matrices on the Calculator(http://openstax.org/l/augmatcalc)\n11.6 SECTION EXERCISES"
	},
	{
		"chunk_id" : "00003301",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.6 SECTION EXERCISES\nVerbal\n1. Can any system of linear 2. Can any matrix be written as 3. Is there only one correct\nequations be written as an a system of linear method of using row\naugmented matrix? Explain equations? Explain why or operations on a matrix? Try\nwhy or why not. Explain how why not. Explain how to to explain two different row\nto write that augmented write that system of operations possible to solve\nmatrix. equations. the augmented matrix"
	},
	{
		"chunk_id" : "00003302",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "matrix. equations. the augmented matrix\n4. Can a matrix whose entry is 5. Can a matrix that has 0\n0 on the diagonal be solved? entries for an entire row\nExplain why or why not. have one solution? Explain\nWhat would you do to why or why not.\nremedy the situation?\nAlgebraic\nFor the following exercises, write the augmented matrix for the linear system.\n6. 7. 8.\nAccess for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1105\n9. 10."
	},
	{
		"chunk_id" : "00003303",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9. 10.\nFor the following exercises, write the linear system from the augmented matrix.\n11. 12. 13.\n14. 15.\nFor the following exercises, solve the system by Gaussian elimination.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n1106 11  Systems of Equations and Inequalities\n43. 44. 45.\n46.\nExtensions\nFor the following exercises, use Gaussian elimination to solve the system.\n47. 48. 49.\n50. 51.\nReal-World Applications"
	},
	{
		"chunk_id" : "00003304",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. 48. 49.\n50. 51.\nReal-World Applications\nFor the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.\n52. Every day, Angeni's 53. At Bakari's competing 54. You invested $10,000 into\ncupcake store sells 5,000 cupcake store, $4,520 two accounts: one that has\ncupcakes in chocolate and worth of cupcakes are sold simple 3% interest, the\nvanilla flavors. If the daily. The chocolate other with 2.5% interest. If"
	},
	{
		"chunk_id" : "00003305",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "chocolate flavor is 3 times cupcakes cost $2.25 and your total interest payment\nas popular as the vanilla the red velvet cupcakes after one year was $283.50,\nflavor, how many of each cost $1.75. If the total how much was in each\ncupcake does the store sell number of cupcakes sold account after the year\nper day? per day is 2,200, how many passed?\nof each flavor are sold each\nday?\nAccess for free at openstax.org\n11.6  Solving Systems with Gaussian Elimination 1107"
	},
	{
		"chunk_id" : "00003306",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "55. You invested $2,300 into 56. BikesRUs manufactures 57. A major appliance store\naccount 1, and $2,700 into bikes, which sell for $250. It has agreed to order\naccount 2. If the total costs the manufacturer vacuums from a startup\namount of interest after $180 per bike, plus a founded by college\none year is $254, and startup fee of $3,500. After engineering students. The\naccount 2 has 1.5 times the how many bikes sold will store would be able to"
	},
	{
		"chunk_id" : "00003307",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interest rate of account 1, the manufacturer break purchase the vacuums for\nwhat are the interest rates? even? $86 each, with a delivery\nAssume simple interest fee of $9,200, regardless of\nrates. how many vacuums are\nsold. If the store needs to\nstart seeing a profit after\n230 units are sold, how\nmuch should they charge\nfor the vacuums?\n58. The three most popular ice 59. At an ice cream shop, three 60. A bag of mixed nuts\ncream flavors are flavors are increasing in contains cashews,"
	},
	{
		"chunk_id" : "00003308",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "chocolate, strawberry, and demand. Last year, pistachios, and almonds.\nvanilla, comprising 83% of banana, pumpkin, and There are 1,000 total nuts\nthe flavors sold at an ice rocky road ice cream made in the bag, and there are\ncream shop. If vanilla sells up 12% of total ice cream 100 less almonds than\n1% more than twice sales. This year, the same pistachios. The cashews\nstrawberry, and chocolate three ice creams made up weigh 3 g, pistachios weigh"
	},
	{
		"chunk_id" : "00003309",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sells 11% more than vanilla, 16.9% of ice cream sales. 4 g, and almonds weigh 5\nhow much of the total ice The rocky road sales g. If the bag weighs 3.7 kg,\ncream consumption are doubled, the banana sales find out how many of each\nthe vanilla, chocolate, and increased by 50%, and the type of nut is in the bag.\nstrawberry flavors? pumpkin sales increased\nby 20%. If the rocky road\nice cream had one less\npercent of sales than the\nbanana ice cream, find out\nthe percentage of ice\ncream sales each individual"
	},
	{
		"chunk_id" : "00003310",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the percentage of ice\ncream sales each individual\nice cream made last year.\n61. A bag of mixed nuts\ncontains cashews,\npistachios, and almonds.\nOriginally there were 900\nnuts in the bag. 30% of the\nalmonds, 20% of the\ncashews, and 10% of the\npistachios were eaten, and\nnow there are 770 nuts left\nin the bag. Originally, there\nwere 100 more cashews\nthan almonds. Figure out\nhow many of each type of\nnut was in the bag to begin\nwith.\n1108 11  Systems of Equations and Inequalities"
	},
	{
		"chunk_id" : "00003311",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1108 11  Systems of Equations and Inequalities\n11.7 Solving Systems with Inverses\nLearning Objectives\nIn this section, you will:\nFind the inverse of a matrix.\nSolve a system of linear equations using an inverse matrix.\nSoriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%,\nand the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much"
	},
	{
		"chunk_id" : "00003312",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "should Soriya invest in each bond? What is the best method to solve this problem?\nThere are several ways we can solve this problem. As we have seen in previous sections, systems of equations and\nmatrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to\nsolve the bond problem using the inverse of a matrix.\nFinding the Inverse of a Matrix\nWe know that the multiplicative inverse of a real number is and For example,"
	},
	{
		"chunk_id" : "00003313",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and Themultiplicative inverse of a matrixis similar in concept, except that the product of matrix and its\ninverse equals theidentity matrix. The identity matrix is a square matrix containing ones down the main diagonal\nand zeros everywhere else. We identify identity matrices by where represents the dimension of the matrix. Observe\nthe following equations.\nThe identity matrix acts as a 1 in matrix algebra. For example,\nA matrix that has a multiplicative inverse has the properties"
	},
	{
		"chunk_id" : "00003314",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A matrix that has a multiplicative inverse is called aninvertible matrix. Only a square matrix may have a multiplicative\ninverse, as the reversibility, is a requirement. Not all square matrices have an inverse, but if is\ninvertible, then is unique. We will look at two methods for finding the inverse of a matrix and a third method\nthat can be used on both and matrices.\nThe Identity Matrix and Multiplicative Inverse"
	},
	{
		"chunk_id" : "00003315",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Identity Matrix and Multiplicative Inverse\nTheidentity matrix, is a square matrix containing ones down the main diagonal and zeros everywhere else.\nIf is an matrix and is an matrix such that then themultiplicative\ninverse of a matrix\nEXAMPLE1\nShowing That the Identity Matrix Acts as a 1\nGiven matrixA, show that\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1109\nSolution"
	},
	{
		"chunk_id" : "00003316",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUse matrix multiplication to show that the product of and the identity is equal to the product of the identity andA.\n...\nHOW TO\nGiven two matrices, show that one is the multiplicative inverse of the other.\n1. Given matrix of order and matrix of order multiply\n2. If then find the product If then and\nEXAMPLE2\nShowing That MatrixAIs the Multiplicative Inverse of MatrixB\nShow that the given matrices are multiplicative inverses of each other.\nSolution"
	},
	{
		"chunk_id" : "00003317",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nMultiply and If both products equal the identity, then the two matrices are inverses of each other.\nand are inverses of each other.\nTRY IT #1 Show that the following two matrices are inverses of each other.\nFinding the Multiplicative Inverse Using Matrix Multiplication\nWe can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since"
	},
	{
		"chunk_id" : "00003318",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting\n1110 11  Systems of Equations and Inequalities\nup an equation usingmatrix multiplication.\nEXAMPLE3\nFinding the Multiplicative Inverse Using Matrix Multiplication\nUse matrix multiplication to find the inverse of the given matrix.\nSolution\nFor this method, we multiply by a matrix containing unknown constants and set it equal to the identity."
	},
	{
		"chunk_id" : "00003319",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the product of the two matrices on the left side of the equal sign.\nNext, set up a system of equations with the entry in row 1, column 1 of the new matrix equal to the first entry of the\nidentity, 1. Set the entry in row 2, column 1 of the new matrix equal to the corresponding entry of the identity, which is 0.\nUsing row operations, multiply and add as follows: Add the equations, and solve for\nBack-substitute to solve for"
	},
	{
		"chunk_id" : "00003320",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Back-substitute to solve for\nWrite another system of equations setting the entry in row 1, column 2 of the new matrix equal to the corresponding\nentry of the identity, 0. Set the entry in row 2, column 2 equal to the corresponding entry of the identity.\nUsing row operations, multiply and add as follows: Add the two equations and solve for\nOnce more, back-substitute and solve for\nFinding the Multiplicative Inverse by Augmenting with the Identity"
	},
	{
		"chunk_id" : "00003321",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Another way to find themultiplicative inverseis by augmenting with the identity. When matrix is transformed into\nthe augmented matrix transforms into\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1111\nFor example, given\naugment with the identity\nPerformrow operationswith the goal of turning into the identity.\n1. Switch row 1 and row 2.\n2. Multiply row 2 by and add to row 1.\n3. Multiply row 1 by and add to row 2.\n4. Add row 2 to row 1.\n5. Multiply row 2 by\nThe matrix we have found is"
	},
	{
		"chunk_id" : "00003322",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5. Multiply row 2 by\nThe matrix we have found is\nFinding the Multiplicative Inverse of 22 Matrices Using a Formula\nWhen we need to find themultiplicative inverseof a matrix, we can use a special formula instead of using matrix\nmultiplication or augmenting with the identity.\nIf is a matrix, such as\nthe multiplicative inverse of is given by the formula\nwhere If then has no inverse.\nEXAMPLE4\nUsing the Formula to Find the Multiplicative Inverse of MatrixA\nUse the formula to find the multiplicative inverse of"
	},
	{
		"chunk_id" : "00003323",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUsing the formula, we have\n1112 11  Systems of Equations and Inequalities\nAnalysis\nWe can check that our formula works by using one of the other methods to calculate the inverse. Lets augment with\nthe identity.\nPerformrow operationswith the goal of turning into the identity.\n1. Multiply row 1 by and add to row 2.\n2. Multiply row 1 by 2 and add to row 1.\nSo, we have verified our original solution.\nTRY IT #2 Use the formula to find the inverse of matrix Verify your answer by augmenting with the"
	},
	{
		"chunk_id" : "00003324",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "identity matrix.\nEXAMPLE5\nFinding the Inverse of the Matrix, If It Exists\nFind the inverse, if it exists, of the given matrix.\nSolution\nWe will use the method of augmenting with the identity.\n1. Switch row 1 and row 2.\n2. Multiply row 1 by 3 and add it to row 2.\n3. There is nothing further we can do. The zeros in row 2 indicate that this matrix has no inverse.\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1113\nFinding the Multiplicative Inverse of 33 Matrices"
	},
	{
		"chunk_id" : "00003325",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Unfortunately, we do not have a formula similar to the one for a matrix to find the inverse of a matrix.\nInstead, we will augment the original matrix with the identity matrix and userow operationsto obtain the inverse.\nGiven a matrix\naugment with the identity matrix\nTo begin, we write theaugmented matrixwith the identity on the right and on the left. Performing elementaryrow\noperationsso that theidentity matrixappears on the left, we will obtain theinverse matrixon the right. We will find the"
	},
	{
		"chunk_id" : "00003326",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inverse of this matrix in the next example.\n...\nHOW TO\nGiven a matrix, find the inverse\n1. Write the original matrix augmented with the identity matrix on the right.\n2. Use elementary row operations so that the identity appears on the left.\n3. What is obtained on the right is the inverse of the original matrix.\n4. Use matrix multiplication to show that and\nEXAMPLE6\nFinding the Inverse of a 3  3 Matrix\nGiven the matrix find the inverse.\nSolution"
	},
	{
		"chunk_id" : "00003327",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given the matrix find the inverse.\nSolution\nAugment with the identity matrix, and then begin row operations until the identity matrix replaces The matrix on\nthe right will be the inverse of\n1114 11  Systems of Equations and Inequalities\nThus,\nAnalysis\nTo prove that lets multiply the two matrices together to see if the product equals the identity, if\nand\nTRY IT #3 Find the inverse of the matrix.\nSolving a System of Linear Equations Using the Inverse of a Matrix"
	},
	{
		"chunk_id" : "00003328",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices: is the\nmatrix representing the variables of the system, and is the matrix representing the constants. Usingmatrix\nmultiplication, we may define a system of equations with the same number of equations as variables as\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1115"
	},
	{
		"chunk_id" : "00003329",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.7  Solving Systems with Inverses 1115\nTo solve a system of linear equations using aninverse matrix, let be thecoefficient matrix, let be the variable\nmatrix, and let be the constant matrix. Thus, we want to solve a system For example, look at the following\nsystem of equations.\nFrom this system, the coefficient matrix is\nThe variable matrix is\nAnd the constant matrix is\nThen looks like\nRecall the discussion earlier in this section regarding multiplying a real number by its inverse, To"
	},
	{
		"chunk_id" : "00003330",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solve a single linear equation for we would simply multiply both sides of the equation by the multiplicative\ninverse (reciprocal) of Thus,\nThe only difference between a solving a linear equation and asystem of equationswritten in matrix form is that finding\nthe inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the\nsameto isolate the variable."
	},
	{
		"chunk_id" : "00003331",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sameto isolate the variable.\nWe will investigate this idea in detail, but it is helpful to begin with a system and then move on to a system.\nSolving a System of Equations Using the Inverse of a Matrix\nGiven a system of equations, write the coefficient matrix the variable matrix and the constant matrix Then\nMultiply both sides by the inverse of to obtain the solution.\nQ&A If the coefficient matrix does not have an inverse, does that mean the system has no solution?"
	},
	{
		"chunk_id" : "00003332",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "No, if the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be\ndependent and have infinitely many solutions.\n1116 11  Systems of Equations and Inequalities\nEXAMPLE7\nSolving a 2  2 System Using the Inverse of a Matrix\nSolve the given system of equations using the inverse of a matrix.\nSolution\nWrite the system in terms of a coefficient matrix, a variable matrix, and a constant matrix.\nThen"
	},
	{
		"chunk_id" : "00003333",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Then\nFirst, we need to calculate Using the formula to calculate the inverse of a 2 by 2 matrix, we have:\nSo,\nNow we are ready to solve. Multiply both sides of the equation by\nThe solution is\nQ&A Can we solve for by finding the product\nNo, recall that matrix multiplication is not commutative, so Consider our steps for solving\nthe matrix equation.\nNotice in the first step we multiplied both sides of the equation by but the was to the left of"
	},
	{
		"chunk_id" : "00003334",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on the left side and to the left of on the right side. Because matrix multiplication is not commutative,\norder matters.\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1117\nEXAMPLE8\nSolving a 3  3 System Using the Inverse of a Matrix\nSolve the following system using the inverse of a matrix.\nSolution\nWrite the equation\nFirst, we will find the inverse of by augmenting with the identity.\nMultiply row 1 by\nMultiply row 1 by 4 and add to row 2.\nAdd row 1 to row 3."
	},
	{
		"chunk_id" : "00003335",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Add row 1 to row 3.\nMultiply row 2 by 3 and add to row 1.\nMultiply row 3 by 5.\nMultiply row 3 by and add to row 1.\n1118 11  Systems of Equations and Inequalities\nMultiply row 3 by and add to row 2.\nSo,\nMultiply both sides of the equation by We want\nThus,\nThe solution is\nTRY IT #4 Solve the system using the inverse of the coefficient matrix.\n...\nHOW TO\nGiven a system of equations, solve with matrix inverses using a calculator.\n1. Save the coefficient matrix and the constant matrix as matrix variables and"
	},
	{
		"chunk_id" : "00003336",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Enter the multiplication into the calculator, calling up each matrix variable as needed.\n3. If the coefficient matrix is invertible, the calculator will present the solution matrix; if the coefficient matrix is not\ninvertible, the calculator will present an error message.\nEXAMPLE9\nUsing a Calculator to Solve a System of Equations with Matrix Inverses\nSolve the system of equations with matrix inverses using a calculator\nSolution"
	},
	{
		"chunk_id" : "00003337",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nOn the matrix page of the calculator, enter thecoefficient matrixas the matrix variable and enter the constant\nmatrix as the matrix variable\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1119\nOn the home screen of the calculator, type in the multiplication to solve for calling up each matrix variable as needed.\nEvaluate the expression.\nMEDIA\nAccess these online resources for additional instruction and practice with solving systems with inverses."
	},
	{
		"chunk_id" : "00003338",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The Identity Matrix(http://openstax.org/l/identmatrix)\nDetermining Inverse Matrices(http://openstax.org/l/inversematrix)\nUsing a Matrix Equation to Solve a System of Equations(http://openstax.org/l/matrixsystem)\n11.7 SECTION EXERCISES\nVerbal\n1. In a previous section, we 2. Does every matrix 3. Can you explain whether a\nshowed that matrix have an inverse? Explain matrix with an entire\nmultiplication is not why or why not. Explain row of zeros can have an"
	},
	{
		"chunk_id" : "00003339",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "commutative, that is, what condition is necessary inverse?\nin most cases. for an inverse to exist.\nCan you explain why matrix\nmultiplication is\ncommutative for matrix\ninverses, that is,\n4. Can a matrix with an entire 5. Can a matrix with zeros on\ncolumn of zeros have an the diagonal have an\ninverse? Explain why or why inverse? If so, find an\nnot. example. If not, prove why\nnot. For simplicity, assume a\nmatrix.\nAlgebraic\nIn the following exercises, show that matrix is the inverse of matrix\n6. 7.\n8. 9."
	},
	{
		"chunk_id" : "00003340",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7.\n8. 9.\n1120 11  Systems of Equations and Inequalities\n10. 11.\n12.\nFor the following exercises, find the multiplicative inverse of each matrix, if it exists.\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22. 23. 24.\n25. 26.\nFor the following exercises, solve the system using the inverse of a matrix.\n27. 28. 29.\n30. 31. 32.\n33. 34.\nFor the following exercises, solve a system using the inverse of a matrix.\n35. 36. 37.\nAccess for free at openstax.org\n11.7  Solving Systems with Inverses 1121\n38. 39. 40.\n41. 42."
	},
	{
		"chunk_id" : "00003341",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "38. 39. 40.\n41. 42.\nTechnology\nFor the following exercises, use a calculator to solve the system of equations with matrix inverses.\n43. 44. 45.\n46.\nExtensions\nFor the following exercises, find the inverse of the given matrix.\n47. 48. 49.\n50. 51.\n1122 11  Systems of Equations and Inequalities\nReal-World Applications\nFor the following exercises, write a system of equations that represents the situation. Then, solve the system using the\ninverse of a matrix."
	},
	{
		"chunk_id" : "00003342",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "inverse of a matrix.\n52. 2,400 tickets were sold for 53. In the previous exercise, if 54. A food drive collected two\na basketball game. If the you were told there were different types of canned\nprices for floor 1 and floor 400 more tickets sold for goods, green beans and\n2 were different, and the floor 2 than floor 1, how kidney beans. The total\ntotal amount of money much was the price of each number of collected cans\nbrought in is $64,000, how ticket? was 350 and the total"
	},
	{
		"chunk_id" : "00003343",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "much was the price of each weight of all donated food\nticket? was 348 lb, 12 oz. If the\ngreen bean cans weigh 2\noz less than the kidney\nbean cans, how many of\neach can was donated?\n55. Students were asked to 56. The nursing club held a 57. A clothing store needs to\nbring their favorite fruit to bake sale to raise money order new inventory. It has\nclass. 95% of the fruits and sold brownies and three different types of\nconsisted of banana, apple, chocolate chip cookies. hats for sale: straw hats,"
	},
	{
		"chunk_id" : "00003344",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and oranges. If oranges They priced the brownies beanies, and cowboy hats.\nwere twice as popular as at $1 and the chocolate The straw hat is priced at\nbananas, and apples were chip cookies at $0.75. They $13.99, the beanie at $7.99,\n5% less popular than raised $700 and sold 850 and the cowboy hat at\nbananas, what are the items. How many brownies $14.49. If 100 hats were\npercentages of each and how many cookies sold this past quarter,\nindividual fruit? were sold? $1,119 was taken in by"
	},
	{
		"chunk_id" : "00003345",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sales, and the amount of\nbeanies sold was 10 more\nthan cowboy hats, how\nmany of each should the\nclothing store order to\nreplace those already sold?\n58. Anna, Percy, and Morgan 59. Three roommates shared a 60. A farmer constructed a\nweigh a combined 370 lb. If package of 12 ice cream chicken coop out of\nMorgan weighs 20 lb more bars, but no one chicken wire, wood, and\nthan Percy, and Anna remembers who ate how plywood. The chicken wire"
	},
	{
		"chunk_id" : "00003346",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "weighs 1.5 times as much many. If Micah ate twice as cost $2 per square foot, the\nas Percy, how much does many ice cream bars as wood $10 per square foot,\neach person weigh? Joe, and Albert ate three and the plywood $5 per\nless than Micah, how many square foot. The farmer\nice cream bars did each spent a total of $51, and\nroommate eat? the total amount of\nmaterials used was\nHe used more chicken\nwire than plywood. How\nmuch of each material in\ndid the farmer use?\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003347",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1123\n61. Jay has lemon, orange, and\npomegranate trees in his\nbackyard. An orange\nweighs 8 oz, a lemon 5 oz,\nand a pomegranate 11 oz.\nJay picked 142 pieces of\nfruit weighing a total of 70\nlb, 10 oz. He picked 15.5\ntimes more oranges than\npomegranates. How many\nof each fruit did Jay pick?\n11.8 Solving Systems with Cramer's Rule\nLearning Objectives\nIn this section, you will:\nEvaluate 2  2 determinants."
	},
	{
		"chunk_id" : "00003348",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Evaluate 2  2 determinants.\nUse Cramers Rule to solve a system of equations in two variables.\nEvaluate 3  3 determinants.\nUse Cramers Rule to solve a system of three equations in three variables.\nKnow the properties of determinants.\nWe have learned how to solve systems of equations in two variables and three variables, and by multiple methods:\nsubstitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are"
	},
	{
		"chunk_id" : "00003349",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "easier to apply than others and are more appropriate in certain situations. In this section, we will study two more\nstrategies for solving systems of equations.\nEvaluating the Determinant of a 22 Matrix\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as\ncalculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by"
	},
	{
		"chunk_id" : "00003350",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "using the entries of asquare matrixto determine whether there is a solution to the system of equations. Perhaps one of\nthe more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent\nencoded in a matrix. The data can only be decrypted with aninvertible matrixand the determinant. For our purposes,\nwe focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix"
	},
	{
		"chunk_id" : "00003351",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "involves following the specific patterns that are outlined in this section.\nFind the Determinant of a 2  2 Matrix\nThedeterminantof a matrix, given\nis defined as\nNotice the change in notation. There are several ways to indicate the determinant, including and replacing the\nbrackets in a matrix with straight lines,\nEXAMPLE1\nFinding the Determinant of a 2  2 Matrix\nFind the determinant of the given matrix.\n1124 11  Systems of Equations and Inequalities\nSolution"
	},
	{
		"chunk_id" : "00003352",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nUsing Cramers Rule to Solve a System of Two Equations in Two Variables\nWe will now introduce a final method for solving systems of equations that uses determinants. Known asCramers Rule,\nthis technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician\nGabriel Cramer (1704-1752), who introduced it in 1750 inIntroduction  l'Analyse des lignes Courbes algbriques."
	},
	{
		"chunk_id" : "00003353",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Cramers Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns,\nprovided that we have the same number of equations as unknowns.\nCramers Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no\nsolution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is\ninconsistent or dependent, another method, such as elimination, will have to be used."
	},
	{
		"chunk_id" : "00003354",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To understand Cramers Rule, lets look closely at how we solve systems of linear equations using basic row operations.\nConsider a system of two equations in two variables.\nWe eliminate one variable using row operations and solve for the other. Say that we wish to solve for If equation (2) is\nmultiplied by the opposite of the coefficient of in equation (1), equation (1) is multiplied by the coefficient of in\nequation (2), and we add the two equations, the variable will be eliminated.\nNow, solve for"
	},
	{
		"chunk_id" : "00003355",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now, solve for\nSimilarly, to solve for we will eliminate\nSolving for gives\nAccess for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1125\nNotice that the denominator for both and is the determinant of the coefficient matrix.\nWe can use these formulas to solve for and but Cramers Rule also introduces new notation:\n determinant of the coefficient matrix\n determinant of the numerator in the solution of\n determinant of the numerator in the solution of"
	},
	{
		"chunk_id" : "00003356",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " determinant of the numerator in the solution of\nThe key to Cramers Rule is replacing the variable column of interest with the constant column and calculating the\ndeterminants. We can then express and as a quotient of two determinants.\nCramers Rule for 22 Systems\nCramers Ruleis a method that uses determinants to solve systems of equations that have the same number of\nequations as variables.\nConsider a system of two linear equations in two variables.\nThe solution using Cramers Rule is given as"
	},
	{
		"chunk_id" : "00003357",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The solution using Cramers Rule is given as\nIf we are solving for the column is replaced with the constant column. If we are solving for the column is\nreplaced with the constant column.\nEXAMPLE2\nUsing Cramers Rule to Solve a 2  2 System\nSolve the following system using Cramers Rule.\nSolution\nSolve for\nSolve for\n1126 11  Systems of Equations and Inequalities\nThe solution is\nTRY IT #1 Use Cramers Rule to solve the 2  2 system of equations.\nEvaluating the Determinant of a 3  3 Matrix"
	},
	{
		"chunk_id" : "00003358",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Evaluating the Determinant of a 3  3 Matrix\nFinding the determinant of a 22 matrix is straightforward, but finding the determinant of a 33 matrix is more\ncomplicated. One method is to augment the 33 matrix with a repetition of the first two columns, giving a 35 matrix.\nThen we calculate the sum of the products of entriesdowneach of the three diagonals (upper left to lower right), and"
	},
	{
		"chunk_id" : "00003359",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "subtract the products of entriesupeach of the three diagonals (lower left to upper right). This is more easily understood\nwith a visual and an example.\nFind thedeterminantof the 33 matrix.\n1. Augment with the first two columns.\n2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries\ndown the second diagonal. Add this result to the product of the entries down the third diagonal."
	},
	{
		"chunk_id" : "00003360",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the\nproduct of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.\nThe algebra is as follows:\nEXAMPLE3\nFinding the Determinant of a 3  3 Matrix\nFind the determinant of the 3  3 matrix given\nSolution\nAugment the matrix with the first two columns and then follow the formula. Thus,\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003361",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1127\nTRY IT #2 Find the determinant of the 3  3 matrix.\nQ&A Can we use the same method to find the determinant of a larger matrix?\nNo, this method only works for and matrices. For larger matrices it is best to use a graphing\nutility or computer software.\nUsing Cramers Rule to Solve a System of Three Equations in Three Variables"
	},
	{
		"chunk_id" : "00003362",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now that we can find thedeterminantof a 3  3 matrix, we can applyCramers Ruleto solve asystem of three equations\nin three variables. Cramers Rule is straightforward, following a pattern consistent with Cramers Rule for 2  2 matrices.\nAs the order of the matrix increases to 3  3, however, there are many more calculations required.\nWhen we calculate the determinant to be zero, Cramers Rule gives no indication as to whether the system has no"
	},
	{
		"chunk_id" : "00003363",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution or an infinite number of solutions. To find out, we have to perform elimination on the system.\nConsider a 3  3 system of equations.\nwhere\nIf we are writing the determinant we replace the column with the constant column. If we are writing the\ndeterminant we replace the column with the constant column. If we are writing the determinant we replace\nthe column with the constant column. Always check the answer.\nEXAMPLE4\nSolving a 3  3 System Using Cramers Rule"
	},
	{
		"chunk_id" : "00003364",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving a 3  3 System Using Cramers Rule\nFind the solution to the given 3  3 system using Cramers Rule.\n1128 11  Systems of Equations and Inequalities\nSolution\nUse Cramers Rule.\nThen,\nThe solution is\nTRY IT #3 Use Cramers Rule to solve the 3  3 matrix.\nEXAMPLE5\nUsing Cramers Rule to Solve an Inconsistent System\nSolve the system of equations using Cramers Rule.\nSolution\nWe begin by finding the determinants"
	},
	{
		"chunk_id" : "00003365",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe begin by finding the determinants\nWe know that a determinant of zero means that either the system has no solution or it has an infinite number of\nsolutions. To see which one, we use the process of elimination. Our goal is to eliminate one of the variables.\n1. Multiply equation (1) by\n2. Add the result to equation\nWe obtain the equation which is false. Therefore, the system has no solution. Graphing the system reveals two\nparallel lines. SeeFigure 1.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003366",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1129\nFigure1\nEXAMPLE6\nUse Cramers Rule to Solve a Dependent System\nSolve the system with an infinite number of solutions.\nSolution\nLets find the determinant first. Set up a matrix augmented by the first two columns.\nThen,\nAs the determinant equals zero, there is either no solution or an infinite number of solutions. We have to perform\nelimination to find out.\n1. Multiply equation (1) by and add the result to equation (3):"
	},
	{
		"chunk_id" : "00003367",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Obtaining an answer of a statement that is always true, means that the system has an infinite number of\nsolutions. Graphing the system, we can see that two of the planes are the same and they both intersect the third\nplane on a line. SeeFigure 2.\nFigure2\nUnderstanding Properties of Determinants\nThere are manyproperties of determinants. Listed here are some properties that may be helpful in calculating the\n1130 11  Systems of Equations and Inequalities\ndeterminant of a matrix.\nProperties of Determinants"
	},
	{
		"chunk_id" : "00003368",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Properties of Determinants\n1. If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\n2. When two rows are interchanged, the determinant changes sign.\n3. If either two rows or two columns are identical, the determinant equals zero.\n4. If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\n5. The determinant of an inverse matrix is the reciprocal of the determinant of the matrix"
	},
	{
		"chunk_id" : "00003369",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.\nEXAMPLE7\nIllustrating Properties of Determinants\nIllustrate each of the properties of determinants.\nSolution\nProperty 1 states that if the matrix is in upper triangular form, the determinant is the product of the entries down the\nmain diagonal.\nAugment with the first two columns.\nThen\nProperty 2 states that interchanging rows changes the sign. Given"
	},
	{
		"chunk_id" : "00003370",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Property 3 states that if two rows or two columns are identical, the determinant equals zero.\nProperty 4 states that if a row or column equals zero, the determinant equals zero. Thus,\nProperty 5 states that the determinant of an inverse matrix is the reciprocal of the determinant Thus,\nAccess for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1131\nProperty 6 states that if any row or column of a matrix is multiplied by a constant, the determinant is multiplied by the\nsame factor. Thus,"
	},
	{
		"chunk_id" : "00003371",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "same factor. Thus,\nEXAMPLE8\nUsing Cramers Rule and Determinant Properties to Solve a System\nFind the solution to the given 3  3 system.\nSolution\nUsingCramers Rule, we have\nNotice that the second and third columns are identical. According to Property 3, the determinant will be zero, so there is\neither no solution or an infinite number of solutions. We have to perform elimination to find out.\n1. Multiply equation (3) by 2 and add the result to equation (1)."
	},
	{
		"chunk_id" : "00003372",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Obtaining a statement that is a contradiction means that the system has no solution.\nMEDIA\nAccess these online resources for additional instruction and practice with Cramers Rule.\nSolve a System of Two Equations Using Cramer's Rule(http://openstax.org/l/system2cramer)\nSolve a Systems of Three Equations using Cramer's Rule(http://openstax.org/l/system3cramer)\n1132 11  Systems of Equations and Inequalities\n11.8 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00003373",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.8 SECTION EXERCISES\nVerbal\n1. Explain why we can always 2. Examining Cramers Rule, 3. Explain what it means in\nevaluate the determinant of explain why there is no terms of an inverse for a\na square matrix. unique solution to the matrix to have a 0\nsystem when the determinant.\ndeterminant of your matrix\nis 0. For simplicity, use a\nmatrix.\n4. The determinant of\nmatrix is 3. If you switch\nthe rows and multiply the\nfirst row by 6 and the\nsecond row by 2, explain\nhow to find the determinant"
	},
	{
		"chunk_id" : "00003374",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "how to find the determinant\nand provide the answer.\nAlgebraic\nFor the following exercises, find the determinant.\n5. 6. 7.\n8. 9. 10.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21. 22.\n23. 24.\nAccess for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1133\nFor the following exercises, solve the system of linear equations using Cramers Rule.\n25. 26. 27.\n28. 29. 30.\n31. 32. 33.\n34.\nFor the following exercises, solve the system of linear equations using Cramers Rule.\n35. 36. 37.\n38. 39. 40."
	},
	{
		"chunk_id" : "00003375",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "35. 36. 37.\n38. 39. 40.\n41. 42. 43.\n44.\nTechnology\nFor the following exercises, use the determinant function on a graphing utility.\n45. 46. 47.\n48.\n1134 11  Systems of Equations and Inequalities\nReal-World Applications\nFor the following exercises, create a system of linear equations to describe the behavior. Then, calculate the\ndeterminant. Will there be a unique solution? If so, find the unique solution.\n49. Two numbers add up to 56. 50. Two numbers add up to 51. Three numbers add up to"
	},
	{
		"chunk_id" : "00003376",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "One number is 20 less than 104. If you add two times 106. The first number is 3\nthe other. the first number plus two less than the second\ntimes the second number, number. The third number\nyour total is 208 is 4 more than the first\nnumber.\n52. Three numbers add to 216.\nThe sum of the first two\nnumbers is 112. The third\nnumber is 8 less than the\nfirst two numbers\ncombined.\nFor the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all"
	},
	{
		"chunk_id" : "00003377",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions using Cramers Rule.\n53. You invest $10,000 into two 54. You invest $80,000 into two 55. A theater needs to know\naccounts, which receive 8% accounts, $22,000 in one how many adult tickets and\ninterest and 5% interest. At account, and $58,000 in the children tickets were sold\nthe end of a year, you had other account. At the end out of the 1,200 total\n$10,710 in your combined of one year, assuming tickets. If childrens tickets"
	},
	{
		"chunk_id" : "00003378",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "accounts. How much was simple interest, you have are $5.95, adult tickets are\ninvested in each account? earned $2,470 in interest. $11.15, and the total\nThe second account amount of revenue was\nreceives half a percent less $12,756, how many\nthan twice the interest on childrens tickets and adult\nthe first account. What are tickets were sold?\nthe interest rates for your\naccounts?\n56. A concert venue sells single 57. You decide to paint your 58. You sold two types of"
	},
	{
		"chunk_id" : "00003379",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tickets for $40 each and kitchen green. You create scarves at a farmers\ncouples tickets for $65. If the color of paint by mixing market and would like to\nthe total revenue was yellow and blue paints. You know which one was more\n$18,090 and the 321 tickets cannot remember how popular. The total number\nwere sold, how many many gallons of each color of scarves sold was 56, the\nsingle tickets and how went into your mix, but you yellow scarf cost $10, and"
	},
	{
		"chunk_id" : "00003380",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "many couples tickets were know there were 10 gal the purple scarf cost $11. If\nsold? total. Additionally, you kept you had total revenue of\nyour receipt, and know the $583, how many yellow\ntotal amount spent was scarves and how many\n$29.50. If each gallon of purple scarves were sold?\nyellow costs $2.59, and\neach gallon of blue costs\n$3.19, how many gallons of\neach color go into your\ngreen mix?\nAccess for free at openstax.org\n11.8  Solving Systems with Cramer's Rule 1135"
	},
	{
		"chunk_id" : "00003381",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.8  Solving Systems with Cramer's Rule 1135\n59. Your garden produced two 60. At a market, the three most 61. At the same market, the\ntypes of tomatoes, one popular vegetables make three most popular fruits\ngreen and one red. The red up 53% of vegetable sales. make up 37% of the total\nweigh 10 oz, and the green Corn has 4% higher sales fruit sold. Strawberries sell\nweigh 4 oz. You have 30 than broccoli, which has 5% twice as much as oranges,"
	},
	{
		"chunk_id" : "00003382",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tomatoes, and a total more sales than onions. and kiwis sell one more\nweight of 13 lb, 14 oz. How What percentage does percentage point than\nmany of each type of each vegetable have in the oranges. For each fruit,\ntomato do you have? market share? find the percentage of total\nfruit sold.\n62. Three artists performed at 63. A movie theatre sold\na concert venue. The first tickets to three movies. The\none charged $15 per ticket, tickets to the first movie\nthe second artist charged were $5, the tickets to the"
	},
	{
		"chunk_id" : "00003383",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$45 per ticket, and the final second movie were $11,\none charged $22 per ticket. and the third movie was\nThere were 510 tickets sold, $12. 100 tickets were sold\nfor a total of $12,700. If the to the first movie. The total\nfirst band had 40 more number of tickets sold was\naudience members than 642, for a total revenue of\nthe second band, how $6,774. How many tickets\nmany tickets were sold for for each movie were sold?\neach band?"
	},
	{
		"chunk_id" : "00003384",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each band?\nFor the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds,\ndried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown inTable 1.\nFat (g) Protein (g) Carbohydrates (g)\nAlmonds (10) 6 2 3\nCranberries (10) 0.02 0 8\nCashews (10) 7 3.5 5.5\nTable1\n64. For the special low- 65. For the hiking mix, there 66. For the energy-booster"
	},
	{
		"chunk_id" : "00003385",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "carbtrail mix, there are are 1,000 pieces in the mix, mix, there are 1,000 pieces\n1,000 pieces of mix. The containing 390.8 g of fat, in the mix, containing 145 g\ntotal number of and 165 g of protein. If of protein and 625 g of\ncarbohydrates is 425 g, and there is the same amount carbohydrates. If the\nthe total amount of fat is of almonds as cashews, number of almonds and\n570.2 g. If there are 200 how many of each item is cashews summed together"
	},
	{
		"chunk_id" : "00003386",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "more pieces of cashews in the trail mix? is equivalent to the amount\nthan cranberries, how of cranberries, how many\nmany of each item is in the of each item is in the trail\ntrail mix? mix?\n1136 11  Chapter Review\nChapter Review\nKey Terms\naddition method an algebraic technique used to solve systems of linear equations in which the equations are added in\na way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable;"
	},
	{
		"chunk_id" : "00003387",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "substitution is then used to solve for the first variable\naugmented matrix a coefficient matrix adjoined with the constant column separated by a vertical line within the\nmatrix brackets\nbreak-even point the point at which a cost function intersects a revenue function; where profit is zero\ncoefficient matrix a matrix that contains only the coefficients from a system of equations\ncolumn a set of numbers aligned vertically in a matrix"
	},
	{
		"chunk_id" : "00003388",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "consistent system a system for which there is a single solution to all equations in the system and it is an independent\nsystem, or if there are an infinite number of solutions and it is a dependent system\ncost function the function used to calculate the costs of doing business; it usually has two parts, fixed costs and\nvariable costs\nCramers Rule a method for solving systems of equations that have the same number of equations as variables using\ndeterminants"
	},
	{
		"chunk_id" : "00003389",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determinants\ndependent system a system of linear equations in which the two equations represent the same line; there are an\ninfinite number of solutions to a dependent system\ndeterminant a number calculated using the entries of a square matrix that determines such information as whether\nthere is a solution to a system of equations\nentry an element, coefficient, or constant in a matrix\nfeasible region the solution to a system of nonlinear inequalities that is the region of the graph where the shaded"
	},
	{
		"chunk_id" : "00003390",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "regions of each inequality intersect\nGaussian elimination using elementary row operations to obtain a matrix in row-echelon form\nidentity matrix a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in\nmatrix algebra\ninconsistent system a system of linear equations with no common solution because they represent parallel lines,\nwhich have no point or line in common\nindependent system a system of linear equations with exactly one solution pair"
	},
	{
		"chunk_id" : "00003391",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "main diagonal entries from the upper left corner diagonally to the lower right corner of a square matrix\nmatrix a rectangular array of numbers\nmultiplicative inverse of a matrix a matrix that, when multiplied by the original, equals the identity matrix\nnonlinear inequality an inequality containing a nonlinear expression\npartial fraction decomposition the process of returning a simplified rational expression to its original form, a sum or\ndifference of simpler rational expressions"
	},
	{
		"chunk_id" : "00003392",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference of simpler rational expressions\npartial fractions the individual fractions that make up the sum or difference of a rational expression before combining\nthem into a simplified rational expression\nprofit function the profit function is written as revenue minus cost\nrevenue function the function that is used to calculate revenue, simply written as where quantity and\nprice\nrow a set of numbers aligned horizontally in a matrix"
	},
	{
		"chunk_id" : "00003393",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "row operations adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with\nthe goal of achieving row-echelon form\nrow-echelon form after performing row operations, the matrix form that contains ones down the main diagonal and\nzeros at every space below the diagonal\nrow-equivalent two matrices and are row-equivalent if one can be obtained from the other by performing basic\nrow operations\nscalar multiple an entry of a matrix that has been multiplied by a scalar"
	},
	{
		"chunk_id" : "00003394",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solution set the set of all ordered pairs or triples that satisfy all equations in a system of equations\nsubstitution method an algebraic technique used to solve systems of linear equations in which one of the two\nequations is solved for one variable and then substituted into the second equation to solve for the second variable\nsystem of linear equations a set of two or more equations in two or more variables that must be considered\nsimultaneously."
	},
	{
		"chunk_id" : "00003395",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "simultaneously.\nsystem of nonlinear equations a system of equations containing at least one equation that is of degree larger than\none\nsystem of nonlinear inequalities a system of two or more inequalities in two or more variables containing at least\none inequality that is not linear\nAccess for free at openstax.org\n11  Chapter Review 1137\nKey Equations\nIdentity matrix for a matrix\nIdentity matrix for a matrix\nMultiplicative inverse of a matrix\nKey Concepts\n11.1Systems of Linear Equations: Two Variables"
	},
	{
		"chunk_id" : "00003396",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.1Systems of Linear Equations: Two Variables\n A system of linear equations consists of two or more equations made up of two or more variables such that all\nequations in the system are considered simultaneously.\n The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation\nindependently. SeeExample 1.\n Systems of equations are classified as independent with one solution, dependent with an infinite number of\nsolutions, or inconsistent with no solution."
	},
	{
		"chunk_id" : "00003397",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solutions, or inconsistent with no solution.\n One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the\nequations on the same set of axes. SeeExample 2.\n Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable\nin one equation and substitute the result into the second equation. SeeExample 3."
	},
	{
		"chunk_id" : "00003398",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding\nopposite coefficients of corresponding variables. SeeExample 4.\n It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when\nadding the two equations together. SeeExample 5,Example 6, andExample 7.\n Either method of solving a system of equations results in a false statement for inconsistent systems because they"
	},
	{
		"chunk_id" : "00003399",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "are made up of parallel lines that never intersect. SeeExample 8.\n The solution to a system of dependent equations will always be true because both equations describe the same line.\nSeeExample 9.\n Systems of equations can be used to solve real-world problems that involve more than one variable, such as those\nrelating to revenue, cost, and profit. SeeExample 10andExample 11.\n11.2Systems of Linear Equations: Three Variables"
	},
	{
		"chunk_id" : "00003400",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.2Systems of Linear Equations: Three Variables\n A solution set is an ordered triple that represents the intersection of three planes in space. See\nExample 1.\n A system of three equations in three variables can be solved by using a series of steps that forces a variable to be\neliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a\nnonzero constant, and adding a nonzero multiple of one equation to another equation. SeeExample 2."
	},
	{
		"chunk_id" : "00003401",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Systems of three equations in three variables are useful for solving many different types of real-world problems. See\nExample 3.\n A system of equations in three variables is inconsistent if no solution exists. After performing elimination\noperations, the result is a contradiction. SeeExample 4.\n Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel"
	},
	{
		"chunk_id" : "00003402",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "planes and one intersecting plane, or three planes that intersect the other two but not at the same location.\n A system of equations in three variables is dependent if it has an infinite number of solutions. After performing\nelimination operations, the result is an identity. SeeExample 5.\n Systems of equations in three variables that are dependent could result from three identical planes, three planes\nintersecting at a line, or two identical planes that intersect the third on a line."
	},
	{
		"chunk_id" : "00003403",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "11.3Systems of Nonlinear Equations and Inequalities: Two Variables\n There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no\nsolution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two\n1138 11  Chapter Review\nsolutions, the line intersects the parabola in two points. SeeExample 1."
	},
	{
		"chunk_id" : "00003404",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution,\nthe line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line\nintersects the circle in two points. SeeExample 2.\n There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:"
	},
	{
		"chunk_id" : "00003405",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to\neach other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and\nellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points. SeeExample 3.\n An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade"
	},
	{
		"chunk_id" : "00003406",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the region containing the solution set. SeeExample 4.\n Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both\ninequalities. SeeExample 5.\n11.4Partial Fractions\n Decompose by writing the partial fractions as Solve by clearing the fractions, expanding the\nright side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and\nsolving a system of equations. SeeExample 1."
	},
	{
		"chunk_id" : "00003407",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "solving a system of equations. SeeExample 1.\n The decomposition of with repeated linear factors must account for the factors of the denominator in\nincreasing powers. SeeExample 2.\n The decomposition of with a nonrepeated irreducible quadratic factor needs a linear numerator over the\nquadratic factor, as in SeeExample 3.\n In the decomposition of where has a repeated irreducible quadratic factor, when the irreducible"
	},
	{
		"chunk_id" : "00003408",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as\nSeeExample 4.\n11.5Matrices and Matrix Operations\n A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.\n The dimensions of a matrix refer to the number of rows and the number of columns. A matrix has three rows\nand two columns. SeeExample 1.\n We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix."
	},
	{
		"chunk_id" : "00003409",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "SeeExample 2,Example 3,Example 4, andExample 5.\n Scalar multiplication involves multiplying each entry in a matrix by a constant. SeeExample 6.\n Scalar multiplication is often required before addition or subtraction can occur. SeeExample 7.\n Multiplying matrices is possible when inner dimensions are the samethe number of columns in the first matrix\nmust match the number of rows in the second.\n The product of two matrices, and is obtained by multiplying each entry in row 1 of by each entry in column 1"
	},
	{
		"chunk_id" : "00003410",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of then multiply each entry of row 1 of by each entry in columns 2 of and so on. SeeExample 8andExample\n9.\n Many real-world problems can often be solved using matrices. SeeExample 10.\n We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. SeeExample 11.\n11.6Solving Systems with Gaussian Elimination\n An augmented matrix is one that contains the coefficients and constants of a system of equations. SeeExample 1."
	},
	{
		"chunk_id" : "00003411",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " A matrix augmented with the constant column can be represented as the original system of equations. SeeExample\n2.\n Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.\n We can use Gaussian elimination to solve a system of equations. SeeExample 3,Example 4, andExample 5.\n Row operations are performed on matrices to obtain row-echelon form. SeeExample 6."
	},
	{
		"chunk_id" : "00003412",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon\nform. Back-substitute to find the solutions. SeeExample 7andExample 8.\n A calculator can be used to solve systems of equations using matrices. SeeExample 9.\n Many real-world problems can be solved using augmented matrices. SeeExample 10andExample 11.\nAccess for free at openstax.org\n11  Exercises 1139\n11.7Solving Systems with Inverses\n An identity matrix has the property SeeExample 1."
	},
	{
		"chunk_id" : "00003413",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " An invertible matrix has the property SeeExample 2.\n Use matrix multiplication and the identity to find the inverse of a matrix. SeeExample 3.\n The multiplicative inverse can be found using a formula. SeeExample 4.\n Another method of finding the inverse is by augmenting with the identity. SeeExample 5.\n We can augment a matrix with the identity on the right and use row operations to turn the original matrix into\nthe identity, and the matrix on the right becomes the inverse. SeeExample 6."
	},
	{
		"chunk_id" : "00003414",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Write the system of equations as and multiply both sides by the inverse of See\nExample 7andExample 8.\n We can also use a calculator to solve a system of equations with matrix inverses. SeeExample 9.\n11.8Solving Systems with Cramer's Rule\n The determinant for is SeeExample 1.\n Cramers Rule replaces a variable column with the constant column. Solutions are SeeExample 2.\n To find the determinant of a 33 matrix, augment with the first two columns. Add the three diagonal entries (upper"
	},
	{
		"chunk_id" : "00003415",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "left to lower right) and subtract the three diagonal entries (lower left to upper right). SeeExample 3.\n To solve a system of three equations in three variables using Cramers Rule, replace a variable column with the\nconstant column for each desired solution: SeeExample 4.\n Cramers Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.\nSeeExample 5andExample 6.\n Certain properties of determinants are useful for solving problems. For example:"
	},
	{
		"chunk_id" : "00003416",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.\n When two rows are interchanged, the determinant changes sign.\n If either two rows or two columns are identical, the determinant equals zero.\n If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.\n The determinant of an inverse matrix is the reciprocal of the determinant of the matrix"
	},
	{
		"chunk_id" : "00003417",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If any row or column is multiplied by a constant, the determinant is multiplied by the same factor. SeeExample 7\nandExample 8.\nExercises\nReview Exercises\nSystems of Linear Equations: Two Variables\nFor the following exercises, determine whether the ordered pair is a solution to the system of equations.\n1. and 2. and\nFor the following exercises, use substitution to solve the system of equations.\n3. 4. 5.\nFor the following exercises, use addition to solve the system of equations.\n6. 7. 8."
	},
	{
		"chunk_id" : "00003418",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "6. 7. 8.\n1140 11  Exercises\nFor the following exercises, write a system of equations to solve each problem. Solve the system of equations.\n9. A factory has a cost of 10. A performer charges\nproduction\nand a where is the total\nrevenue function number of attendees at a\nWhat is the show. The venue charges\nbreak-even point? $75 per ticket. After how\nmany people buy tickets\ndoes the venue break even,\nand what is the value of\nthe total tickets sold at that\npoint?\nSystems of Linear Equations: Three Variables"
	},
	{
		"chunk_id" : "00003419",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Systems of Linear Equations: Three Variables\nFor the following exercises, solve the system of three equations using substitution or addition.\n11. 12. 13.\n14. 15. 16.\n17. 18.\nFor the following exercises, write a system of equations to solve each problem. Solve the system of equations.\n19. Three odd numbers sum 20. A local theatre sells out for\nup to 61. The smaller is their show. They sell all 500\none-third the larger and tickets for a total purse of\nthe middle number is 16 $8,070.00. The tickets were"
	},
	{
		"chunk_id" : "00003420",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "less than the larger. What priced at $15 for students,\nare the three numbers? $12 for children, and $18\nfor adults. If the band sold\nthree times as many adult\ntickets as childrens tickets,\nhow many of each type\nwas sold?\nSystems of Nonlinear Equations and Inequalities: Two Variables\nFor the following exercises, solve the system of nonlinear equations.\n21. 22. 23.\nAccess for free at openstax.org\n11  Exercises 1141\n24. 25.\nFor the following exercises, graph the inequality.\n26. 27."
	},
	{
		"chunk_id" : "00003421",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "26. 27.\nFor the following exercises, graph the system of inequalities.\n28. 29. 30.\nPartial Fractions\nFor the following exercises, decompose into partial fractions.\n31. 32. 33.\n34. 35. 36.\n37. 38.\nMatrices and Matrix Operations\nFor the following exercises, perform the requested operations on the given matrices.\n39. 40. 41.\n42. 43. 44.\n45. 46. 47.\n48. 49. 50.\nSolving Systems with Gaussian Elimination"
	},
	{
		"chunk_id" : "00003422",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Systems with Gaussian Elimination\nFor the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will\nbe a unique solution.\n51. 52.\n1142 11  Exercises\nFor the following exercises, write the augmented matrix from the system of linear equations.\n53. 54. 55.\nFor the following exercises, solve the system of linear equations using Gaussian elimination.\n56. 57. 58.\n59. 60.\nSolving Systems with Inverses"
	},
	{
		"chunk_id" : "00003423",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "56. 57. 58.\n59. 60.\nSolving Systems with Inverses\nFor the following exercises, find the inverse of the matrix.\n61. 62. 63.\n64.\nFor the following exercises, find the solutions by computing the inverse of the matrix.\n65. 66. 67.\n68.\nFor the following exercises, write a system of equations to solve each problem. Solve the system of equations.\n69. Students were asked to bring their favorite fruit to 70. A school club held a bake sale to raise money and"
	},
	{
		"chunk_id" : "00003424",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "class. 90% of the fruits consisted of banana, apple, sold brownies and chocolate chip cookies. They\nand oranges. If oranges were half as popular as priced the brownies at $2 and the chocolate chip\nbananas and apples were 5% more popular than cookies at $1. They raised $250 and sold 175\nbananas, what are the percentages of each items. How many brownies and how many cookies\nindividual fruit? were sold?\nAccess for free at openstax.org\n11  Exercises 1143\nSolving Systems with Cramer's Rule"
	},
	{
		"chunk_id" : "00003425",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Systems with Cramer's Rule\nFor the following exercises, find the determinant.\n71. 72. 73.\n74.\nFor the following exercises, use Cramers Rule to solve the linear systems of equations.\n75. 76. 77.\n78. 79. 80.\nPractice Test\nIs the following ordered pair a solution to the system of equations?\n1. with\nFor the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination.\nIndicate if no solution exists.\n2. 3. 4.\n5. 6. 7.\n8. 9."
	},
	{
		"chunk_id" : "00003426",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. 3. 4.\n5. 6. 7.\n8. 9.\nFor the following exercises, graph the following inequalities.\n10. 11.\n1144 11  Exercises\nFor the following exercises, write the partial fraction decomposition.\n12. 13. 14.\nFor the following exercises, perform the given matrix operations.\n15. 16. 17.\n18. 19. 20. If what would\nbe the determinant if you\nswitched rows 1 and 3,\nmultiplied the second row\nby 12, and took the\ninverse?\n21. Rewrite the system of 22. Rewrite the augmented\nlinear equations as an matrix as a system of linear"
	},
	{
		"chunk_id" : "00003427",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "augmented matrix. equations.\nFor the following exercises, use Gaussian elimination to solve the systems of equations.\n23. 24.\nFor the following exercises, use the inverse of a matrix to solve the systems of equations.\n25. 26.\nFor the following exercises, use Cramers Rule to solve the systems of equations.\n27. 28.\nAccess for free at openstax.org\n11  Exercises 1145\nFor the following exercises, solve using a system of linear equations."
	},
	{
		"chunk_id" : "00003428",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "29. A factory producing cell phones has the following 30. A small fair charges $1.50 for students, $1 for\ncost and revenue functions: children, and $2 for adults. In one day, three times\nand as many children as adults attended. A total of 800\nWhat is the range of cell phones they should tickets were sold for a total revenue of $1,050.\nproduce each day so there is profit? Round to the How many of each type of ticket was sold?\nnearest number that generates profit.\n1146 11  Exercises"
	},
	{
		"chunk_id" : "00003429",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1146 11  Exercises\nAccess for free at openstax.org\n12  Introduction 1147\n12 ANALYTIC GEOMETRY\nThe rings of Saturn have produced wonder, as well as misunderstanding, since Galileo first discovered them (he initially\nthought they were moons). Though they appear to be a series of solid discs even in this 2004 closeup from the Cassini\nprobe, 19th century mathematicians proved that they are made up of billions of small objects clustered together. (credit:"
	},
	{
		"chunk_id" : "00003430",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "modificaion of \"Saturn\"\" by NASA/JPL-Caltech/SSI/Kevin M. Gill/flickr)"
	},
	{
		"chunk_id" : "00003432",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that\nthe sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.\nOther objects in the solar system (and perhaps other systems) follow a similar elliptical path, including the spectacular\nrings of Saturn. Using this understanding as a basis, 19th century mathematicians like James ClerkMaxwelland Sofya"
	},
	{
		"chunk_id" : "00003433",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Kovalevskayashowed that despite their appearance through the telescopes of the day (and even in current telescopes),\nthe rings are not solid and continuous, but are rather composed of small particles. Even after the Voyager and Cassini\nmissions have provided close-up and detailed data regarding the ring structures, full understanding of their construction\nrelies heavily on mathematical analysis. Of particular interest are the influences of Saturn's moons and moonlets, and"
	},
	{
		"chunk_id" : "00003434",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the ways they both disrupt and preserve the ring structure.\nIn this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected\nby a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for\neach figure and then learn how to use these equations to solve a variety of problems.\n1148 12  Analytic Geometry\n12.1 The Ellipse\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00003435",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nWrite equations of ellipses in standard form.\nGraph ellipses centered at the origin.\nGraph ellipses not centered at the origin.\nSolve applied problems involving ellipses.\nFigure1 The National Statuary Hall in Washington, D.C. (credit: Greg Palmer, Flickr)\nCan you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at"
	},
	{
		"chunk_id" : "00003436",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the other end? The National Statuary Hall in Washington, D.C., shown inFigure 1, is such a room.1 It is an semi-circular\nroom called awhispering chamberbecause the shape makes it possible for sound to travel along the walls and dome. In\nthis section, we will investigate the shape of this room and its real-world applications, including how far apart two people\nin Statuary Hall can stand and still hear each other whisper.\nWriting Equations of Ellipses in Standard Form"
	},
	{
		"chunk_id" : "00003437",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Writing Equations of Ellipses in Standard Form\nA conic section, orconic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the\nplane intersects the cone determines the shape, as shown inFigure 2.\nFigure2\nConic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the\ngraph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the"
	},
	{
		"chunk_id" : "00003438",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for\nthe ellipse. Anellipseis the set of all points in a plane such that the sum of their distances from two fixed points is\na constant. Each fixed point is called afocus(plural:foci).\n1 Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014.\nAccess for free at openstax.org\n12.1  The Ellipse 1149"
	},
	{
		"chunk_id" : "00003439",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.1  The Ellipse 1149\nWe can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the\ncardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the\nlength of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a\ncurve with a pencil held taut against the string. The result is an ellipse. SeeFigure 3.\nFigure3"
	},
	{
		"chunk_id" : "00003440",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure3\nEvery ellipse has two axes of symmetry. The longer axis is called themajor axis, and the shorter axis is called theminor\naxis. Each endpoint of the major axis is thevertexof the ellipse (plural:vertices), and each endpoint of the minor axis is\naco-vertexof the ellipse. Thecenter of an ellipseis the midpoint of both the major and minor axes. The axes are\nperpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point"
	},
	{
		"chunk_id" : "00003441",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on the ellipse (the constant sum) is greater than the distance between the foci. SeeFigure 4.\nFigure4\nIn this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is,\nthe axes will either lie on or be parallel to thex- andy-axes. Later in the chapter, we will see ellipses that are rotated in\nthe coordinate plane.\nTo work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at"
	},
	{
		"chunk_id" : "00003442",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of\nellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to\ndraw the graphs.\nDeriving the Equation of an Ellipse Centered at the Origin\nTo derive the equation of anellipsecentered at the origin, we begin with the foci and The ellipse is the set"
	},
	{
		"chunk_id" : "00003443",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of all points such that the sum of the distances from to the foci is constant, as shown inFigure 5.\n1150 12  Analytic Geometry\nFigure5\nIf is avertexof the ellipse, the distance from to is The distance from to is\n. The sum of the distances from thefocito the vertex is\nIf is a point on the ellipse, then we can define the following variables:\nBy the definition of an ellipse, is constant for any point on the ellipse. We know that the sum of these"
	},
	{
		"chunk_id" : "00003444",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distances is for the vertex It follows that for any point on the ellipse. We will begin the derivation\nby applying the distance formula. The rest of the derivation is algebraic.\nThus, the standard equation of an ellipse is This equation defines an ellipse centered at the origin. If\nthe ellipse is stretched further in the horizontal direction, and if the ellipse is stretched further in the\nvertical direction.\nWriting Equations of Ellipses Centered at the Origin in Standard Form"
	},
	{
		"chunk_id" : "00003445",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of\nequations weve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to\ninterpret standard forms of equations, we are bridging the relationship between algebraic and geometric\nrepresentations of mathematical phenomena.\nAccess for free at openstax.org\n12.1  The Ellipse 1151"
	},
	{
		"chunk_id" : "00003446",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.1  The Ellipse 1151\nThe key features of theellipseare its center,vertices,co-vertices,foci, and lengths and positions of themajor and minor\naxes. Just as with other equations, we can identify all of these features just by looking at the standard form of the\nequation. There are four variations of the standard form of the ellipse. These variations are categorized first by the\nlocation of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented"
	},
	{
		"chunk_id" : "00003447",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form\na mental picture of the ellipse.\nStandard Forms of the Equation of an Ellipse with Center (0,0)\nThe standard form of the equation of an ellipse with center and major axis on thex-axisis\nwhere\n\n the length of the major axis is\n the coordinates of the vertices are\n the length of the minor axis is\n the coordinates of the co-vertices are"
	},
	{
		"chunk_id" : "00003448",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the coordinates of the co-vertices are\n the coordinates of the foci are , where SeeFigure 6a\nThe standard form of the equation of an ellipse with center and major axis on they-axisis\nwhere\n\n the length of the major axis is\n the coordinates of the vertices are\n the length of the minor axis is\n the coordinates of the co-vertices are\n the coordinates of the foci are , where SeeFigure 6b\nNote that the vertices, co-vertices, and foci are related by the equation When we are given the"
	},
	{
		"chunk_id" : "00003449",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in\nstandard form.\nFigure6 (a) Horizontal ellipse with center (b) Vertical ellipse with center\n1152 12  Analytic Geometry\n...\nHOW TO\nGiven the vertices and foci of an ellipse centered at the origin, write its equation in standard form.\n1. Determine whether the major axis lies on thex- ory-axis.\na. If the given coordinates of the vertices and foci have the form and respectively, then the"
	},
	{
		"chunk_id" : "00003450",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "major axis is thex-axis. Use the standard form\nb. If the given coordinates of the vertices and foci have the form and respectively, then the\nmajor axis is they-axis. Use the standard form\n2. Use the equation along with the given coordinates of the vertices and foci, to solve for\n3. Substitute the values for and into the standard form of the equation determined in Step 1.\nEXAMPLE1\nWriting the Equation of an Ellipse Centered at the Origin in Standard Form"
	},
	{
		"chunk_id" : "00003451",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "What is the standard form equation of the ellipse that has vertices and foci\nSolution\nThe foci are on thex-axis, so the major axis is thex-axis. Thus, the equation will have the form\nThe vertices are so and\nThe foci are so and\nWe know that the vertices and foci are related by the equation Solving for we have:\nNow we need only substitute and into the standard form of the equation. The equation of the ellipse is\nTRY IT #1 What is the standard form equation of the ellipse that has vertices and foci"
	},
	{
		"chunk_id" : "00003452",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Can we write the equation of an ellipse centered at the origin given coordinates of just one focus\nand vertex?\nYes. Ellipses are symmetrical, so the coordinates of the vertices of an ellipse centered around the origin\nwill always have the form or Similarly, the coordinates of the foci will always have the\nform or Knowing this, we can use and from the given points, along with the equation\nto find\nWriting Equations of Ellipses Not Centered at the Origin"
	},
	{
		"chunk_id" : "00003453",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Like the graphs of other equations, the graph of anellipsecan be translated. If an ellipse is translated units\nhorizontally and units vertically, the center of the ellipse will be Thistranslationresults in the standard form of\nthe equation we saw previously, with replaced by andyreplaced by\nAccess for free at openstax.org\n12.1  The Ellipse 1153\nStandard Forms of the Equation of an Ellipse with Center (h,k)\nThe standard form of the equation of an ellipse with center andmajor axisparallel to thex-axis is"
	},
	{
		"chunk_id" : "00003454",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where\n\n the length of the major axis is\n the coordinates of the vertices are\n the length of the minor axis is\n the coordinates of the co-vertices are\n the coordinates of the foci are where SeeFigure 7a\nThe standard form of the equation of an ellipse with center and major axis parallel to they-axis is\nwhere\n\n the length of the major axis is\n the coordinates of the vertices are\n the length of the minor axis is\n the coordinates of the co-vertices are"
	},
	{
		"chunk_id" : "00003455",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the coordinates of the co-vertices are\n the coordinates of the foci are where SeeFigure 7b\nJust as with ellipses centered at the origin, ellipses that are centered at a point have vertices, co-vertices, and\nfoci that are related by the equation We can use this relationship along with the midpoint and distance\nformulas to find the equation of the ellipse in standard form when the vertices and foci are given.\nFigure7 (a) Horizontal ellipse with center (b) Vertical ellipse with center\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003456",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the vertices and foci of an ellipse not centered at the origin, write its equation in standard form.\n1. Determine whether the major axis is parallel to thex- ory-axis.\na. If they-coordinates of the given vertices and foci are the same, then the major axis is parallel to thex-axis.\n1154 12  Analytic Geometry\nUse the standard form\nb. If thex-coordinates of the given vertices and foci are the same, then the major axis is parallel to they-axis.\nUse the standard form"
	},
	{
		"chunk_id" : "00003457",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use the standard form\n2. Identify the center of the ellipse using the midpoint formula and the given coordinates for the vertices.\n3. Find by solving for the length of the major axis, which is the distance between the given vertices.\n4. Find using and found in Step 2, along with the given coordinates for the foci.\n5. Solve for using the equation\n6. Substitute the values for and into the standard form of the equation determined in Step 1.\nEXAMPLE2"
	},
	{
		"chunk_id" : "00003458",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE2\nWriting the Equation of an Ellipse Centered at a Point Other Than the Origin\nWhat is the standard form equation of the ellipse that has vertices and\nand foci and\nSolution\nThex-coordinates of the vertices and foci are the same, so the major axis is parallel to they-axis. Thus, the equation of\nthe ellipse will have the form\nFirst, we identify the center, The center is halfway between the vertices, and Applying the\nmidpoint formula, we have:"
	},
	{
		"chunk_id" : "00003459",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "midpoint formula, we have:\nNext, we find The length of the major axis, is bounded by the vertices. We solve for by finding the distance\nbetween they-coordinates of the vertices.\nSo\nNow we find The foci are given by So, and We substitute\nusing either of these points to solve for\nSo\nNext, we solve for using the equation\nFinally, we substitute the values found for and into the standard form equation for an ellipse:\nAccess for free at openstax.org\n12.1  The Ellipse 1155"
	},
	{
		"chunk_id" : "00003460",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.1  The Ellipse 1155\nTRY IT #2 What is the standard form equation of the ellipse that has vertices and and foci\nand\nGraphing Ellipses Centered at the Origin\nJust as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph\nellipses centered at the origin, we use the standard form for horizontal ellipses and\nfor vertical ellipses.\n...\nHOW TO\nGiven the standard form of an equation for an ellipse centered at sketch the graph."
	},
	{
		"chunk_id" : "00003461",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.\na. If the equation is in the form where then\n the major axis is thex-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\nb. If the equation is in the form where then\n the major axis is they-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are"
	},
	{
		"chunk_id" : "00003462",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the coordinates of the foci are\n2. Solve for using the equation\n3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the\nellipse.\nEXAMPLE3\nGraphing an Ellipse Centered at the Origin\nGraph the ellipse given by the equation, Identify and label the center, vertices, co-vertices, and foci.\nSolution\nFirst, we determine the position of the major axis. Because the major axis is on they-axis. Therefore, the\nequation is in the form where and It follows that:"
	},
	{
		"chunk_id" : "00003463",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the center of the ellipse is\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are where Solving for we have:\nTherefore, the coordinates of the foci are\n1156 12  Analytic Geometry\nNext, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. SeeFigure\n8.\nFigure8\nTRY IT #3 Graph the ellipse given by the equation Identify and label the center, vertices, co-\nvertices, and foci.\nEXAMPLE4"
	},
	{
		"chunk_id" : "00003464",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertices, and foci.\nEXAMPLE4\nGraphing an Ellipse Centered at the Origin from an Equation Not in Standard Form\nGraph the ellipse given by the equation Rewrite the equation in standard form. Then identify and\nlabel the center, vertices, co-vertices, and foci.\nSolution\nFirst, use algebra to rewrite the equation in standard form.\nNext, we determine the position of the major axis. Because the major axis is on thex-axis. Therefore, the\nequation is in the form where and It follows that:"
	},
	{
		"chunk_id" : "00003465",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the center of the ellipse is\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are where Solving for we have:\nTherefore the coordinates of the foci are\nNext, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.\nAccess for free at openstax.org\n12.1  The Ellipse 1157\nFigure9\nTRY IT #4 Graph the ellipse given by the equation Rewrite the equation in standard"
	},
	{
		"chunk_id" : "00003466",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form. Then identify and label the center, vertices, co-vertices, and foci.\nGraphing Ellipses Not Centered at the Origin\nWhen anellipseis not centered at the origin, we can still use the standard forms to find the key features of the graph.\nWhen the ellipse is centered at some point, we use the standard forms for horizontal\nellipses and for vertical ellipses. From these standard equations, we can easily determine\nthe center, vertices, co-vertices, foci, and positions of the major and minor axes.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003467",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the standard form of an equation for an ellipse centered at sketch the graph.\n1. Use the standard forms of the equations of an ellipse to determine the center, position of the major axis,\nvertices, co-vertices, and foci.\na. If the equation is in the form where then\n the center is\n the major axis is parallel to thex-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\nb. If the equation is in the form where then"
	},
	{
		"chunk_id" : "00003468",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "b. If the equation is in the form where then\n the center is\n the major axis is parallel to they-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\n2. Solve for using the equation\n3. Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the\nellipse.\nEXAMPLE5\nGraphing an Ellipse Centered at (h,k)"
	},
	{
		"chunk_id" : "00003469",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE5\nGraphing an Ellipse Centered at (h,k)\nGraph the ellipse given by the equation, Identify and label the center, vertices, co-vertices, and foci.\n1158 12  Analytic Geometry\nSolution\nFirst, we determine the position of the major axis. Because the major axis is parallel to they-axis. Therefore, the\nequation is in the form where and It follows that:\n the center of the ellipse is\n the coordinates of the vertices are or and\n the coordinates of the co-vertices are or and"
	},
	{
		"chunk_id" : "00003470",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the coordinates of the co-vertices are or and\n the coordinates of the foci are where Solving for we have:\nTherefore, the coordinates of the foci are and\nNext, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.\nFigure10\nTRY IT #5 Graph the ellipse given by the equation Identify and label the center,\nvertices, co-vertices, and foci.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003471",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertices, co-vertices, and foci.\n...\nHOW TO\nGiven the general form of an equation for an ellipse centered at (h,k), express the equation in standard form.\n1. Recognize that an ellipse described by an equation in the form is in general form.\n2. Rearrange the equation by grouping terms that contain the same variable. Move the constant term to the\nopposite side of the equation.\n3. Factor out the coefficients of the and terms in preparation for completing the square."
	},
	{
		"chunk_id" : "00003472",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Complete the square for each variable to rewrite the equation in the form of the sum of multiples of two\nbinomials squared set equal to a constant, where and are constants.\n5. Divide both sides of the equation by the constant term to express the equation in standard form.\nEXAMPLE6\nGraphing an Ellipse Centered at (h,k) by First Writing It in Standard Form\nGraph the ellipse given by the equation Identify and label the center, vertices, co-\nAccess for free at openstax.org\n12.1  The Ellipse 1159"
	},
	{
		"chunk_id" : "00003473",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.1  The Ellipse 1159\nvertices, and foci.\nSolution\nWe must begin by rewriting the equation in standard form.\nGroup terms that contain the same variable, and move the constant to the opposite side of the equation.\nFactor out the coefficients of the squared terms.\nComplete the square twice. Remember to balance the equation by adding the same constants to each side.\nRewrite as perfect squares.\nDivide both sides by the constant term to place the equation in standard form."
	},
	{
		"chunk_id" : "00003474",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Now that the equation is in standard form, we can determine the position of the major axis. Because the major\naxis is parallel to thex-axis. Therefore, the equation is in the form where and It\nfollows that:\n the center of the ellipse is\n the coordinates of the vertices are or and\n the coordinates of the co-vertices are or and\n the coordinates of the foci are where Solving for we have:\nTherefore, the coordinates of the foci are and"
	},
	{
		"chunk_id" : "00003475",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Therefore, the coordinates of the foci are and\nNext we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in\nFigure 11.\nFigure11\nTRY IT #6 Express the equation of the ellipse given in standard form. Identify the center, vertices, co-\nvertices, and foci of the ellipse.\n1160 12  Analytic Geometry\nSolving Applied Problems Involving Ellipses"
	},
	{
		"chunk_id" : "00003476",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Applied Problems Involving Ellipses\nMany real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and\nshapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to\nbreak up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with"
	},
	{
		"chunk_id" : "00003477",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus.\nThis occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering\nchamber, the sound wave will be reflected off the elliptical dome and back to the other focus. SeeFigure 12. In the\nwhisper chamber at the Museum of Science and Industry in Chicago, two people standing at the fociabout 43 feet"
	},
	{
		"chunk_id" : "00003478",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "apartcan hear each other whisper. When these chambers are placed in unexpected places, such as the ones inside\nBush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions\namong travelers.\nFigure12 Sound waves are reflected between foci in an elliptical room, called a whispering chamber.\nEXAMPLE7\nLocating the Foci of a Whispering Chamber"
	},
	{
		"chunk_id" : "00003479",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Locating the Foci of a Whispering Chamber\nA large room in an art gallery is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in\nFigure 13.\na. What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a\nhorizontal ellipse, and let the center of the room be the point\nb. If two visitors standing at the foci of this room can hear each other whisper, how far apart are the two visitors?\nRound to the nearest foot.\nFigure13\nSolution"
	},
	{
		"chunk_id" : "00003480",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Round to the nearest foot.\nFigure13\nSolution\na. We are assuming a horizontal ellipse with center so we need to find an equation of the form\nwhere We know that the length of the major axis, is longer than the length of the minor axis, So the\nlength of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor\naxis.\n Solving for we have so and\n Solving for we have so and\nTherefore, the equation of the ellipse is"
	},
	{
		"chunk_id" : "00003481",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Therefore, the equation of the ellipse is\nb. To find the distance between the senators, we must find the distance between the foci, where\nAccess for free at openstax.org\n12.1  The Ellipse 1161\nSolving for we have:\nThe points represent the foci. Thus, the distance between the senators is feet.\nTRY IT #7 Suppose a whispering chamber is 480 feet long and 320 feet wide.\n What is the standard form of the equation of the ellipse representing the room? Hint: assume"
	},
	{
		"chunk_id" : "00003482",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a horizontal ellipse, and let the center of the room be the point\n If two people are standing at the foci of this room and can hear each other whisper, how far\napart are the people? Round to the nearest foot.\nMEDIA\nAccess these online resources for additional instruction and practice with ellipses.\nConic Sections: The Ellipse(http://openstax.org/l/conicellipse)\nGraph an Ellipse with Center at the Origin(http://openstax.org/l/grphellorigin)"
	},
	{
		"chunk_id" : "00003483",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph an Ellipse with Center Not at the Origin(http://openstax.org/l/grphellnot)\n12.1 SECTION EXERCISES\nVerbal\n1. Define an ellipse in terms of 2. Where must the foci of an 3. What special case of the\nits foci. ellipse lie? ellipse do we have when the\nmajor and minor axis are of\nthe same length?\n4. For the special case 5. What can be said about the\nmentioned in the previous symmetry of the graph of an\nquestion, what would be ellipse with center at the\ntrue about the foci of that origin and foci along the"
	},
	{
		"chunk_id" : "00003484",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ellipse? y-axis?\nAlgebraic\nFor the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.\n6. 7. 8.\n9. 10.\n1162 12  Analytic Geometry\nFor the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major\nand minor axes as well as the foci.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21.\n22. 23.\n24. 25.\n26.\nFor the following exercises, find the foci for the given ellipses.\n27. 28. 29.\n30. 31.\nGraphical"
	},
	{
		"chunk_id" : "00003485",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "27. 28. 29.\n30. 31.\nGraphical\nFor the following exercises, graph the given ellipses, noting center, vertices, and foci.\n32. 33. 34.\n35. 36. 37.\n38. 39. 40.\n41. 42.\n43. 44.\n45.\nAccess for free at openstax.org\n12.1  The Ellipse 1163\nFor the following exercises, use the given information about the graph of each ellipse to determine its equation.\n46. Center at the origin, 47. Center at the origin, 48. Center at the origin,\nsymmetric with respect to symmetric with respect to symmetric with respect to"
	},
	{
		"chunk_id" : "00003486",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "thex- andy-axes, focus at thex- andy-axes, focus at thex- andy-axes, focus at\nand point on graph and point on graph and major axis is\ntwice as long as minor axis.\n49. Center ; vertex ; 50. Center ; vertex 51. Center ; vertex\none focus: . ; one focus: ; one focus:\nFor the following exercises, given the graph of the ellipse, determine its equation.\n52. 53. 54.\n55. 56.\nExtensions\nFor the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula\n57. 58. 59.\n60. 61."
	},
	{
		"chunk_id" : "00003487",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "57. 58. 59.\n60. 61.\n1164 12  Analytic Geometry\nReal-World Applications\n62. Find the equation of the 63. Find the equation of the 64. An arch has the shape of a\nellipse that will just fit ellipse that will just fit semi-ellipse (the top half of\ninside a box that is 8 units inside a box that is four an ellipse). The arch has a\nwide and 4 units high. times as wide as it is high. height of 8 feet and a span\nExpress in terms of the of 20 feet. Find an equation\nheight. for the ellipse, and use that"
	},
	{
		"chunk_id" : "00003488",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "height. for the ellipse, and use that\nto find the height to the\nnearest 0.01 foot of the\narch at a distance of 4 feet\nfrom the center.\n65. An arch has the shape of a 66. A bridge is to be built in the 67. A person in a whispering\nsemi-ellipse. The arch has a shape of a semi-elliptical gallery standing at one\nheight of 12 feet and a arch and is to have a span focus of the ellipse can\nspan of 40 feet. Find an of 120 feet. The height of whisper and be heard by a"
	},
	{
		"chunk_id" : "00003489",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation for the ellipse, the arch at a distance of 40 person standing at the\nand use that to find the feet from the center is to other focus because all the\ndistance from the center to be 8 feet. Find the height of sound waves that reach the\na point at which the height the arch at its center. ceiling are reflected to the\nis 6 feet. Round to the other person. If a\nnearest hundredth. whispering gallery has a\nlength of 120 feet, and the\nfoci are located 30 feet\nfrom the center, find the"
	},
	{
		"chunk_id" : "00003490",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from the center, find the\nheight of the ceiling at the\ncenter.\n68. A person is standing 8 feet\nfrom the nearest wall in a\nwhispering gallery. If that\nperson is at one focus, and\nthe other focus is 80 feet\naway, what is the length\nand height at the center of\nthe gallery?\n12.2 The Hyperbola\nLearning Objectives\nIn this section, you will:\nLocate a hyperbolas vertices and foci.\nWrite equations of hyperbolas in standard form.\nGraph hyperbolas centered at the origin.\nGraph hyperbolas not centered at the origin."
	},
	{
		"chunk_id" : "00003491",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph hyperbolas not centered at the origin.\nSolve applied problems involving hyperbolas.\nWhat do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common?\nThey can all be modeled by the same type ofconic. For instance, when something moves faster than the speed of sound,\na shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground,\nresulting in a sonic boom. SeeFigure 1.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003492",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n12.2  The Hyperbola 1165\nFigure1 A shock wave intersecting the ground forms a portion of a conic and results in a sonic boom.\nMost people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound\nbarrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of\nsound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually"
	},
	{
		"chunk_id" : "00003493",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "supersedes the sound of the sonic boom.\nLocating the Vertices and Foci of a Hyperbola\nIn analytic geometry, ahyperbolais a conic section formed by intersecting a right circular cone with a plane at an angle\nsuch that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are\nmirror images of each other. SeeFigure 2.\nFigure2 A hyperbola\nLike the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all"
	},
	{
		"chunk_id" : "00003494",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "points in a plane such that the difference of the distances between and the foci is a positive constant.\nNotice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is\ndefined in terms of thedifferenceof two distances, whereas the ellipse is defined in terms of thesumof two distances.\nAs with the ellipse, every hyperbola has twoaxes of symmetry. Thetransverse axisis a line segment that passes"
	},
	{
		"chunk_id" : "00003495",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse\naxis. Theconjugate axisis perpendicular to the transverse axis and has the co-vertices as its endpoints. Thecenter of a\nhyperbolais the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has\ntwoasymptotesthat pass through its center. As a hyperbola recedes from the center, its branches approach these"
	},
	{
		"chunk_id" : "00003496",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptotes. Thecentral rectangleof the hyperbola is centered at the origin with sides that pass through each vertex\nand co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the\nhyperbola, simply sketch and extend the diagonals of the central rectangle. SeeFigure 3.\n1166 12  Analytic Geometry\nFigure3 Key features of the hyperbola\nIn this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate"
	},
	{
		"chunk_id" : "00003497",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "plane; the axes will either lie on or be parallel to thex- andy-axes. We will consider two cases: those that are centered at\nthe origin, and those that are centered at a point other than the origin.\nDeriving the Equation of a Hyperbola Centered at the Origin\nLet and be thefociof a hyperbola centered at the origin. The hyperbola is the set of all points such\nthat the difference of the distances from to the foci is constant. SeeFigure 4.\nFigure4"
	},
	{
		"chunk_id" : "00003498",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure4\nIf is a vertex of the hyperbola, the distance from to is The distance from to\nis The difference of the distances from the foci to the vertex is\nIf is a point on the hyperbola, we can define the following variables:\nBy definition of a hyperbola, is constant for any point on the hyperbola. We know that the difference of\nthese distances is for the vertex It follows that for any point on the hyperbola. As with the"
	},
	{
		"chunk_id" : "00003499",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "derivation of the equation of an ellipse, we will begin by applying thedistance formula. The rest of the derivation is\nalgebraic. Compare this derivation with the one from the previous section for ellipses.\nAccess for free at openstax.org\n12.2  The Hyperbola 1167\nThis equation defines a hyperbola centered at the origin with vertices and co-vertices\nStandard Forms of the Equation of a Hyperbola with Center (0,0)"
	},
	{
		"chunk_id" : "00003500",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The standard form of the equation of a hyperbola with center and transverse axis on thex-axis is\nwhere\n the length of the transverse axis is\n the coordinates of the vertices are\n the length of the conjugate axis is\n the coordinates of the co-vertices are\n the distance between the foci is where\n the coordinates of the foci are\n the equations of the asymptotes are\nSeeFigure 5a.\nThe standard form of the equation of a hyperbola with center and transverse axis on they-axis is\nwhere"
	},
	{
		"chunk_id" : "00003501",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where\n the length of the transverse axis is\n the coordinates of the vertices are\n the length of the conjugate axis is\n the coordinates of the co-vertices are\n1168 12  Analytic Geometry\n the distance between the foci is where\n the coordinates of the foci are\n the equations of the asymptotes are\nSeeFigure 5b.\nNote that the vertices, co-vertices, and foci are related by the equation When we are given the equation\nof a hyperbola, we can use this relationship to identify its vertices and foci."
	},
	{
		"chunk_id" : "00003502",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure5 (a) Horizontal hyperbola with center (b) Vertical hyperbola with center\n...\nHOW TO\nGiven the equation of a hyperbola in standard form, locate its vertices and foci.\n1. Determine whether the transverse axis lies on thex- ory-axis. Notice that is always under the variable with\nthe positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the\ncase where the hyperbola is centered at the origin, the intercepts coincide with the vertices."
	},
	{
		"chunk_id" : "00003503",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If the equation has the form then the transverse axis lies on thex-axis. The vertices are\nlocated at and the foci are located at\nb. If the equation has the form then the transverse axis lies on they-axis. The vertices are\nlocated at and the foci are located at\n2. Solve for using the equation\n3. Solve for using the equation\nEXAMPLE1\nLocating a Hyperbolas Vertices and Foci\nIdentify the vertices and foci of thehyperbolawith equation\nSolution"
	},
	{
		"chunk_id" : "00003504",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe equation has the form so the transverse axis lies on they-axis. The hyperbola is centered at the\nAccess for free at openstax.org\n12.2  The Hyperbola 1169\norigin, so the vertices serve as they-intercepts of the graph. To find the vertices, set and solve for\nThe foci are located at Solving for\nTherefore, the vertices are located at and the foci are located at\nTRY IT #1 Identify the vertices and foci of the hyperbola with equation\nWriting Equations of Hyperbolas in Standard Form"
	},
	{
		"chunk_id" : "00003505",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Writing Equations of Hyperbolas in Standard Form\nJust as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its\ncenter, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes.\nConversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for"
	},
	{
		"chunk_id" : "00003506",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered\nat some point other than the origin.\nHyperbolas Centered at the Origin\nReviewing the standard forms given for hyperbolas centered at we see that the vertices, co-vertices, and foci are\nrelated by the equation Note that this equation can also be rewritten as This relationship is\nused to write the equation for a hyperbola when given the coordinates of its foci and vertices.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003507",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven the vertices and foci of a hyperbola centered at write its equation in standard form.\n1. Determine whether the transverse axis lies on thex- ory-axis.\na. If the given coordinates of the vertices and foci have the form and respectively, then the\ntransverse axis is thex-axis. Use the standard form\nb. If the given coordinates of the vertices and foci have the form and respectively, then the\ntransverse axis is they-axis. Use the standard form\n2. Find using the equation"
	},
	{
		"chunk_id" : "00003508",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Find using the equation\n3. Substitute the values for and into the standard form of the equation determined in Step 1.\nEXAMPLE2\nFinding the Equation of a Hyperbola Centered at (0,0) Given its Foci and Vertices\nWhat is the standard form equation of thehyperbolathat has vertices and foci\nSolution\nThe vertices and foci are on thex-axis. Thus, the equation for the hyperbola will have the form\nThe vertices are so and\n1170 12  Analytic Geometry\nThe foci are so and\nSolving for we have"
	},
	{
		"chunk_id" : "00003509",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The foci are so and\nSolving for we have\nFinally, we substitute and into the standard form of the equation, The equation of the\nhyperbola is as shown inFigure 6.\nFigure6\nTRY IT #2 What is the standard form equation of the hyperbola that has vertices and foci\nHyperbolas Not Centered at the Origin\nLike the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated units"
	},
	{
		"chunk_id" : "00003510",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontally and units vertically, the center of thehyperbolawill be This translation results in the standard form\nof the equation we saw previously, with replaced by and replaced by\nStandard Forms of the Equation of a Hyperbola with Center (h,k)\nThe standard form of the equation of a hyperbola with center and transverse axis parallel to thex-axis is\nwhere\n the length of the transverse axis is\n the coordinates of the vertices are\n the length of the conjugate axis is"
	},
	{
		"chunk_id" : "00003511",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the length of the conjugate axis is\n the coordinates of the co-vertices are\n the distance between the foci is where\n the coordinates of the foci are\nThe asymptotes of the hyperbola coincide with the diagonals of the central rectangle. The length of the rectangle is\nand its width is The slopes of the diagonals are and each diagonal passes through the center Using\nthepoint-slope formula, it is simple to show that the equations of the asymptotes are SeeFigure\n7a\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003512",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "7a\nAccess for free at openstax.org\n12.2  The Hyperbola 1171\nThe standard form of the equation of a hyperbola with center and transverse axis parallel to they-axis is\nwhere\n the length of the transverse axis is\n the coordinates of the vertices are\n the length of the conjugate axis is\n the coordinates of the co-vertices are\n the distance between the foci is where\n the coordinates of the foci are\nUsing the reasoning above, the equations of the asymptotes are SeeFigure 7b."
	},
	{
		"chunk_id" : "00003513",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure7 (a) Horizontal hyperbola with center (b) Vertical hyperbola with center\nLike hyperbolas centered at the origin, hyperbolas centered at a point have vertices, co-vertices, and foci that are\nrelated by the equation We can use this relationship along with the midpoint and distance formulas to find\nthe standard equation of a hyperbola when the vertices and foci are given.\n...\nHOW TO\nGiven the vertices and foci of a hyperbola centered at write its equation in standard form."
	},
	{
		"chunk_id" : "00003514",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Determine whether the transverse axis is parallel to thex- ory-axis.\na. If they-coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the\nx-axis. Use the standard form\nb. If thex-coordinates of the given vertices and foci are the same, then the transverse axis is parallel to the\ny-axis. Use the standard form\n2. Identify the center of the hyperbola, using the midpoint formula and the given coordinates for the\nvertices."
	},
	{
		"chunk_id" : "00003515",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertices.\n3. Find by solving for the length of the transverse axis, , which is the distance between the given vertices.\n1172 12  Analytic Geometry\n4. Find using and found in Step 2 along with the given coordinates for the foci.\n5. Solve for using the equation\n6. Substitute the values for and into the standard form of the equation determined in Step 1.\nEXAMPLE3\nFinding the Equation of a Hyperbola Centered at (h,k) Given its Foci and Vertices"
	},
	{
		"chunk_id" : "00003516",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "What is the standard form equation of thehyperbolathat has vertices at and and foci at and\nSolution\nThey-coordinates of the vertices and foci are the same, so the transverse axis is parallel to thex-axis. Thus, the equation\nof the hyperbola will have the form\nFirst, we identify the center, The center is halfway between the vertices and Applying the midpoint\nformula, we have\nNext, we find The length of the transverse axis, is bounded by the vertices. So, we can find by finding the"
	},
	{
		"chunk_id" : "00003517",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distance between thex-coordinates of the vertices.\nNow we need to find The coordinates of the foci are So and We\ncan use thex-coordinate from either of these points to solve for Using the point and substituting\nNext, solve for using the equation\nFinally, substitute the values found for and into the standard form of the equation.\nTRY IT #3 What is the standard form equation of the hyperbola that has vertices and and foci\nand\nGraphing Hyperbolas Centered at the Origin"
	},
	{
		"chunk_id" : "00003518",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and\nGraphing Hyperbolas Centered at the Origin\nWhen we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify\nthe key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the\ntransverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form for\nAccess for free at openstax.org\n12.2  The Hyperbola 1173"
	},
	{
		"chunk_id" : "00003519",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.2  The Hyperbola 1173\nhorizontal hyperbolas and the standard form for vertical hyperbolas.\n...\nHOW TO\nGiven a standard form equation for a hyperbola centered at sketch the graph.\n1. Determine which of the standard forms applies to the given equation.\n2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the\nvertices, co-vertices, and foci; and the equations for the asymptotes.\na. If the equation is in the form then"
	},
	{
		"chunk_id" : "00003520",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If the equation is in the form then\n the transverse axis is on thex-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\n the equations of the asymptotes are\nb. If the equation is in the form then\n the transverse axis is on they-axis\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\n the equations of the asymptotes are"
	},
	{
		"chunk_id" : "00003521",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the equations of the asymptotes are\n3. Solve for the coordinates of the foci using the equation\n4. Plot the vertices, co-vertices, foci, and asymptotes in the coordinate plane, and draw a smooth curve to form the\nhyperbola.\nEXAMPLE4\nGraphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form\nGraph thehyperbolagiven by the equation Identify and label the vertices, co-vertices, foci, and\nasymptotes.\nSolution"
	},
	{
		"chunk_id" : "00003522",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptotes.\nSolution\nThe standard form that applies to the given equation is Thus, the transverse axis is on they-axis\nThe coordinates of the vertices are\nThe coordinates of the co-vertices are\nThe coordinates of the foci are where Solving for we have\nTherefore, the coordinates of the foci are\nThe equations of the asymptotes are\nPlot and label the vertices and co-vertices, and then sketch the central rectangle. Sides of the rectangle are parallel to the"
	},
	{
		"chunk_id" : "00003523",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "axes and pass through the vertices and co-vertices. Sketch and extend the diagonals of the central rectangle to show the\nasymptotes. The central rectangle and asymptotes provide the framework needed to sketch an accurate graph of the\nhyperbola. Label the foci and asymptotes, and draw a smooth curve to form the hyperbola, as shown inFigure 8.\n1174 12  Analytic Geometry\nFigure8\nTRY IT #4 Graph the hyperbola given by the equation Identify and label the vertices, co-\nvertices, foci, and asymptotes."
	},
	{
		"chunk_id" : "00003524",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertices, foci, and asymptotes.\nGraphing Hyperbolas Not Centered at the Origin\nGraphing hyperbolas centered at a point other than the origin is similar to graphing ellipses centered at a point\nother than the origin. We use the standard forms for horizontal hyperbolas, and\nfor vertical hyperbolas. From these standard form equations we can easily calculate and plot key\nfeatures of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and"
	},
	{
		"chunk_id" : "00003525",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the positions of the transverse and conjugate axes.\n...\nHOW TO\nGiven a general form for a hyperbola centered at sketch the graph.\n1. Convert the general form to that standard form. Determine which of the standard forms applies to the given\nequation.\n2. Use the standard form identified in Step 1 to determine the position of the transverse axis; coordinates for the\ncenter, vertices, co-vertices, foci; and equations for the asymptotes.\na. If the equation is in the form then"
	},
	{
		"chunk_id" : "00003526",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If the equation is in the form then\n the transverse axis is parallel to thex-axis\n the center is\n the coordinates of the vertices are\n the coordinates of the co-vertices are\n the coordinates of the foci are\n the equations of the asymptotes are\nAccess for free at openstax.org\n12.2  The Hyperbola 1175\nb. If the equation is in the form then\n the transverse axis is parallel to they-axis\n the center is\n the coordinates of the vertices are\n the coordinates of the co-vertices are"
	},
	{
		"chunk_id" : "00003527",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the coordinates of the co-vertices are\n the coordinates of the foci are\n the equations of the asymptotes are\n3. Solve for the coordinates of the foci using the equation\n4. Plot the center, vertices, co-vertices, foci, and asymptotes in the coordinate plane and draw a smooth curve to\nform the hyperbola.\nEXAMPLE5\nGraphing a Hyperbola Centered at (h,k) Given an Equation in General Form\nGraph thehyperbolagiven by the equation Identify and label the center, vertices, co-\nvertices, foci, and asymptotes."
	},
	{
		"chunk_id" : "00003528",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "vertices, foci, and asymptotes.\nSolution\nStart by expressing the equation in standard form. Group terms that contain the same variable, and move the constant\nto the opposite side of the equation.\nFactor the leading coefficient of each expression.\nComplete the square twice. Remember to balance the equation by adding the same constants to each side.\nRewrite as perfect squares.\nDivide both sides by the constant term to place the equation in standard form."
	},
	{
		"chunk_id" : "00003529",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The standard form that applies to the given equation is where and or and\nThus, the transverse axis is parallel to thex-axis. It follows that:\n the center of the ellipse is\n the coordinates of the vertices are or and\n the coordinates of the co-vertices are or and\n the coordinates of the foci are where Solving for we have\nTherefore, the coordinates of the foci are and\nThe equations of the asymptotes are"
	},
	{
		"chunk_id" : "00003530",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The equations of the asymptotes are\nNext, we plot and label the center, vertices, co-vertices, foci, and asymptotes and draw smooth curves to form the\nhyperbola, as shown inFigure 9.\n1176 12  Analytic Geometry\nFigure9\nTRY IT #5 Graph the hyperbola given by the standard form of an equation Identify and\nlabel the center, vertices, co-vertices, foci, and asymptotes.\nSolving Applied Problems Involving Hyperbolas"
	},
	{
		"chunk_id" : "00003531",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Applied Problems Involving Hyperbolas\nAs we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as\nastronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly\ninteresting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to"
	},
	{
		"chunk_id" : "00003532",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while\nrequiring less material than any other forms of their size and strength. SeeFigure 10. For example, a 500-foot tower can\nbe made of a reinforced concrete shell only 6 or 8 inches wide!\nFigure10 Cooling towers at the Drax power station in North Yorkshire, United Kingdom (credit: Les Haines, Flickr)"
	},
	{
		"chunk_id" : "00003533",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in\nFrance, standing a remarkable 170 meters tall. InExample 6we will use the design layout of a cooling tower to find a\nhyperbolic equation that models its sides.\nAccess for free at openstax.org\n12.2  The Hyperbola 1177\nEXAMPLE6\nSolving Applied Problems Involving Hyperbolas\nThe design layout of a cooling tower is shown inFigure 11. The tower stands 179.6 meters tall. The diameter of the top is"
	},
	{
		"chunk_id" : "00003534",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "72 meters. At their closest, the sides of the tower are 60 meters apart.\nFigure11 Project design for a natural draft cooling tower\nFind the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the\nhyperbolaindicated by the intersection of dashed perpendicular lines in the figureis the origin of the coordinate\nplane. Round final values to four decimal places.\nSolution"
	},
	{
		"chunk_id" : "00003535",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola\ncentered at the origin: where the branches of the hyperbola form the sides of the cooling tower. We must\nfind the values of and to complete the model.\nFirst, we find Recall that the length of the transverse axis of a hyperbola is This length is represented by the\ndistance where the sides are closest, which is given as meters. So, Therefore, and"
	},
	{
		"chunk_id" : "00003536",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To solve for we need to substitute for and in our equation using a known point. To do this, we can use the\ndimensions of the tower to find some point that lies on the hyperbola. We will use the top right corner of the tower\nto represent that point. Since they-axis bisects the tower, ourx-value can be represented by the radius of the top, or 36\nmeters. They-value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,"
	},
	{
		"chunk_id" : "00003537",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The sides of the tower can be modeled by the hyperbolic equation\nTRY IT #6 A design for a cooling tower project is shown inFigure 12. Find the equation of the hyperbola that\nmodels the sides of the cooling tower. Assume that the center of the hyperbolaindicated by the\n1178 12  Analytic Geometry\nintersection of dashed perpendicular lines in the figureis the origin of the coordinate plane.\nRound final values to four decimal places.\nFigure12\nMEDIA"
	},
	{
		"chunk_id" : "00003538",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure12\nMEDIA\nAccess these online resources for additional instruction and practice with hyperbolas.\nConic Sections: The Hyperbola Part 1 of 2(http://openstax.org/l/hyperbola1)\nConic Sections: The Hyperbola Part 2 of 2(http://openstax.org/l/hyperbola2)\nGraph a Hyperbola with Center at Origin(http://openstax.org/l/hyperbolaorigin)\nGraph a Hyperbola with Center not at Origin(http://openstax.org/l/hbnotorigin)\n12.2 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00003539",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.2 SECTION EXERCISES\nVerbal\n1. Define a hyperbola in terms 2. What can we conclude 3. What must be true of the\nof its foci. about a hyperbola if its foci of a hyperbola?\nasymptotes intersect at the\norigin?\n4. If the transverse axis of a 5. Where must the center of\nhyperbola is vertical, what hyperbola be relative to its\ndo we know about the foci?\ngraph?\nAlgebraic\nFor the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard\nform.\n6. 7. 8.\n9. 10."
	},
	{
		"chunk_id" : "00003540",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form.\n6. 7. 8.\n9. 10.\nAccess for free at openstax.org\n12.2  The Hyperbola 1179\nFor the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the\nvertices and foci, and write equations of asymptotes.\n11. 12. 13.\n14. 15. 16.\n17. 18. 19.\n20. 21.\n22. 23.\n24. 25.\nFor the following exercises, find the equations of the asymptotes for each hyperbola.\n26. 27. 28.\n29. 30.\nGraphical"
	},
	{
		"chunk_id" : "00003541",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "26. 27. 28.\n29. 30.\nGraphical\nFor the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41.\n42. 43.\n44.\nFor the following exercises, given information about the graph of the hyperbola, find its equation.\n45. Vertices at and 46. Vertices at and 47. Vertices at and\nand one focus at and one focus at and one focus at\n1180 12  Analytic Geometry\n48. Center: vertex: 49. Center: vertex: 50. Center: vertex:\none focus: one focus: one focus:"
	},
	{
		"chunk_id" : "00003542",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "one focus: one focus: one focus:\nFor the following exercises, given the graph of the hyperbola, find its equation.\n51. 52. 53.\n54. 55.\nExtensions\nFor the following exercises, express the equation for the hyperbola as two functions, with as a function of Express as\nsimply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.\n56. 57. 58.\n59. 60.\nReal-World Applications"
	},
	{
		"chunk_id" : "00003543",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "56. 57. 58.\n59. 60.\nReal-World Applications\nFor the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the\nyard. Find the equation of the hyperbola and sketch the graph.\n61. The hedge will follow the 62. The hedge will follow the 63. The hedge will follow the\nasymptotes asymptotes asymptotes and\nand its and and its closest\nclosest distance to the its closest distance to the\ndistance to the center"
	},
	{
		"chunk_id" : "00003544",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distance to the center\ncenter fountain is 5 yards. center fountain is 6 yards.\nfountain is 10 yards.\nAccess for free at openstax.org\n12.3  The Parabola 1181\n64. The hedge will follow the 65. The hedge will follow the\nasymptotes and asymptotes and\nand its closest and its closest\ndistance to the center distance to the center\nfountain is 12 yards. fountain is 20 yards.\nFor the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate"
	},
	{
		"chunk_id" : "00003545",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the\nflight path of each object using the given information.\n66. The object enters along a 67. The object enters along a 68. The object enters along a\npath approximated by the path approximated by the path approximated by the\nline and passes line and passes line and\nwithin 1 au (astronomical within 0.5 au of the sun at passes within 1 au of the"
	},
	{
		"chunk_id" : "00003546",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "unit) of the sun at its its closest approach, so the sun at its closest approach,\nclosest approach, so that sun is one focus of the so the sun is one focus of\nthe sun is one focus of the hyperbola. It then departs the hyperbola. It then\nhyperbola. It then departs the solar system along a departs the solar system\nthe solar system along a path approximated by the along a path approximated\npath approximated by the line by the line\nline\n69. The object enters along a 70. The object enters along a"
	},
	{
		"chunk_id" : "00003547",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "path approximated by the path approximated by the\nline and passes line and passes\nwithin 1 au of the sun at its within 1 au of the sun at its\nclosest approach, so the closest approach, so the\nsun is one focus of the sun is one focus of the\nhyperbola. It then departs hyperbola. It then departs\nthe solar system along a the solar system along a\npath approximated by the path approximated by the\nline\nline\n12.3 The Parabola\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00003548",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nGraph parabolas with vertices at the origin.\nWrite equations of parabolas in standard form.\nGraph parabolas with vertices not at the origin.\nSolve applied problems involving parabolas.\n1182 12  Analytic Geometry\nFigure1 KatherineJohnson's pioneering mathematical work in the area of parabolic and other orbital calculations\nplayed a significant role in the development of U.S space flight. (credit: NASA)"
	},
	{
		"chunk_id" : "00003549",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return\nof many human missions as well as satellites. Prior to the work featured in the movieHidden Figures, she had already\nmade major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan\nShepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper"
	},
	{
		"chunk_id" : "00003550",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she\ndetermined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented\nlevel of precision. Johnson said, \"You tell me when you want it and where you want it to land"
	},
	{
		"chunk_id" : "00003551",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science.\nParabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property\nare evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes,\nmicrophones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy"
	},
	{
		"chunk_id" : "00003552",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little\nmaintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.\nGraphing Parabolas with Vertices at the Origin\nInThe Ellipse, we saw that anellipseis formed when a plane cuts through a right circular cone. If the plane is parallel to\nthe edge of the cone, an unbounded curve is formed. This curve is aparabola. SeeFigure 2."
	},
	{
		"chunk_id" : "00003553",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n12.3  The Parabola 1183\nFigure2 Parabola\nLike the ellipse andhyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is\nthe set of all points in a plane that are the same distance from a fixed line, called thedirectrix, and a fixed point\n(thefocus) not on the directrix.\nInQuadratic Functions(http://openstax.org/books/precalculus-2e/pages/3-3-power-functions-and-polynomial-"
	},
	{
		"chunk_id" : "00003554",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "functions), we learned about a parabolas vertex and axis of symmetry. Now we extend the discussion to include other\nkey features of the parabola. SeeFigure 3. Notice that the axis of symmetry passes through the focus and vertex and is\nperpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.\nThe line segment that passes through the focus and is parallel to the directrix is called thelatus rectum. The endpoints"
	},
	{
		"chunk_id" : "00003555",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the latus rectum lie on the curve. By definition, the distance from the focus to any point on the parabola is equal\nto the distance from to the directrix.\nFigure3 Key features of the parabola\nTo work with parabolas in thecoordinate plane, we consider two cases: those with a vertex at the origin and those with a\nvertexat a point other than the origin. We begin with the former.\n1184 12  Analytic Geometry\nFigure4\nLet be a point on the parabola with vertex focus and directrix as shown inFigure 4. The"
	},
	{
		"chunk_id" : "00003556",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "distance from point to point on the directrix is the difference of they-values: The distance from\nthe focus to the point is also equal to and can be expressed using thedistance formula.\nSet the two expressions for equal to each other and solve for to derive the equation of the parabola. We do this\nbecause the distance from to equals the distance from to\nWe then square both sides of the equation, expand the squared terms, and simplify by combining like terms."
	},
	{
		"chunk_id" : "00003557",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The equations of parabolas with vertex are when thex-axis is the axis of symmetry and when\nthey-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key\nfeatures.\nStandard Forms of Parabolas with Vertex (0, 0)\nTable 1andFigure 5summarize the standard features of parabolas with a vertex at the origin.\nAxis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum\nx-axis\ny-axis\nTable1\nAccess for free at openstax.org\n12.3  The Parabola 1185"
	},
	{
		"chunk_id" : "00003558",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.3  The Parabola 1185\nFigure5 (a) When and the axis of symmetry is thex-axis, the parabola opens right. (b) When and the\naxis of symmetry is thex-axis, the parabola opens left. (c) When and the axis of symmetry is they-axis, the\nparabola opens up. (d) When and the axis of symmetry is they-axis, the parabola opens down.\nThe key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. SeeFigure 5. When"
	},
	{
		"chunk_id" : "00003559",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the\nparabola.\nA line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the\nparabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown inFigure 6.\n1186 12  Analytic Geometry\nFigure6\n...\nHOW TO\nGiven a standard form equation for a parabola centered at (0, 0), sketch the graph."
	},
	{
		"chunk_id" : "00003560",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Determine which of the standard forms applies to the given equation: or\n2. Use the standard form identified in Step 1 to determine the axis of symmetry, focus, equation of the directrix,\nand endpoints of the latus rectum.\na. If the equation is in the form then\n the axis of symmetry is thex-axis,\n set equal to the coefficient ofxin the given equation to solve for If the parabola opens right.\nIf the parabola opens left.\n use to find the coordinates of the focus,"
	},
	{
		"chunk_id" : "00003561",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " use to find the coordinates of the focus,\n use to find the equation of the directrix,\n use to find the endpoints of the latus rectum, Alternately, substitute into the original\nequation.\nb. If the equation is in the form then\n the axis of symmetry is they-axis,\n set equal to the coefficient ofyin the given equation to solve for If the parabola opens up. If\nthe parabola opens down.\n use to find the coordinates of the focus,\n use to find equation of the directrix,"
	},
	{
		"chunk_id" : "00003562",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " use to find equation of the directrix,\n use to find the endpoints of the latus rectum,\n3. Plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola.\nEXAMPLE1\nGraphing a Parabola with Vertex (0, 0) and thex-axis as the Axis of Symmetry\nGraph Identify and label thefocus,directrix, and endpoints of thelatus rectum.\nAccess for free at openstax.org\n12.3  The Parabola 1187\nSolution"
	},
	{
		"chunk_id" : "00003563",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.3  The Parabola 1187\nSolution\nThe standard form that applies to the given equation is Thus, the axis of symmetry is thex-axis. It follows that:\n so Since the parabola opens right\n the coordinates of the focus are\n the equation of the directrix is\n the endpoints of the latus rectum have the samex-coordinate at the focus. To find the endpoints, substitute\ninto the original equation:\nNext we plot the focus, directrix, and latus rectum, and draw a smooth curve to form theparabola.Figure 7\nFigure7"
	},
	{
		"chunk_id" : "00003564",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure7\nTRY IT #1 Graph Identify and label the focus, directrix, and endpoints of the latus rectum.\nEXAMPLE2\nGraphing a Parabola with Vertex (0, 0) and they-axis as the Axis of Symmetry\nGraph Identify and label thefocus,directrix, and endpoints of thelatus rectum.\nSolution\nThe standard form that applies to the given equation is Thus, the axis of symmetry is they-axis. It follows that:\n so Since the parabola opens down.\n the coordinates of the focus are\n the equation of the directrix is"
	},
	{
		"chunk_id" : "00003565",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the equation of the directrix is\n the endpoints of the latus rectum can be found by substituting into the original equation,\nNext we plot the focus, directrix, and latus rectum, and draw a smooth curve to form theparabola.\n1188 12  Analytic Geometry\nFigure8\nTRY IT #2 Graph Identify and label the focus, directrix, and endpoints of the latus rectum.\nWriting Equations of Parabolas in Standard Form"
	},
	{
		"chunk_id" : "00003566",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Writing Equations of Parabolas in Standard Form\nIn the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features.\nWe can also use the calculations in reverse to write an equation for a parabola when given its key features.\n...\nHOW TO\nGiven its focus and directrix, write the equation for a parabola in standard form.\n1. Determine whether the axis of symmetry is thex- ory-axis."
	},
	{
		"chunk_id" : "00003567",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. If the given coordinates of the focus have the form then the axis of symmetry is thex-axis. Use the\nstandard form\nb. If the given coordinates of the focus have the form then the axis of symmetry is they-axis. Use the\nstandard form\n2. Multiply\n3. Substitute the value from Step 2 into the equation determined in Step 1.\nEXAMPLE3\nWriting the Equation of a Parabola in Standard Form Given its Focus and Directrix\nWhat is the equation for theparabolawithfocus anddirectrix\nSolution"
	},
	{
		"chunk_id" : "00003568",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe focus has the form so the equation will have the form\n Multiplying we have\n Substituting for we have\nTherefore, the equation for the parabola is\nTRY IT #3 What is the equation for the parabola with focus and directrix\nGraphing Parabolas with Vertices Not at the Origin\nLike other graphs weve worked with, the graph of a parabola can be translated. If a parabola is translated units"
	},
	{
		"chunk_id" : "00003569",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "horizontally and units vertically, the vertex will be This translation results in the standard form of the equation\nAccess for free at openstax.org\n12.3  The Parabola 1189\nwe saw previously with replaced by and replaced by\nTo graph parabolas with a vertex other than the origin, we use the standard form for\nparabolas that have an axis of symmetry parallel to thex-axis, and for parabolas that have an axis"
	},
	{
		"chunk_id" : "00003570",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of symmetry parallel to they-axis. These standard forms are given below, along with their general graphs and key\nfeatures.\nStandard Forms of Parabolas with Vertex (h,k)\nTable 2andFigure 9summarize the standard features of parabolas with a vertex at a point\nAxis of Symmetry Equation Focus Directrix Endpoints of Latus Rectum\nTable2\n1190 12  Analytic Geometry\nFigure9 (a) When the parabola opens right. (b) When the parabola opens left. (c) When the\nparabola opens up. (d) When the parabola opens down.\n..."
	},
	{
		"chunk_id" : "00003571",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven a standard form equation for a parabola centered at (h,k), sketch the graph.\n1. Determine which of the standard forms applies to the given equation: or\n2. Use the standard form identified in Step 1 to determine the vertex, axis of symmetry, focus, equation of the\ndirectrix, and endpoints of the latus rectum.\na. If the equation is in the form then:\n use the given equation to identify and for the vertex,\n use the value of to determine the axis of symmetry,"
	},
	{
		"chunk_id" : "00003572",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " set equal to the coefficient of in the given equation to solve for If the parabola opens\nright. If the parabola opens left.\nAccess for free at openstax.org\n12.3  The Parabola 1191\n use and to find the coordinates of the focus,\n use and to find the equation of the directrix,\n use and to find the endpoints of the latus rectum,\nb. If the equation is in the form then:\n use the given equation to identify and for the vertex,\n use the value of to determine the axis of symmetry,"
	},
	{
		"chunk_id" : "00003573",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " set equal to the coefficient of in the given equation to solve for If the parabola opens\nup. If the parabola opens down.\n use and to find the coordinates of the focus,\n use and to find the equation of the directrix,\n use and to find the endpoints of the latus rectum,\n3. Plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the\nparabola.\nEXAMPLE4\nGraphing a Parabola with Vertex (h,k) and Axis of Symmetry Parallel to thex-axis"
	},
	{
		"chunk_id" : "00003574",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graph Identify and label thevertex,axis of symmetry,focus,directrix, and endpoints of thelatus\nrectum.\nSolution\nThe standard form that applies to the given equation is Thus, the axis of symmetry is parallel to\nthex-axis. It follows that:\n the vertex is\n the axis of symmetry is\n so Since the parabola opens left.\n the coordinates of the focus are\n the equation of the directrix is\n the endpoints of the latus rectum are or and"
	},
	{
		"chunk_id" : "00003575",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the endpoints of the latus rectum are or and\nNext we plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the\nparabola. SeeFigure 10.\nFigure10\n1192 12  Analytic Geometry\nTRY IT #4 Graph Identify and label the vertex, axis of symmetry, focus, directrix, and\nendpoints of the latus rectum.\nEXAMPLE5\nGraphing a Parabola from an Equation Given in General Form\nGraph Identify and label the vertex, axis of symmetry, focus, directrix, and endpoints of the"
	},
	{
		"chunk_id" : "00003576",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "latus rectum.\nSolution\nStart by writing the equation of theparabolain standard form. The standard form that applies to the given equation is\nThus, the axis of symmetry is parallel to they-axis. To express the equation of the parabola in this\nform, we begin by isolating the terms that contain the variable in order to complete the square.\nIt follows that:\n the vertex is\n the axis of symmetry is\n since and so the parabola opens up\n the coordinates of the focus are\n the equation of the directrix is"
	},
	{
		"chunk_id" : "00003577",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the equation of the directrix is\n the endpoints of the latus rectum are or and\nNext we plot the vertex, axis of symmetry, focus, directrix, and latus rectum, and draw a smooth curve to form the\nparabola. SeeFigure 11.\nFigure11\nTRY IT #5 Graph Identify and label the vertex, axis of symmetry, focus, directrix, and\nendpoints of the latus rectum.\nSolving Applied Problems Involving Parabolas\nAs we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as"
	},
	{
		"chunk_id" : "00003578",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the\nOlympic torch, have a very unique reflecting property. When rays of light parallel to the parabolasaxis of symmetryare\ndirected toward any surface of the mirror, the light is reflected directly to the focus. SeeFigure 12. This is why the\nOlympic torch is ignited when it is held at the focus of the parabolic mirror.\nAccess for free at openstax.org\n12.3  The Parabola 1193"
	},
	{
		"chunk_id" : "00003579",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.3  The Parabola 1193\nFigure12 Reflecting property of parabolas\nParabolic mirrors have the ability to focus the suns energy to a single point, raising the temperature hundreds of\ndegrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products,\nsuch as solar cookers, solar heaters, and even travel-sized fire starters.\nEXAMPLE6\nSolving Applied Problems Involving Parabolas"
	},
	{
		"chunk_id" : "00003580",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Applied Problems Involving Parabolas\nA cross-section of a design for a travel-sized solar fire starter is shown inFigure 13. The suns rays reflect off the\nparabolic mirror toward an object attached to the igniter. Because the igniter is located at the focus of the parabola, the\nreflected rays cause the object to burn in just seconds.\n Find the equation of the parabola that models the fire starter. Assume that the vertex of the parabolic mirror is the\norigin of the coordinate plane."
	},
	{
		"chunk_id" : "00003581",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "origin of the coordinate plane.\n Use the equation found in part  to find the depth of the fire starter.\nFigure13 Cross-section of a travel-sized solar fire starter\nSolution\n The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form\nwhere The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have\n The dish extends inches on either side of the origin. We can substitute 2.25 for in the equation from"
	},
	{
		"chunk_id" : "00003582",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "part (a) to find the depth of the dish.\nThe dish is about 0.74 inches deep.\n1194 12  Analytic Geometry\nTRY IT #6 Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a\ndiameter of 1600 mm. The suns rays reflect off the parabolic mirror toward the cooker, which is\nplaced 320 mm from the base.\n Find an equation that models a cross-section of the solar cooker. Assume that the vertex of"
	},
	{
		"chunk_id" : "00003583",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the parabolic mirror is the origin of the coordinate plane, and that the parabola opens to the right\n(i.e., has thex-axis as its axis of symmetry).\n Use the equation found in part to find the depth of the cooker.\nMEDIA\nAccess these online resources for additional instruction and practice with parabolas.\nConic Sections: The Parabola Part 1 of 2(http://openstax.org/l/parabola1)\nConic Sections: The Parabola Part 2 of 2(http://openstax.org/l/parabola2)"
	},
	{
		"chunk_id" : "00003584",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Parabola with Vertical Axis(http://openstax.org/l/parabolavertcal)\nParabola with Horizontal Axis(http://openstax.org/l/parabolahoriz)\n12.3 SECTION EXERCISES\nVerbal\n1. Define a parabola in terms 2. If the equation of a parabola 3. If the equation of a parabola\nof its focus and directrix. is written in standard form is written in standard form\nand is positive and the and is negative and the\ndirectrix is a vertical line, directrix is a horizontal line,\nthen what can we conclude then what can we conclude"
	},
	{
		"chunk_id" : "00003585",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "about its graph? about its graph?\n4. What is the effect on the 5. As the graph of a parabola\ngraph of a parabola if its becomes wider, what will\nequation in standard form happen to the distance\nhas increasing values of between the focus and\ndirectrix?\nAlgebraic\nFor the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard\nform.\n6. 7. 8.\n9. 10."
	},
	{
		"chunk_id" : "00003586",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form.\n6. 7. 8.\n9. 10.\nFor the following exercises, rewrite the given equation in standard form, and then determine the vertex focus\nand directrix of the parabola.\n11. 12. 13.\n14. 15. 16.\nAccess for free at openstax.org\n12.3  The Parabola 1195\n17. 18. 19.\n20. 21. 22.\n23. 24. 25.\n26. 27. 28.\n29. 30.\nGraphical\nFor the following exercises, graph the parabola, labeling the focus and the directrix.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n43. 44."
	},
	{
		"chunk_id" : "00003587",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "34. 35. 36.\n37. 38. 39.\n40. 41. 42.\n43. 44.\nFor the following exercises, find the equation of the parabola given information about its graph.\n45. Vertex is directrix is 46. Vertex is directrix is 47. Vertex is directrix is\nfocus is focus is focus is\n48. Vertex is directrix 49. Vertex is 50. Vertex is directrix is\nis focus is focus is\ndirectrix is focus\nis\n1196 12  Analytic Geometry\nFor the following exercises, determine the equation for the parabola from its graph.\n51. 52. 53.\n54. 55.\nExtensions"
	},
	{
		"chunk_id" : "00003588",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "51. 52. 53.\n54. 55.\nExtensions\nFor the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.\n56. , Endpoints , 57. , Endpoints , 58. , Endpoints ,\n59. , Endpoints 60. , Endpoints\n, ,\nReal-World Applications\n61. The mirror in an 62. If we want to construct the 63. A satellite dish is shaped\nautomobile headlight has a mirror from the previous like a paraboloid of\nparabolic cross-section exercise such that the revolution. This means that"
	},
	{
		"chunk_id" : "00003589",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with the light bulb at the focus is located at it can be formed by\nfocus. On a schematic, the what should the rotating a parabola around\nequation of the parabola is equation of the parabola its axis of symmetry. The\ngiven as At what be? receiver is to be located at\ncoordinates should you the focus. If the dish is 12\nplace the light bulb? feet across at its opening\nand 4 feet deep at its\ncenter, where should the\nreceiver be placed?\nAccess for free at openstax.org\n12.4  Rotation of Axes 1197"
	},
	{
		"chunk_id" : "00003590",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.4  Rotation of Axes 1197\n64. Consider the satellite dish 65. The reflector in a 66. If the reflector in the\nfrom the previous exercise. searchlight is shaped like a searchlight from the\nIf the dish is 8 feet across paraboloid of revolution. A previous exercise has the\nat the opening and 2 feet light source is located 1 light source located 6\ndeep, where should we foot from the base along inches from the base along\nplace the receiver? the axis of symmetry. If the the axis of symmetry and"
	},
	{
		"chunk_id" : "00003591",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opening of the searchlight the opening is 4 feet, find\nis 3 feet across, find the the depth.\ndepth.\n67. An arch is in the shape of a 68. If the arch from the 69. An object is projected so as\nparabola. It has a span of previous exercise has a to follow a parabolic path\n100 feet and a maximum span of 160 feet and a given by\nheight of 20 feet. Find the maximum height of 40 where is the horizontal\nequation of the parabola, feet, find the equation of distance traveled in feet"
	},
	{
		"chunk_id" : "00003592",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and determine the height the parabola, and and is the height.\nof the arch 40 feet from the determine the distance Determine the maximum\ncenter. from the center at which height the object reaches.\nthe height is 20 feet.\n70. For the object from the\nprevious exercise, assume\nthe path followed is given\nby\nDetermine how far along\nthe horizontal the object\ntraveled to reach\nmaximum height.\n12.4 Rotation of Axes\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00003593",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nIdentify nondegenerate conic sections given their general form equations.\nUse rotation of axes formulas.\nWrite equations of rotated conics in standard form.\nIdentify conics without rotating axes.\nAs we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and\nextending infinitely far in opposite directions, which we also call acone. The way in which we slice the cone will"
	},
	{
		"chunk_id" : "00003594",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane\nperpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not\nperpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the\ndouble-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. SeeFigure 1.\n1198 12  Analytic Geometry"
	},
	{
		"chunk_id" : "00003595",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1198 12  Analytic Geometry\nFigure1 The nondegenerate conic sections\nEllipses, circles, hyperbolas, and parabolas are sometimes called thenondegenerate conic sections, in contrast to the\ndegenerate conic sections, which are shown inFigure 2. A degenerate conic results when a plane intersects the double\ncone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are\npossible: a point, a line, or two intersecting lines.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003596",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n12.4  Rotation of Axes 1199\nFigure2 Degenerate conic sections\nIdentifying Nondegenerate Conics in General Form\nIn previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections.\nIn this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is\nset equal to zero, and the terms and coefficients are given in a particular order, as shown below."
	},
	{
		"chunk_id" : "00003597",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "where and are not all zero. We can use the values of the coefficients to identify which type conic is represented\nby a given equation.\nYou may notice that the general form equation has an term that we have not seen in any of the standard form\nequations. As we will discuss later, the term rotates the conic whenever is not equal to zero.\nConic Sections Example\nellipse\ncircle\nhyperbola\nparabola\none line\nTable1\n1200 12  Analytic Geometry\nConic Sections Example\nintersecting lines\nparallel lines\na point"
	},
	{
		"chunk_id" : "00003598",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "intersecting lines\nparallel lines\na point\nno graph\nTable1\nGeneral Form of Conic Sections\nAconic sectionhas the general form\nwhere and are not all zero.\nTable 2summarizes the different conic sections where and and are nonzero real numbers. This indicates\nthat the conic has not been rotated.\nellipse\ncircle\nhyperbola where and are positive\nparabola\nTable2\n...\nHOW TO\nGiven the equation of a conic, identify the type of conic.\n1. Rewrite the equation in the general form,"
	},
	{
		"chunk_id" : "00003599",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Rewrite the equation in the general form,\n2. Identify the values of and from the general form.\na. If and are nonzero, have the same sign, and are not equal to each other, then the graph may be an\nellipse.\nb. If and are equal and nonzero and have the same sign, then the graph may be a circle.\nc. If and are nonzero and have opposite signs, then the graph may be a hyperbola.\nd. If either or is zero, then the graph may be a parabola."
	},
	{
		"chunk_id" : "00003600",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "IfB= 0, the conic section will have a vertical and/or horizontal axes. IfBdoes not equal 0, as shown below, the\nconic section is rotated. Notice the phrase may be in the definitions. That is because the equation may not\nrepresent a conic section at all, depending on the values ofA,B,C,D,E, andF. For example, the degenerate case\nof a circle or an ellipse is a point:\nwhenAandBhave the same sign.\nThe degenerate case of a hyperbola is two intersecting straight lines: when A and B have\nopposite signs."
	},
	{
		"chunk_id" : "00003601",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "opposite signs.\nOn the other hand, the equation, when A and B are positive does not represent a graph at\nAccess for free at openstax.org\n12.4  Rotation of Axes 1201\nall, since there are no real ordered pairs which satisfy it.\nEXAMPLE1\nIdentifying a Conic from Its General Form\nIdentify the graph of each of the following nondegenerate conic sections.\n  \n\nSolution\n Rewriting the general form, we have\nand so we observe that and have opposite signs. The graph of this equation is a hyperbola."
	},
	{
		"chunk_id" : "00003602",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Rewriting the general form, we have\nand We can determine that the equation is a parabola, since is zero.\n Rewriting the general form, we have\nand Because the graph of this equation is a circle.\n Rewriting the general form, we have\nand Because and the graph of this equation is an ellipse.\nTRY IT #1 Identify the graph of each of the following nondegenerate conic sections.\n \nFinding a New Representation of the Given Equation after Rotating through a Given Angle"
	},
	{
		"chunk_id" : "00003603",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Until now, we have looked at equations of conic sections without an term, which aligns the graphs with thex- and\ny-axes. When we add an term, we are rotating the conic about the origin. If thex- andy-axes are rotated through an\nangle, say then every point on the plane may be thought of as having two representations: on the Cartesian\nplane with the originalx-axis andy-axis, and on the new plane defined by the new, rotated axes, called the\nx'-axis andy'-axis. SeeFigure 3."
	},
	{
		"chunk_id" : "00003604",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "x'-axis andy'-axis. SeeFigure 3.\nFigure3 The graph of the rotated ellipse\n1202 12  Analytic Geometry\nWe will find the relationships between and on the Cartesian plane with and on the new rotated plane. See\nFigure 4.\nFigure4 The Cartesian plane withx- andy-axes and the resultingx andyaxes formed by a rotation by an angle\nThe original coordinatex- andy-axes have unit vectors and The rotated coordinate axes have unit vectors and"
	},
	{
		"chunk_id" : "00003605",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The angle is known as theangle of rotation. SeeFigure 5. We may write the new unit vectors in terms of the original\nones.\nFigure5 Relationship between the old and new coordinate planes.\nConsider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.\nBecause we have representations of and in terms of the new coordinate system.\nAccess for free at openstax.org\n12.4  Rotation of Axes 1203\nEquations of Rotation"
	},
	{
		"chunk_id" : "00003606",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Equations of Rotation\nIf a point on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are\nformed by rotating an angle from the positivex-axis, then the coordinates of the point with respect to the new axes\nare We can use the following equations of rotation to define the relationship between and\nand\n...\nHOW TO\nGiven the equation of a conic, find a new representation after rotating through an angle.\n1. Find and where and"
	},
	{
		"chunk_id" : "00003607",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Find and where and\n2. Substitute the expression for and into in the given equation, then simplify.\n3. Write the equations with and in standard form.\nEXAMPLE2\nFinding a New Representation of an Equation after Rotating through a Given Angle\nFind a new representation of the equation after rotating through an angle of\nSolution\nFind and where and\nBecause\nand\nSubstitute and into\nSimplify.\n1204 12  Analytic Geometry\nWrite the equations with and in the standard form."
	},
	{
		"chunk_id" : "00003608",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "This equation is an ellipse.Figure 6shows the graph.\nFigure6\nWriting Equations of Rotated Conics in Standard Form\nNow that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to\ntransform the equation of a conic given in the form into standard form by\nrotating the axes. To do so, we will rewrite the general form as an equation in the and coordinate system without\nthe term, by rotating the axes by a measure of that satisfies"
	},
	{
		"chunk_id" : "00003609",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We have learned already that any conic may be represented by the second degree equation\nwhere and are not all zero. However, if then we have an term that prevents us from rewriting the\nequation in standard form. To eliminate it, we can rotate the axes by an acute angle where\n If then is in the first quadrant, and is between\n If then is in the second quadrant, and is between\n If then\nAccess for free at openstax.org\n12.4  Rotation of Axes 1205\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003610",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.4  Rotation of Axes 1205\n...\nHOW TO\nGiven an equation for a conic in the system, rewrite the equation without the term in terms of and\nwhere the and axes are rotations of the standard axes by degrees.\n1. Find\n2. Find and\n3. Substitute and into and\n4. Substitute the expression for and into in the given equation, and then simplify.\n5. Write the equations with and in the standard form with respect to the rotated axes.\nEXAMPLE3\nRewriting an Equation with respect to thexandyaxes without thexyTerm"
	},
	{
		"chunk_id" : "00003611",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Rewrite the equation in the system without an term.\nSolution\nFirst, we find SeeFigure 7.\nFigure7\nSo the hypotenuse is\nNext, we find and\n1206 12  Analytic Geometry\nSubstitute the values of and into and\nand\nSubstitute the expressions for and into in the given equation, and then simplify.\nWrite the equations with and in the standard form with respect to the new coordinate system.\nFigure 8shows the graph of the ellipse.\nFigure8\nAccess for free at openstax.org\n12.4  Rotation of Axes 1207"
	},
	{
		"chunk_id" : "00003612",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.4  Rotation of Axes 1207\nTRY IT #2 Rewrite the in the system without the term.\nEXAMPLE4\nGraphing an Equation That Has NoxyTerms\nGraph the following equation relative to the system:\nSolution\nFirst, we find\nBecause we can draw a reference triangle as inFigure 9.\nFigure9\nThus, the hypotenuse is\nNext, we find and We will use half-angle identities.\n1208 12  Analytic Geometry\nNow we find and\nand\nNow we substitute and into\nFigure 10shows the graph of the hyperbola\nFigure10\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003613",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure10\nAccess for free at openstax.org\n12.4  Rotation of Axes 1209\nIdentifying Conics without Rotating Axes\nNow we have come full circle. How do we identify the type of conic described by an equation? What happens when the\naxes are rotated? Recall, the general form of a conic is\nIf we apply the rotation formulas to this equation we get the form\nIt may be shown that The expression does not vary after rotation, so we call the expression"
	},
	{
		"chunk_id" : "00003614",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "invariant.The discriminant, is invariant and remains unchanged after rotation. Because the discriminant\nremains unchanged, observing the discriminant enables us to identify the conic section.\nUsing the Discriminant to Identify a Conic\nIf the equation is transformed by rotating axes into the equation\nthen\nThe equation is an ellipse, a parabola, or a hyperbola, or a degenerate case of\none of these.\nIf the discriminant, is\n the conic section is an ellipse\n the conic section is a parabola"
	},
	{
		"chunk_id" : "00003615",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " the conic section is a parabola\n the conic section is a hyperbola\nEXAMPLE5\nIdentifying the Conic without Rotating Axes\nIdentify the conic for each of the following without rotating axes.\n \nSolution\n Lets begin by determining and\nNow, we find the discriminant.\nTherefore, represents an ellipse.\n Again, lets begin by determining and\nNow, we find the discriminant.\n1210 12  Analytic Geometry\nTherefore, represents an ellipse.\nTRY IT #3 Identify the conic for each of the following without rotating axes."
	},
	{
		"chunk_id" : "00003616",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nMEDIA\nAccess this online resource for additional instruction and practice with conic sections and rotation of axes.\nIntroduction to Conic Sections(http://openstax.org/l/introconic)\n12.4 SECTION EXERCISES\nVerbal\n1. What effect does the term have on the graph of 2. If the equation of a conic section is written in the\na conic section? form and\nwhat can we conclude?\n3. If the equation of a conic section is written in the 4. Given the equation what\nform and can we conclude if\nwhat can we conclude?"
	},
	{
		"chunk_id" : "00003617",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form and can we conclude if\nwhat can we conclude?\n5. For the equation\nthe value\nof that satisfies gives us what\ninformation?\nAlgebraic\nFor the following exercises, determine which conic section is represented based on the given equation.\n6. 7. 8.\n9. 10. 11.\n12. 13.\n14. 15.\n16. 17.\nAccess for free at openstax.org\n12.4  Rotation of Axes 1211\nFor the following exercises, find a new representation of the given equation after rotating through the given angle.\n18. 19. 20.\n21. 22."
	},
	{
		"chunk_id" : "00003618",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "18. 19. 20.\n21. 22.\nFor the following exercises, determine the angle that will eliminate the term and write the corresponding equation\nwithout the term.\n23. 24.\n25. 26.\n27. 28.\n29. 30.\nGraphical\nFor the following exercises, rotate through the given angle based on the given equation. Give the new equation and\ngraph the original and rotated equation.\n31. 32. 33.\n34. 35. 36.\n37. 38.\nFor the following exercises, graph the equation relative to the system in which the equation has no term.\n39. 40. 41."
	},
	{
		"chunk_id" : "00003619",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "39. 40. 41.\n42. 43. 44.\n45. 46.\n47. 48.\n49.\nFor the following exercises, determine the angle of rotation in order to eliminate the term. Then graph the new set of\naxes.\n50. 51.\n1212 12  Analytic Geometry\n52. 53.\n54. 55.\nFor the following exercises, determine the value of based on the given equation.\n56. Given 57. Given\nfind for the graph to be a parabola. find for the graph to be an ellipse.\n58. Given 59. Given\nfind for the graph to be a hyperbola. find for the graph to be a parabola.\n60. Given"
	},
	{
		"chunk_id" : "00003620",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "60. Given\nfind for the graph to be an ellipse.\n12.5 Conic Sections in Polar Coordinates\nLearning Objectives\nIn this section, you will:\nIdentify a conic in polar form.\nGraph the polar equations of conics.\nDefine conics in terms of a focus and a directrix.\nFigure1 Planets orbiting the sun follow elliptical paths. (credit: NASA Blueshift, Flickr)\nMost of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an"
	},
	{
		"chunk_id" : "00003621",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are\noften elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics\nof the planets orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the\ndistance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to\nrepresent these orbits."
	},
	{
		"chunk_id" : "00003622",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "represent these orbits.\nIn an elliptical orbit, theperiapsisis the point at which the two objects are closest, and theapoapsisis the point at which\nthey are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and\ndecrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These"
	},
	{
		"chunk_id" : "00003623",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial bodys\ngravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate\nsystem.\nIdentifying a Conic in Polar Form\nAny conic may be determined by three characteristics: a singlefocus, a fixed line called thedirectrix, and the ratio of the\nAccess for free at openstax.org\n12.5  Conic Sections in Polar Coordinates 1213"
	},
	{
		"chunk_id" : "00003624",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.5  Conic Sections in Polar Coordinates 1213\ndistances of each to a point on the graph. Consider theparabola shown inFigure 2.\nFigure2\nInThe Parabola, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this\nsection, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus at\nthe pole, and a line, the directrix, which is perpendicular to the polar axis."
	},
	{
		"chunk_id" : "00003625",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "If is a fixed point, the focus, and is a fixed line, the directrix, then we can let be a fixed positive number, called the\neccentricity, which we can define as the ratio of the distances from a point on the graph to the focus and the point on\nthe graph to the directrix. Then the set of all points such that is a conic. In other words, we can define a conic\nas the set of all points with the property that the ratio of the distance from to to the distance from to is equal\nto the constant"
	},
	{
		"chunk_id" : "00003626",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to the constant\nFor a conic with eccentricity\n if the conic is an ellipse\n if the conic is a parabola\n if the conic is an hyperbola\nWith this definition, we may now define a conic in terms of the directrix, the eccentricity and the angle Thus,\neach conic may be written as apolar equation, an equation written in terms of and\nThe Polar Equation for a Conic\nFor a conic with a focus at the origin, if the directrix is where is a positive real number, and theeccentricity"
	},
	{
		"chunk_id" : "00003627",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is a positive real number the conic has apolar equation\nFor a conic with a focus at the origin, if the directrix is where is a positive real number, and the eccentricity\nis a positive real number the conic has a polar equation\n...\nHOW TO\nGiven the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.\n1. Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the\nequation in standard form."
	},
	{
		"chunk_id" : "00003628",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "equation in standard form.\n2. Identify the eccentricity as the coefficient of the trigonometric function in the denominator.\n3. Compare with 1 to determine the shape of the conic.\n4. Determine the directrix as if cosine is in the denominator and if sine is in the denominator. Set\n1214 12  Analytic Geometry\nequal to the numerator in standard form to solve for or\nEXAMPLE1\nIdentifying a Conic Given the Polar Form"
	},
	{
		"chunk_id" : "00003629",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nIdentifying a Conic Given the Polar Form\nFor each of the following equations, identify the conic with focus at the origin, thedirectrix, and theeccentricity.\na.\nb.\nc.\nSolution\nFor each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the\ndenominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the\nreciprocal of the constant of the original equation, where is that constant."
	},
	{
		"chunk_id" : "00003630",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a. Multiply the numerator and denominator by\nBecause is in the denominator, the directrix is Comparing to standard form, note that Therefore,\nfrom the numerator,\nSince the conic is anellipse. The eccentricity is and the directrix is\nb. Multiply the numerator and denominator by\nBecause is in the denominator, the directrix is Comparing to standard form, Therefore, from\nthe numerator,\nSince the conic is ahyperbola. The eccentricity is and the directrix is\nc. Multiply the numerator and denominator by"
	},
	{
		"chunk_id" : "00003631",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "c. Multiply the numerator and denominator by\nAccess for free at openstax.org\n12.5  Conic Sections in Polar Coordinates 1215\nBecause sine is in the denominator, the directrix is Comparing to standard form, Therefore, from the\nnumerator,\nBecause the conic is aparabola. The eccentricity is and the directrix is\nTRY IT #1 Identify the conic with focus at the origin, the directrix, and the eccentricity for\nGraphing the Polar Equations of Conics"
	},
	{
		"chunk_id" : "00003632",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing the Polar Equations of Conics\nWhen graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in\npolar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then\ndetermine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the"
	},
	{
		"chunk_id" : "00003633",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables\nus to determine and, therefore, the shape of the curve. The next step is to substitute values for and solve for to plot\na few key points. Setting equal to and provides the vertices so we can create a rough sketch of the graph.\nEXAMPLE2\nGraphing a Parabola in Polar Form\nGraph\nSolution"
	},
	{
		"chunk_id" : "00003634",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Graphing a Parabola in Polar Form\nGraph\nSolution\nFirst, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which\nis\nBecause we will graph aparabolawith a focus at the origin. The function has a and there is an addition\nsign in the denominator, so the directrix is\n1216 12  Analytic Geometry\nThe directrix is\nPlotting a few key points as inTable 1will enable us to see the vertices. SeeFigure 3.\nA B C D\nundefined\nTable1\nFigure3\nAnalysis"
	},
	{
		"chunk_id" : "00003635",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A B C D\nundefined\nTable1\nFigure3\nAnalysis\nWe can check our result with a graphing utility. SeeFigure 4.\nFigure4\nEXAMPLE3\nGraphing a Hyperbola in Polar Form\nGraph\nAccess for free at openstax.org\n12.5  Conic Sections in Polar Coordinates 1217\nSolution\nFirst, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which\nis\nBecause so we will graph ahyperbolawith a focus at the origin. The function has a term and there is"
	},
	{
		"chunk_id" : "00003636",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a subtraction sign in the denominator, so the directrix is\nThe directrix is\nPlotting a few key points as inTable 2will enable us to see the vertices. SeeFigure 5.\nA B C D\nTable2\nFigure5\n1218 12  Analytic Geometry\nEXAMPLE4\nGraphing an Ellipse in Polar Form\nGraph\nSolution\nFirst, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which\nis\nBecause so we will graph anellipsewith afocusat the origin. The function has a and there is a"
	},
	{
		"chunk_id" : "00003637",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "subtraction sign in the denominator, so thedirectrixis\nThe directrix is\nPlotting a few key points as inTable 3will enable us to see the vertices. SeeFigure 6.\nA B C D\nTable3\nFigure6\nAccess for free at openstax.org\n12.5  Conic Sections in Polar Coordinates 1219\nAnalysis\nWe can check our result using a graphing utility. SeeFigure 7.\nFigure7 graphed on a viewing window of by and\nTRY IT #2 Graph\nDefining Conics in Terms of a Focus and a Directrix"
	},
	{
		"chunk_id" : "00003638",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we\nwill use information about the origin, eccentricity, and directrix to determine the polar equation.\n...\nHOW TO\nGiven the focus, eccentricity, and directrix of a conic, determine the polar equation.\n1. Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of we use the general"
	},
	{
		"chunk_id" : "00003639",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "polar form in terms of sine. If the directrix is given in terms of we use the general polar form in terms of\ncosine.\n2. Determine the sign in the denominator. If use subtraction. If use addition.\n3. Write the coefficient of the trigonometric function as the given eccentricity.\n4. Write the absolute value of in the numerator, and simplify the equation.\nEXAMPLE5\nFinding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix"
	},
	{
		"chunk_id" : "00003640",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the polar form of theconicgiven afocusat the origin, anddirectrix\nSolution\nThe directrix is so we know the trigonometric function in the denominator is sine.\nBecause so we know there is a subtraction sign in the denominator. We use the standard form of\nand and\n1220 12  Analytic Geometry\nTherefore,\nEXAMPLE6\nFinding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix\nFind thepolar form of a conicgiven afocusat the origin, anddirectrix\nSolution"
	},
	{
		"chunk_id" : "00003641",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nBecause the directrix is we know the function in the denominator is cosine. Because so we know\nthere is an addition sign in the denominator. We use the standard form of\nand and\nTherefore,\nTRY IT #3 Find the polar form of the conic given a focus at the origin, and directrix\nEXAMPLE7\nConverting a Conic in Polar Form to Rectangular Form\nConvert the conic to rectangular form.\nSolution\nWe will rearrange the formula to use the identities\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003642",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n12.5  Conic Sections in Polar Coordinates 1221\nTRY IT #4 Convert the conic to rectangular form.\nMEDIA\nAccess these online resources for additional instruction and practice with conics in polar coordinates.\nPolar Equations of Conic Sections(http://openstax.org/l/determineconic)\nGraphing Polar Equations of Conics - 1(http://openstax.org/l/graphconic1)\nGraphing Polar Equations of Conics - 2(http://openstax.org/l/graphconic2)\n12.5 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00003643",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.5 SECTION EXERCISES\nVerbal\n1. Explain how eccentricity 2. If a conic section is written 3. If a conic section is written\ndetermines which conic as a polar equation, what as a polar equation, and the\nsection is given. must be true of the denominator involves\ndenominator? what conclusion can be\ndrawn about the directrix?\n4. If the directrix of a conic 5. What do we know about the\nsection is perpendicular to focus/foci of a conic section\nthe polar axis, what do we if it is written as a polar"
	},
	{
		"chunk_id" : "00003644",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "know about the equation of equation?\nthe graph?\nAlgebraic\nFor the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15. 16. 17.\n1222 12  Analytic Geometry\nFor the following exercises, convert the polar equation of a conic section to a rectangular equation.\n18. 19. 20.\n21. 22. 23.\n24. 25. 26.\n27. 28. 29.\n30."
	},
	{
		"chunk_id" : "00003645",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "21. 22. 23.\n24. 25. 26.\n27. 28. 29.\n30.\nFor the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is\nan ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.\n31. 32. 33.\n34. 35. 36.\n37. 38. 39.\n40. 41. 42.\nFor the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and\ndirectrix.\n43. Directrix: 44. Directrix: 45. Directrix:"
	},
	{
		"chunk_id" : "00003646",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "43. Directrix: 44. Directrix: 45. Directrix:\n46. Directrix: 47. Directrix: 48. Directrix:\n49. Directrix: 50. Directrix: 51. Directrix:\n52. Directrix: 53. Directrix: 54. Directrix:\n55. Directrix:\nExtensions\nRecall fromRotation of Axesthat equations of conics with an term have rotated graphs. For the following exercises,\nexpress each equation in polar form with as a function of\n56. 57. 58.\n59. 60.\nAccess for free at openstax.org\n12  Chapter Review 1223\nChapter Review\nKey Terms"
	},
	{
		"chunk_id" : "00003647",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12  Chapter Review 1223\nChapter Review\nKey Terms\nangle of rotation an acute angle formed by a set of axes rotated from the Cartesian plane where, if then\nis between if then is between and if then\ncenter of a hyperbola the midpoint of both the transverse and conjugate axes of a hyperbola\ncenter of an ellipse the midpoint of both the major and minor axes\nconic section any shape resulting from the intersection of a right circular cone with a plane"
	},
	{
		"chunk_id" : "00003648",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "conjugate axis the axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its\nendpoints\ndegenerate conic sections any of the possible shapes formed when a plane intersects a double cone through the\napex. Types of degenerate conic sections include a point, a line, and intersecting lines.\ndirectrix a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between"
	},
	{
		"chunk_id" : "00003649",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the points on the conic and the focus to the distance to the directrix is constant\neccentricity the ratio of the distances from a point on the graph to the focus and to the directrix represented\nby where is a positive real number\nellipse the set of all points in a plane such that the sum of their distances from two fixed points is a constant\nfoci plural of focus\nfocus (of a parabola) a fixed point in the interior of a parabola that lies on the axis of symmetry"
	},
	{
		"chunk_id" : "00003650",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "focus (of an ellipse) one of the two fixed points on the major axis of an ellipse such that the sum of the distances from\nthese points to any point on the ellipse is a constant\nhyperbola the set of all points in a plane such that the difference of the distances between and the foci is a\npositive constant\nlatus rectum the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on\nthe parabola\nmajor axis the longer of the two axes of an ellipse"
	},
	{
		"chunk_id" : "00003651",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "minor axis the shorter of the two axes of an ellipse\nnondegenerate conic section a shape formed by the intersection of a plane with a double right cone such that the\nplane does not pass through the apex; nondegenerate conics include circles, ellipses, hyperbolas, and parabolas\nparabola the set of all points in a plane that are the same distance from a fixed line, called the directrix, and a\nfixed point (the focus) not on the directrix\npolar equation an equation of a curve in polar coordinates and"
	},
	{
		"chunk_id" : "00003652",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "transverse axis the axis of a hyperbola that includes the foci and has the vertices as its endpoints\nKey Equations\nHorizontal ellipse, center at origin\nVertical ellipse, center at origin\nHorizontal ellipse, center\nVertical ellipse, center\nHyperbola, center at origin, transverse axis onx-axis\nHyperbola, center at origin, transverse axis ony-axis\nHyperbola, center at transverse axis parallel tox-axis\n1224 12  Chapter Review\nHyperbola, center at transverse axis parallel toy-axis"
	},
	{
		"chunk_id" : "00003653",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Parabola, vertex at origin, axis of symmetry onx-axis\nParabola, vertex at origin, axis of symmetry ony-axis\nParabola, vertex at axis of symmetry onx-axis\nParabola, vertex at axis of symmetry ony-axis\nGeneral Form equation of a conic section\nRotation of a conic section\nAngle of rotation\nKey Concepts\n12.1The Ellipse\n An ellipse is the set of all points in a plane such that the sum of their distances from two fixed points is a\nconstant. Each fixed point is called a focus (plural: foci)."
	},
	{
		"chunk_id" : "00003654",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard\nform. SeeExample 1andExample 2.\n When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-\nvertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. SeeExample 3\nandExample 4."
	},
	{
		"chunk_id" : "00003655",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "andExample 4.\n When given the equation for an ellipse centered at some point other than the origin, we can identify its key features\nand graph the ellipse. SeeExample 5andExample 6.\n Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key\nfeatures, such as lengths of axes and distance between foci. SeeExample 7.\n12.2The Hyperbola\n A hyperbola is the set of all points in a plane such that the difference of the distances between and the"
	},
	{
		"chunk_id" : "00003656",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "foci is a positive constant.\n The standard form of a hyperbola can be used to locate its vertices and foci. SeeExample 1.\n When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in\nstandard form. SeeExample 2andExample 3.\n When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and\npositions of the transverse and conjugate axes in order to graph the hyperbola. SeeExample 4andExample 5."
	},
	{
		"chunk_id" : "00003657",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the\ndimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. SeeExample 6.\n12.3The Parabola\n A parabola is the set of all points in a plane that are the same distance from a fixed line, called the directrix,\nand a fixed point (the focus) not on the directrix."
	},
	{
		"chunk_id" : "00003658",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The standard form of a parabola with vertex and thex-axis as its axis of symmetry can be used to graph the\nparabola. If the parabola opens right. If the parabola opens left. SeeExample 1.\n The standard form of a parabola with vertex and they-axis as its axis of symmetry can be used to graph the\nparabola. If the parabola opens up. If the parabola opens down. SeeExample 2.\n When given the focus and directrix of a parabola, we can write its equation in standard form. SeeExample 3."
	},
	{
		"chunk_id" : "00003659",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The standard form of a parabola with vertex and axis of symmetry parallel to thex-axis can be used to graph\nAccess for free at openstax.org\n12  Exercises 1225\nthe parabola. If the parabola opens right. If the parabola opens left. SeeExample 4.\n The standard form of a parabola with vertex and axis of symmetry parallel to they-axis can be used to graph\nthe parabola. If the parabola opens up. If the parabola opens down. SeeExample 5."
	},
	{
		"chunk_id" : "00003660",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter\nand focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. SeeExample 6.\n12.4Rotation of Axes\n Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail.\nThey include an ellipse, a circle, a hyperbola, and a parabola.\n A nondegenerate conic section has the general form where and are"
	},
	{
		"chunk_id" : "00003661",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "not all zero. The values of and determine the type of conic. SeeExample 1.\n Equations of conic sections with an term have been rotated about the origin. SeeExample 2.\n The general form can be transformed into an equation in the and coordinate system without the term.\nSeeExample 3andExample 4.\n An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant,\nobserving it enables us to identify the conic section. SeeExample 5."
	},
	{
		"chunk_id" : "00003662",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12.5Conic Sections in Polar Coordinates\n Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define\na conic in terms of a fixed point, the focus at the pole, and a line, the directrix, which is perpendicular to the\npolar axis.\n A conic is the set of all points where eccentricity is a positive real number. Each conic may be written in\nterms of its polar equation. SeeExample 1."
	},
	{
		"chunk_id" : "00003663",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms of its polar equation. SeeExample 1.\n The polar equations of conics can be graphed. SeeExample 2,Example 3, andExample 4.\n Conics can be defined in terms of a focus, a directrix, and eccentricity. SeeExample 5andExample 6.\n We can use the identities and to convert the equation for a conic from\npolar to rectangular form. SeeExample 7.\nExercises\nReview Exercises\nThe Ellipse\nFor the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci."
	},
	{
		"chunk_id" : "00003664",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. 2. 3.\n4.\nFor the following exercises, graph the ellipse, noting center, vertices, and foci.\n5. 6. 7.\n8.\n1226 12  Exercises\nFor the following exercises, use the given information to find the equation for the ellipse.\n9. Center at focus at 10. Center at vertex at 11. A whispering gallery is to\nvertex at focus at be constructed such that\nthe foci are located 35 feet\nfrom the center. If the\nlength of the gallery is to\nbe 100 feet, what should\nthe height of the ceiling\nbe?\nThe Hyperbola"
	},
	{
		"chunk_id" : "00003665",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the height of the ceiling\nbe?\nThe Hyperbola\nFor the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.\n12. 13. 14.\n15.\nFor the following exercises, graph the hyperbola, labeling vertices and foci.\n16. 17. 18.\n19.\nFor the following exercises, find the equation of the hyperbola.\n20. Center at vertex at 21. Foci at and\nfocus at vertex at\nThe Parabola"
	},
	{
		"chunk_id" : "00003666",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "focus at vertex at\nThe Parabola\nFor the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and\ndirectrix.\n22. 23. 24.\n25.\nFor the following exercises, graph the parabola, labeling vertex, focus, and directrix.\n26. 27. 28.\n29.\nAccess for free at openstax.org\n12  Exercises 1227\nFor the following exercises, write the equation of the parabola using the given information.\n30. Focus at directrix 31. Focus at directrix is 32. A cable TV receiving dish is"
	},
	{
		"chunk_id" : "00003667",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the shape of a paraboloid\nof revolution. Find the\nlocation of the receiver,\nwhich is placed at the\nfocus, if the dish is 5 feet\nacross at its opening and\n1.5 feet deep.\nRotation of Axes\nFor the following exercises, determine which of the conic sections is represented.\n33. 34.\n35.\nFor the following exercises, determine the angle that will eliminate the term, and write the corresponding equation\nwithout the term.\n36. 37."
	},
	{
		"chunk_id" : "00003668",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "without the term.\n36. 37.\nFor the following exercises, graph the equation relative to the system in which the equation has no term.\n38. 39.\n40.\nConic Sections in Polar Coordinates\nFor the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and\ndirectrix.\n41. 42. 43.\n44.\nFor the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If"
	},
	{
		"chunk_id" : "00003669",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "it is an ellipse or a hyperbola, label the vertices and foci.\n45. 46. 47.\n48.\n1228 12  Exercises\nFor the following exercises, given information about the graph of a conic with focus at the origin, find the equation in\npolar form.\n49. Directrix is and 50. Directrix is and\neccentricity eccentricity\nPractice Test\nFor the following exercises, write the equation in standard form and state the center, vertices, and foci.\n1. 2."
	},
	{
		"chunk_id" : "00003670",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. 2.\nFor the following exercises, sketch the graph, identifying the center, vertices, and foci.\n3. 4. 5. Write the standard form\nequation of an ellipse with a\ncenter at vertex at\nand focus at\n6. A whispering gallery is to be\nconstructed with a length of\n150 feet. If the foci are to be\nlocated 20 feet away from\nthe wall, how high should\nthe ceiling be?\nFor the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and\nasymptotes.\n7. 8."
	},
	{
		"chunk_id" : "00003671",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "asymptotes.\n7. 8.\nFor the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the\nasymptotes.\n9. 10. 11. Write the standard form\nequation of a hyperbola\nwith foci at and\nand a vertex at\nFor the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation\nof the directrix.\n12. 13.\nFor the following exercises, graph the parabola, labeling the vertex, focus, and directrix."
	},
	{
		"chunk_id" : "00003672",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "14. 15. 16. Write the equation of a\nparabola with a focus at\nand directrix\nAccess for free at openstax.org\n12  Exercises 1229\n17. A searchlight is shaped like\na paraboloid of revolution.\nIf the light source is\nlocated 1.5 feet from the\nbase along the axis of\nsymmetry, and the depth\nof the searchlight is 3 feet,\nwhat should the width of\nthe opening be?\nFor the following exercises, determine which conic section is represented by the given equation, and then determine the\nangle that will eliminate the term."
	},
	{
		"chunk_id" : "00003673",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "angle that will eliminate the term.\n18. 19.\nFor the following exercises, rewrite in the system without the term, and graph the rotated graph.\n20. 21.\nFor the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.\n22. 23.\nFor the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an\nellipse or a hyperbola, label vertices and foci.\n24. 25. 26. Find a polar equation of"
	},
	{
		"chunk_id" : "00003674",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "24. 25. 26. Find a polar equation of\nthe conic with focus at the\norigin, eccentricity of\nand directrix:\n1230 12  Exercises\nAccess for free at openstax.org\n13  Introduction 1231\n13 SEQUENCES, PROBABILITY, AND COUNTING THEORY\n(credit: Robert Couse-Baker, Flickr.)\nChapter Outline\n13.1Sequences and Their Notations\n13.2Arithmetic Sequences\n13.3Geometric Sequences\n13.4Series and Their Notations\n13.5Counting Principles\n13.6Binomial Theorem\n13.7Probability"
	},
	{
		"chunk_id" : "00003675",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.6Binomial Theorem\n13.7Probability\nIntroduction to Sequences, Probability and Counting Theory\nA lottery winner has some big decisions to make regarding what to do with the winnings. Buy a new home? A luxury\nconvertible? A cruise around the world?\nThe likelihood of winning the lottery is slim, but we all love to fantasize about what we could buy with the winnings. One\nof the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series"
	},
	{
		"chunk_id" : "00003676",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of regular payments, called an annuity, over an extended period of time.\nThis decision is often based on many factors, such as tax implications, interest rates, and investment strategies. There\nare also personal reasons to consider when making the choice, and one can make many arguments for either decision.\nHowever, most lottery winners opt for the lump sum.\nIn this chapter, we will explore the mathematics behind situations such as these. We will take an in-depth look at"
	},
	{
		"chunk_id" : "00003677",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "annuities. We will also look at the branch of mathematics that would allow us to calculate the number of ways to choose\nlottery numbers and the probability of winning.\n13.1 Sequences and Their Notations\nLearning Objectives\nIn this section, you will:\nWrite the terms of a sequence defined by an explicit formula.\nWrite the terms of a sequence defined by a recursive formula.\nUse factorial notation."
	},
	{
		"chunk_id" : "00003678",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use factorial notation.\nA video game company launches an exciting new advertising campaign. They predict the number of online visits to their\nwebsite, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the\n1232 13  Sequences, Probability, and Counting Theory\nthird day, and so on. SeeTable 1.\nDay 1 2 3 4 5 \nHits 2 4 8 16 32 \nTable1"
	},
	{
		"chunk_id" : "00003679",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Day 1 2 3 4 5 \nHits 2 4 8 16 32 \nTable1\nIf their model continues, how many hits will there be at the end of the month? To answer this question, well first need to\nknow how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of\nordered lists.\nWriting the Terms of a Sequence Defined by an Explicit Formula\nOne way to describe an ordered list of numbers is as asequence. A sequence is a function whose domain is a subset of"
	},
	{
		"chunk_id" : "00003680",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the counting numbers. The sequence established by the number of hits on the website is\nTheellipsis() indicates that the sequence continues indefinitely. Each number in the sequence is called aterm. The\nfirst five terms of this sequence are 2, 4, 8, 16, and 32.\nListing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at\nthe end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is"
	},
	{
		"chunk_id" : "00003681",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "by writing a formula to define the sequence.\nOne type of formula is anexplicit formula, which defines the terms of a sequence using their position in the sequence.\nExplicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We\ncan use the formula to find thenth term of the sequence, where is any positive number. In our example, each number\nin the sequence is double the previous number, so we can use powers of 2 to write a formula for the term."
	},
	{
		"chunk_id" : "00003682",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first term of the sequence is the second term is the third term is and so on. The term of\nthe sequence can be found by raising 2 to the power. An explicit formula for a sequence is named by a lower case\nletter with the subscript The explicit formula for this sequence is\nNow that we have a formula for the term of the sequence, we can answer the question posed at the beginning of this\nsection. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the"
	},
	{
		"chunk_id" : "00003683",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number of hits on the last day of the month, we need to find the 31stterm of the sequence. We will substitute 31 for in\nthe formula.\nIf the doubling trend continues, the company will get hits on the last day of the month. That is over 2.1\nbillion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition\ninto account. It does, however, give the company a starting point from which to consider business decisions."
	},
	{
		"chunk_id" : "00003684",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Another way to represent the sequence is by using a table. The first five terms of the sequence and the term of the\nsequence are shown inTable 2.\n1 2 3 4 5\nterm of the sequence, 2 4 8 16 32\nTable2\nGraphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in\nFigure 1that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function.\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1233"
	},
	{
		"chunk_id" : "00003685",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.1  Sequences and Their Notations 1233\nFigure1\nLastly, we can write this particular sequence as\nA sequence that continues indefinitely is called aninfinite sequence. The domain of an infinite sequence is the set of\ncounting numbers. If we consider only the first 10 terms of the sequence, we could write\nThis sequence is called afinite sequencebecause it does not continue indefinitely.\nSequence\nAsequenceis a function whose domain is the set of positive integers. Afinite sequenceis a sequence whose domain"
	},
	{
		"chunk_id" : "00003686",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "consists of only the first positive integers. The numbers in a sequence are calledterms. The variable with a\nnumber subscript is used to represent the terms in a sequence and to indicate the position of the term in the\nsequence.\nWe call the first term of the sequence, the second term of the sequence, the third term of the sequence, and\nso on. The term is called thenth term of the sequence, or the general term of the sequence. Anexplicit formula"
	},
	{
		"chunk_id" : "00003687",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "defines the term of a sequence using the position of the term. A sequence that continues indefinitely is an\ninfinite sequence.\nQ&A Does a sequence always have to begin with\nNo. In certain problems, it may be useful to define the initial term as instead of In these problems,\nthe domain of the function includes 0.\n...\nHOW TO\nGiven an explicit formula, write the first terms of a sequence.\n1. Substitute each value of into the formula. Begin with to find the first term,\n2. To find the second term, use"
	},
	{
		"chunk_id" : "00003688",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. To find the second term, use\n3. Continue in the same manner until you have identified all terms.\n1234 13  Sequences, Probability, and Counting Theory\nEXAMPLE1\nWriting the Terms of a Sequence Defined by an Explicit Formula\nWrite the first five terms of the sequence defined by the explicit formula\nSolution\nSubstitute into the formula. Repeat with values 2 through 5 for\nThe first five terms are\nAnalysis"
	},
	{
		"chunk_id" : "00003689",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first five terms are\nAnalysis\nThe sequence values can be listed in a table. A table, such asTable 3, is a convenient way to input the function into a\ngraphing utility.\n1 2 3 4 5\n5 2 1 4 7\nTable3\nA graph can be made from this table of values. From the graph inFigure 2, we can see that this sequence represents a\nlinear function, but notice the graph is not continuous because the domain is over the positive integers only.\nFigure2"
	},
	{
		"chunk_id" : "00003690",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure2\nTRY IT #1 Write the first five terms of the sequence defined by theexplicit formula\nInvestigating Alternating Sequences\nSometimes sequences have terms that are alternate. In fact, the terms may actually alternate in sign. The steps to\nfinding terms of the sequence are the same as if the signs did not alternate. However, the resulting terms will not show\nincrease or decrease as increases. Lets take a look at the following sequence."
	},
	{
		"chunk_id" : "00003691",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Notice the first term is greater than the second term, the second term is less than the third term, and the third term is\ngreater than the fourth term. This trend continues forever. Do not rearrange the terms in numerical order to interpret\nthe sequence.\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1235\n...\nHOW TO\nGiven an explicit formula with alternating terms, write the first terms of a sequence."
	},
	{
		"chunk_id" : "00003692",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Substitute each value of into the formula. Begin with to find the first term, The sign of the term is\ngiven by the in the explicit formula.\n2. To find the second term, use\n3. Continue in the same manner until you have identified all terms.\nEXAMPLE2\nWriting the Terms of an Alternating Sequence Defined by an Explicit Formula\nWrite the first five terms of the sequence.\nSolution\nSubstitute and so on in the formula.\nThe first five terms are\nAnalysis"
	},
	{
		"chunk_id" : "00003693",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first five terms are\nAnalysis\nThe graph of this function, shown inFigure 3, looks different from the ones we have seen previously in this section\nbecause the terms of the sequence alternate between positive and negative values.\nFigure3\nQ&A InExample 2, does the (1) to the power of account for the oscillations of signs?\n1236 13  Sequences, Probability, and Counting Theory\nYes, the power might be and so on, but any odd powers will result in a negative term, and"
	},
	{
		"chunk_id" : "00003694",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "any even power will result in a positive term.\nTRY IT #2 Write the first five terms of the sequence.\nInvestigating Piecewise Explicit Formulas\nWeve learned that sequences are functions whose domain is over the positive integers. This is true for other types of\nfunctions, including somepiecewise functions. Recall that a piecewise function is a function defined by multiple\nsubsections. A different formula might represent each individual subsection.\n...\nHOW TO"
	},
	{
		"chunk_id" : "00003695",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "...\nHOW TO\nGiven an explicit formula for a piecewise function, write the first terms of a sequence\n1. Identify the formula to which applies.\n2. To find the first term, use in the appropriate formula.\n3. Identify the formula to which applies.\n4. To find the second term, use in the appropriate formula.\n5. Continue in the same manner until you have identified all terms.\nEXAMPLE3\nWriting the Terms of a Sequence Defined by a Piecewise Explicit Formula\nWrite the first six terms of the sequence.\nSolution"
	},
	{
		"chunk_id" : "00003696",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSubstitute and so on in the appropriate formula. Use when is not a multiple of 3. Use when is a\nmultiple of 3.\nThe first six terms are\nAnalysis\nEvery third point on the graph shown inFigure 4stands out from the two nearby points. This occurs because the\nsequence was defined by a piecewise function.\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1237\nFigure4\nTRY IT #3 Write the first six terms of the sequence.\nFinding an Explicit Formula"
	},
	{
		"chunk_id" : "00003697",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding an Explicit Formula\nThus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the\nexplicit formula for the term of a sequence is not given. Instead, we are given several terms from the sequence.\nWhen this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to\nfinding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating"
	},
	{
		"chunk_id" : "00003698",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms, formulas for numerators, formulas for denominators, exponents, or bases.\n...\nHOW TO\nGiven the first few terms of a sequence, find an explicit formula for the sequence.\n1. Look for a pattern among the terms.\n2. If the terms are fractions, look for a separate pattern among the numerators and denominators.\n3. Look for a pattern among the signs of the terms.\n4. Write a formula for in terms of Test your formula for and\nEXAMPLE4\nWriting an Explicit Formula for thenth Term of a Sequence"
	},
	{
		"chunk_id" : "00003699",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write an explicit formula for the term of each sequence.\n  \nSolution\nLook for the pattern in each sequence.\n The terms alternate between positive and negative. We can use to make the terms alternate. The\nnumerator can be represented by The denominator can be represented by\n1238 13  Sequences, Probability, and Counting Theory\n\nThe terms are all negative.\nSo we know that the fraction is negative, the numerator is 2, and the denominator can be represented by\n"
	},
	{
		"chunk_id" : "00003700",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "\nThe terms are powers of For the first term is so the exponent must be\nTRY IT #4 Write an explicit formula for the term of the sequence.\nTRY IT #5 Write an explicit formula for the term of the sequence.\nTRY IT #6 Write an explicit formula for the term of the sequence.\nWriting the Terms of a Sequence Defined by a Recursive Formula\nSequences occur naturally in the growth patterns of nautilus shells, pinecones, tree branches, and many other natural"
	},
	{
		"chunk_id" : "00003701",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "structures. We may see the sequence in the leaf or branch arrangement, the number of petals of a flower, or the pattern\nof the chambers in a nautilus shell. Their growth follows the Fibonacci sequence, a famous sequence in which each term\ncan be found by adding the preceding two terms. The numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34,. Other\nexamples from the natural world that exhibit the Fibonacci sequence are the Calla Lily, which has just one petal, the"
	},
	{
		"chunk_id" : "00003702",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Black-Eyed Susan with 13 petals, and different varieties of daisies that may have 21 or 34 petals.\nEach term of the Fibonacci sequence depends on the terms that come before it. The Fibonacci sequence cannot easily be\nwritten using an explicit formula. Instead, we describe the sequence using arecursive formula, a formula that defines\nthe terms of a sequence using previous terms.\nA recursive formula always has two parts: the value of an initial term (or terms), and an equation defining in terms of"
	},
	{
		"chunk_id" : "00003703",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "preceding terms. For example, suppose we know the following:\nWe can find the subsequent terms of the sequence using the first term.\nSo the first four terms of the sequence are .\nThe recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1239\nof the preceding two terms."
	},
	{
		"chunk_id" : "00003704",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the preceding two terms.\nTo find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told\npreviously that the eighth and ninth terms are 21 and 34, so\nRecursive Formula\nArecursive formulais a formula that defines each term of a sequence using preceding term(s). Recursive formulas\nmust always state the initial term, or terms, of the sequence.\nQ&A Must the first two terms always be given in a recursive formula?"
	},
	{
		"chunk_id" : "00003705",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "No. The Fibonacci sequence defines each term using the two preceding terms, but many recursive\nformulas define each term using only one preceding term. These sequences need only the first term to be\ndefined.\n...\nHOW TO\nGiven a recursive formula with only the first term provided, write the first terms of a sequence.\n1. Identify the initial term, which is given as part of the formula. This is the first term.\n2. To find the second term, substitute the initial term into the formula for Solve."
	},
	{
		"chunk_id" : "00003706",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. To find the third term, substitute the second term into the formula. Solve.\n4. Repeat until you have solved for the term.\nEXAMPLE5\nWriting the Terms of a Sequence Defined by a Recursive Formula\nWrite the first five terms of the sequence defined by the recursive formula.\nSolution\nThe first term is given in the formula. For each subsequent term, we replace with the value of the preceding term.\nThe first five terms are SeeFigure 5.\n1240 13  Sequences, Probability, and Counting Theory\nFigure5"
	},
	{
		"chunk_id" : "00003707",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure5\nTRY IT #7 Write the first five terms of the sequence defined by the recursive formula.\n...\nHOW TO\nGiven a recursive formula with two initial terms, write the first terms of a sequence.\n1. Identify the initial term, which is given as part of the formula.\n2. Identify the second term, which is given as part of the formula.\n3. To find the third term, substitute the initial term and the second term into the formula. Evaluate.\n4. Repeat until you have evaluated the term.\nEXAMPLE6"
	},
	{
		"chunk_id" : "00003708",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE6\nWriting the Terms of a Sequence Defined by a Recursive Formula\nWrite the first six terms of the sequence defined by the recursive formula.\nSolution\nThe first two terms are given. For each subsequent term, we replace and with the values of the two preceding\nterms.\nThe first six terms are SeeFigure 6.\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1241\nFigure6\nTRY IT #8 Write the first 8 terms of the sequence defined by the recursive formula.\nUsing Factorial Notation"
	},
	{
		"chunk_id" : "00003709",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Factorial Notation\nThe formulas for some sequences include products of consecutive positive integers. factorial, written as is the\nproduct of the positive integers from 1 to For example,\nAn example of formula containing afactorialis The sixth term of the sequence can be found by\nsubstituting 6 for\nThe factorial of any whole number is We can therefore also think of as\nn Factorial\nnfactorialis a mathematical operation that can be defined using a recursive formula. The factorial of denoted"
	},
	{
		"chunk_id" : "00003710",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is defined for a positive integer as:\nThe special case is defined as\nQ&A Can factorials always be found using a calculator?\nNo. Factorials get large very quicklyfaster than even exponential functions! When the output gets too\nlarge for the calculator, it will not be able to calculate the factorial.\n1242 13  Sequences, Probability, and Counting Theory\nEXAMPLE7\nWriting the Terms of a Sequence Using Factorials\nWrite the first five terms of the sequence defined by the explicit formula\nSolution"
	},
	{
		"chunk_id" : "00003711",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nSubstitute and so on in the formula.\nThe first five terms are\nAnalysis\nFigure 7shows the graph of the sequence. Notice that, since factorials grow very quickly, the presence of the factorial\nterm in the denominator results in the denominator becoming much larger than the numerator as increases. This\nmeans the quotient gets smaller and, as the plot of the terms shows, the terms are decreasing and nearing zero.\nFigure7"
	},
	{
		"chunk_id" : "00003712",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure7\nTRY IT #9 Write the first five terms of the sequence defined by the explicit formula\nMEDIA\nAccess this online resource for additional instruction and practice with sequences.\nFinding Terms in a Sequence(http://openstax.org/l/findingterms)\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1243\n13.1 SECTION EXERCISES\nVerbal\n1. Discuss the meaning of a 2. Describe three ways that a 3. Is the ordered set of even\nsequence. If a finite sequence can be defined. numbers an infinite"
	},
	{
		"chunk_id" : "00003713",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sequence is defined by a sequence? What about the\nformula, what is its domain? ordered set of odd\nWhat about an infinite numbers? Explain why or\nsequence? why not.\n4. What happens to the terms 5. What is a factorial, and how\nof a sequence when is it denoted? Use an\nthere is a negative factor in example to illustrate how\nthe formula that is raised to factorial notation can be\na power that includes beneficial.\nWhat is the term used to\ndescribe this phenomenon?\nAlgebraic"
	},
	{
		"chunk_id" : "00003714",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "describe this phenomenon?\nAlgebraic\nFor the following exercises, write the first four terms of the sequence.\n6. 7. 8.\n9. 10. 11.\n12. 13. 14.\n15.\nFor the following exercises, write the first eight terms of the piecewise sequence.\n16. 17.\n18. 19.\n20.\nFor the following exercises, write an explicit formula for each sequence.\n21. 22. 23.\n1244 13  Sequences, Probability, and Counting Theory\n24. 25.\nFor the following exercises, write the first five terms of the sequence.\n26. 27. 28.\n29. 30."
	},
	{
		"chunk_id" : "00003715",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "26. 27. 28.\n29. 30.\nFor the following exercises, write the first eight terms of the sequence.\n31. 32.\n33.\nFor the following exercises, write a recursive formula for each sequence.\n34. 35. 36.\n37. 38.\nFor the following exercises, evaluate the factorial.\n39. 40. 41.\n42.\nFor the following exercises, write the first four terms of the sequence.\n43. 44. 45.\n46.\nGraphical\nFor the following exercises, graph the first five terms of the indicated sequence\n47. 48. 49.\n50. 51.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003716",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "50. 51.\nAccess for free at openstax.org\n13.1  Sequences and Their Notations 1245\nFor the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.\n52. 53. 54.\nFor the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.\n55. 56.\nTechnology\nFollow these steps to evaluate a sequence defined recursively using a graphing calculator:"
	},
	{
		"chunk_id" : "00003717",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " On the home screen, key in the value for the initial term and press[ENTER].\n Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes[2ND]\nANSfor the previous term Press[ENTER].\n Continue pressing[ENTER]to calculate the values for each successive term.\nFor the following exercises, use the steps above to find the indicated term or terms for the sequence.\n57. Find the first five terms of 58. Find the 15thterm of the 59. Find the first five terms of"
	},
	{
		"chunk_id" : "00003718",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the sequence , sequence , the sequence ,\nUse the\n>Fracfeature to give\nfractional results.\n60. Find the first ten terms of 61. Find the tenth term of the\nthe sequence , sequence ,\nFollow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.\n In the home screen, press[2ND] LIST.\n Scroll over toOPSand chooseseq(from the dropdown list. Press[ENTER].\n In the line headedExpr:type in the explicit formula, using the button for"
	},
	{
		"chunk_id" : "00003719",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1246 13  Sequences, Probability, and Counting Theory\n In the line headedVariable:type in the variable used on the previous step.\n In the line headedstart:key in the value of that begins the sequence.\n In the line headedend:key in the value of that ends the sequence.\n Press[ENTER]3 times to return to the home screen. You will see the sequence syntax on the screen. Press[ENTER]\nto see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms."
	},
	{
		"chunk_id" : "00003720",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using a TI-83, do the following.\n In the home screen, press[2ND] LIST.\n Scroll over toOPSand chooseseq(from the dropdown list. Press[ENTER].\n Enter the items in the orderExpr,Variable,start,endseparated by commas. See the instructions above\nfor the description of each item.\n Press[ENTER]to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the\nlist of terms."
	},
	{
		"chunk_id" : "00003721",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "list of terms.\nFor the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest\nthousandth when necessary.\n62. List the first five terms of 63. List the first six terms of 64. List the first five terms of\nthe sequence the sequence the sequence\n65. List the first four terms of 66. List the first six terms of\nthe sequence the sequence\nExtensions\n67. Consider the sequence 68. What term in the sequence 69. Find a recursive formula"
	},
	{
		"chunk_id" : "00003722",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "defined by for the sequence 1, 0, 1,\nhas the\nIs a term in the 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,\nvalue Verify the result.\nsequence? Verify the .... (Hint: find a pattern for\nresult. based on the first two\nterms.)\n70. Calculate the first eight terms of the sequences 71. Prove the conjecture made in the preceding\nand and then exercise.\nmake a conjecture about the relationship between\nthese two sequences.\n13.2 Arithmetic Sequences\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00003723",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nFind the common difference for an arithmetic sequence.\nWrite terms of an arithmetic sequence.\nUse a recursive formula for an arithmetic sequence.\nUse an explicit formula for an arithmetic sequence.\nCompanies often make large purchases, such as computers and vehicles, for business use. The book-value of these\nsupplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating"
	},
	{
		"chunk_id" : "00003724",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1247\nAs an example, consider a woman who starts a small contracting business. She purchases a new truck for $25,000. After\nfive years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be"
	},
	{
		"chunk_id" : "00003725",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "$17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two\nyears; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will\nconsider specific kinds of sequences that will allow us to calculate depreciation, such as the trucks value.\nFinding Common Differences\nThe values of the truck in the example are said to form anarithmetic sequencebecause they change by a constant"
	},
	{
		"chunk_id" : "00003726",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "amount each year. Each term increases or decreases by the same constant value called thecommon differenceof the\nsequence. For this sequence, the common difference is 3,400.\nThe sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can\nchoose anytermof thesequence, and add 3 to find the subsequent term.\nArithmetic Sequence\nAnarithmetic sequenceis a sequence that has the property that the difference between any two consecutive terms"
	},
	{
		"chunk_id" : "00003727",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is a constant. This constant is called thecommon difference. If is the first term of an arithmetic sequence and is\nthe common difference, the sequence will be:\nEXAMPLE1\nFinding Common Differences\nIs each sequence arithmetic? If so, find the common difference.\n \nSolution\nSubtract each term from the subsequent term to determine whether a common difference exists.\n The sequence is not arithmetic because there is no common difference."
	},
	{
		"chunk_id" : "00003728",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The sequence is arithmetic because there is a common difference. The common difference is 4.\nAnalysis\nThe graph of each of these sequences is shown inFigure 1. We can see from the graphs that, although both sequences\nshow growth, is not linear whereas is linear. Arithmetic sequences have a constant rate of change so their graphs will\nalways be points on a line.\n1248 13  Sequences, Probability, and Counting Theory\nFigure1"
	},
	{
		"chunk_id" : "00003729",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nQ&A If we are told that a sequence is arithmetic, do we have to subtract every term from the following\nterm to find the common difference?\nNo. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract\nit from the subsequent term to find the common difference.\nTRY IT #1 Is the given sequence arithmetic? If so, find the common difference.\nTRY IT #2 Is the given sequence arithmetic? If so, find the common difference.\nWriting Terms of Arithmetic Sequences"
	},
	{
		"chunk_id" : "00003730",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Writing Terms of Arithmetic Sequences\nNow that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common\ndifference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In\naddition, any term can also be found by plugging in the values of and into formula below.\n...\nHOW TO\nGiven the first term and the common difference of an arithmetic sequence, find the first several terms."
	},
	{
		"chunk_id" : "00003731",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Add the common difference to the first term to find the second term.\n2. Add the common difference to the second term to find the third term.\n3. Continue until all of the desired terms are identified.\n4. Write the terms separated by commas within brackets.\nEXAMPLE2\nWriting Terms of Arithmetic Sequences\nWrite the first five terms of thearithmetic sequencewith and .\nSolution\nAdding is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term."
	},
	{
		"chunk_id" : "00003732",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The first five terms are\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1249\nAnalysis\nAs expected, the graph of the sequence consists of points on a line as shown inFigure 2.\nFigure2\nTRY IT #3 List the first five terms of the arithmetic sequence with and .\n...\nHOW TO\nGiven any first term and any other term in an arithmetic sequence, find a given term.\n1. Substitute the values given for into the formula to solve for"
	},
	{
		"chunk_id" : "00003733",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Find a given term by substituting the appropriate values for and into the formula\nEXAMPLE3\nWriting Terms of Arithmetic Sequences\nGiven and , find .\nSolution\nThe sequence can be written in terms of the initial term 8 and the common difference .\nWe know the fourth term equals 14; we know the fourth term has the form .\nWe can find the common difference .\nFind the fifth term by adding the common difference to the fourth term.\nAnalysis"
	},
	{
		"chunk_id" : "00003734",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nNotice that the common difference is added to the first term once to find the second term, twice to find the third term,\nthree times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the\nfirst term nine times or by using the equation\nTRY IT #4 Given and , find .\n1250 13  Sequences, Probability, and Counting Theory\nUsing Recursive Formulas for Arithmetic Sequences"
	},
	{
		"chunk_id" : "00003735",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Recursive Formulas for Arithmetic Sequences\nSome arithmetic sequences are defined in terms of the previous term using arecursive formula. The formula provides\nan algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an\narithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common"
	},
	{
		"chunk_id" : "00003736",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive\nformula, the first term must be given.\nRecursive Formula for an Arithmetic Sequence\nThe recursive formula for an arithmetic sequence with common difference is:\n...\nHOW TO\nGiven an arithmetic sequence, write its recursive formula.\n1. Subtract any term from the subsequent term to find the common difference."
	},
	{
		"chunk_id" : "00003737",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.\nEXAMPLE4\nWriting a Recursive Formula for an Arithmetic Sequence\nWrite arecursive formulafor thearithmetic sequence.\nSolution\nThe first term is given as . The common difference can be found by subtracting the first term from the second term.\nSubstitute the initial term and the common difference into the recursive formula for arithmetic sequences.\nAnalysis"
	},
	{
		"chunk_id" : "00003738",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown\ninFigure 3. The growth pattern of the sequence shows the constant difference of 11 units.\nFigure3\nQ&A Do we have to subtract the first term from the second term to find the common difference?\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1251\nNo. We can subtract any term in the sequence from the subsequent term. It is, however, most common to"
	},
	{
		"chunk_id" : "00003739",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "subtract the first term from the second term because it is often the easiest method of finding the\ncommon difference.\nTRY IT #5 Write a recursive formula for the arithmetic sequence.\nUsing Explicit Formulas for Arithmetic Sequences\nWe can think of anarithmetic sequenceas a function on the domain of the natural numbers; it is a linear function\nbecause it has a constant rate of change. The common difference is the constant rate of change, or the slope of the"
	},
	{
		"chunk_id" : "00003740",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function. We can construct the linear function if we know the slope and the vertical intercept.\nTo find they-intercept of the function, we can subtract the common difference from the first term of the sequence.\nConsider the following sequence.\nThe common difference is , so the sequence represents a linear function with a slope of . To find the\n-intercept, we subtract from . You can also find the -intercept by graphing"
	},
	{
		"chunk_id" : "00003741",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown\ninFigure 4.\nFigure4\nRecall the slope-intercept form of a line is When dealing with sequences, we use in place of and in\nplace of If we know the slope and vertical intercept of the function, we can substitute them for and in the slope-\nintercept form of a line. Substituting for the slope and for the vertical intercept, we get the following equation:"
	},
	{
		"chunk_id" : "00003742",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We do not need to find the vertical intercept to write anexplicit formulafor an arithmetic sequence. Another explicit\nformula for this sequence is , which simplifies to\nExplicit Formula for an Arithmetic Sequence\nAn explicit formula for the term of an arithmetic sequence is given by\n...\nHOW TO\nGiven the first several terms for an arithmetic sequence, write an explicit formula.\n1. Find the common difference,\n1252 13  Sequences, Probability, and Counting Theory"
	},
	{
		"chunk_id" : "00003743",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Substitute the common difference and the first term into\nEXAMPLE5\nWriting thenth Term Explicit Formula for an Arithmetic Sequence\nWrite an explicit formula for the arithmetic sequence.\nSolution\nThe common difference can be found by subtracting the first term from the second term.\nThe common difference is 10. Substitute the common difference and the first term of the sequence into the formula and\nsimplify.\nAnalysis"
	},
	{
		"chunk_id" : "00003744",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "simplify.\nAnalysis\nThe graph of this sequence, represented inFigure 5, shows a slope of 10 and a vertical intercept of .\nFigure5\nTRY IT #6 Write an explicit formula for the following arithmetic sequence.\nFinding the Number of Terms in a Finite Arithmetic Sequence\nExplicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the\ncommon difference, and then determine how many times the common difference must be added to the first term to"
	},
	{
		"chunk_id" : "00003745",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "obtain the final term of the sequence.\n...\nHOW TO\nGiven the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.\n1. Find the common difference\n2. Substitute the common difference and the first term into\n3. Substitute the last term for and solve for\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1253\nEXAMPLE6\nFinding the Number of Terms in a Finite Arithmetic Sequence\nFind the number of terms in thefinite arithmetic sequence.\nSolution"
	},
	{
		"chunk_id" : "00003746",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThe common difference can be found by subtracting the first term from the second term.\nThe common difference is . Substitute the common difference and the initial term of the sequence into the term\nformula and simplify.\nSubstitute for and solve for\nThere are eight terms in the sequence.\nTRY IT #7 Find the number of terms in the finite arithmetic sequence.\nSolving Application Problems with Arithmetic Sequences"
	},
	{
		"chunk_id" : "00003747",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In many application problems, it often makes sense to use an initial term of instead of In these problems, we alter\nthe explicit formula slightly to account for the difference in initial terms. We use the following formula:\nEXAMPLE7\nSolving Application Problems with Arithmetic Sequences\nA five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.\n Write a formula for the childs weekly allowance in a given year."
	},
	{
		"chunk_id" : "00003748",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " What will the childs allowance be when he is 16 years old?\nSolution\n\nThe situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.\nLet be the amount of the allowance and be the number of years after age 5. Using the altered explicit formula for\nan arithmetic sequence we get:\n\nWe can find the number of years since age 5 by subtracting.\nWe are looking for the childs allowance after 11 years. Substitute 11 into the formula to find the childs allowance at"
	},
	{
		"chunk_id" : "00003749",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "age 16.\nThe childs allowance at age 16 will be $23 per week.\n1254 13  Sequences, Probability, and Counting Theory\nTRY IT #8 A woman decides to go for a 10-minute run every day this week and plans to increase the time of\nher daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How\nlong will her daily run be 8 weeks from today?\nMEDIA\nAccess this online resource for additional instruction and practice with arithmetic sequences."
	},
	{
		"chunk_id" : "00003750",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Arithmetic Sequences(http://openstax.org/l/arithmeticseq)\n13.2 SECTION EXERCISES\nVerbal\n1. What is an arithmetic 2. How is the common 3. How do we determine\nsequence? difference of an arithmetic whether a sequence is\nsequence found? arithmetic?\n4. What are the main 5. Describe how linear\ndifferences between using a functions and arithmetic\nrecursive formula and using sequences are similar. How\nan explicit formula to are they different?\ndescribe an arithmetic\nsequence?\nAlgebraic"
	},
	{
		"chunk_id" : "00003751",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "describe an arithmetic\nsequence?\nAlgebraic\nFor the following exercises, find the common difference for the arithmetic sequence provided.\n6. 7.\nFor the following exercises, determine whether the sequence is arithmetic. If so find the common difference.\n8. 9.\nFor the following exercises, write the first five terms of the arithmetic sequence given the first term and common\ndifference.\n10. , 11. ,\nFor the following exercises, write the first five terms of the arithmetic series given two terms.\n12. 13."
	},
	{
		"chunk_id" : "00003752",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "12. 13.\nFor the following exercises, find the specified term for the arithmetic sequence given the first term and common\ndifference.\n14. First term is 3, common 15. First term is 4, common 16. First term is 5, common\ndifference is 4, find the 5th difference is 5, find the 4th difference is 6, find the 8th\nterm. term. term.\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1255\n17. First term is 6, common 18. First term is 7, common\ndifference is 7, find the 6th difference is 8, find the 7th"
	},
	{
		"chunk_id" : "00003753",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "term. term.\nFor the following exercises, find the first term given two terms from an arithmetic sequence.\n19. Find the first term or of 20. Find the first term or of 21. Find the first term or of\nan arithmetic sequence if an arithmetic sequence if an arithmetic sequence if\nand and and\n22. Find the first term or of 23. Find the first term or of\nan arithmetic sequence if an arithmetic sequence if\nand and\nFor the following exercises, find the specified term given two terms from an arithmetic sequence."
	},
	{
		"chunk_id" : "00003754",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "24. and 25. and\nFind Find\nFor the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.\n26. 27.\nFor the following exercises, write a recursive formula for each arithmetic sequence.\n28. 29. 30.\n31. 32. 33.\n34. 35. 36.\n37.\nFor the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.\n38. Find the 39. Find 40. Find the\n17thterm. the 14thterm. 12thterm."
	},
	{
		"chunk_id" : "00003755",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "17thterm. the 14thterm. 12thterm.\nFor the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.\n41. 42.\nFor the following exercises, write an explicit formula for each arithmetic sequence.\n43. 44. 45.\n46. 47. 48.\n49. 50. 51.\n1256 13  Sequences, Probability, and Counting Theory\n52.\nFor the following exercises, find the number of terms in the given finite arithmetic sequence.\n53. 54. 55.\nGraphical"
	},
	{
		"chunk_id" : "00003756",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "53. 54. 55.\nGraphical\nFor the following exercises, determine whether the graph shown represents an arithmetic sequence.\n56.\nAccess for free at openstax.org\n13.2  Arithmetic Sequences 1257\n57.\nFor the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.\n58. 59. 60.\nTechnology\nFor the following exercises, follow the steps to work with the arithmetic sequence using a graphing\ncalculator:\n Press[MODE]\n Select SEQ in the fourth line"
	},
	{
		"chunk_id" : "00003757",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Press[MODE]\n Select SEQ in the fourth line\n Select DOT in the fifth line\n Press[ENTER]\n Press[Y=]\n is the first counting number for the sequence. Set\n is the pattern for the sequence. Set\n is the first number in the sequence. Set\n Press[2ND]then[WINDOW]to go toTBLSET\n Set\n Set\n Set Indpnt: Auto and Depend: Auto\n Press[2ND]then[GRAPH]to go to theTABLE\n1258 13  Sequences, Probability, and Counting Theory\n61. What are the first seven 62. Use the scroll-down arrow 63. Press[WINDOW]. Set"
	},
	{
		"chunk_id" : "00003758",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms shown in the column to scroll to What , ,\nwith the heading value is given for , ,\n, and\nThen press\n[GRAPH]. Graph the\nsequence as it appears on\nthe graphing calculator.\nFor the following exercises, follow the steps given above to work with the arithmetic sequence using a\ngraphing calculator.\n64. What are the first seven 65. Graph the sequence as it\nterms shown in the column appears on the graphing\nwith the heading in calculator. Be sure to\nthe TABLE feature? adjust the WINDOW\nsettings as needed."
	},
	{
		"chunk_id" : "00003759",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "settings as needed.\nExtensions\n66. Give two examples of 67. Give two examples of 68. Find the 5thterm of the\narithmetic sequences arithmetic sequences arithmetic sequence\nwhose 4thterms are whose 10thterms are\n69. Find the 11thterm of the 70. At which term does the 71. At which term does the\narithmetic sequence sequence sequence\nexceed begin to have negative\n151? values?\n72. For which terms does the 73. Write an arithmetic 74. Write an arithmetic"
	},
	{
		"chunk_id" : "00003760",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "finite arithmetic sequence sequence using a recursive sequence using an explicit\nhave formula. Show the first 4 formula. Show the first 4\nterms, and then find the terms, and then find the\ninteger values?\n31stterm. 28thterm.\n13.3 Geometric Sequences\nLearning Objectives\nIn this section, you will:\nFind the common ratio for a geometric sequence.\nList the terms of a geometric sequence.\nUse a recursive formula for a geometric sequence.\nUse an explicit formula for a geometric sequence."
	},
	{
		"chunk_id" : "00003761",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use an explicit formula for a geometric sequence.\nMany jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a\nrecent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2%\ncost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the"
	},
	{
		"chunk_id" : "00003762",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years;\nand so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we\nwill review sequences that grow in this way.\nFinding Common Ratios\nThe yearly salary values described form ageometric sequencebecause they change by a constant factor each year."
	},
	{
		"chunk_id" : "00003763",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Each term of a geometric sequence increases or decreases by a constant factor called thecommon ratio. The sequence\nAccess for free at openstax.org\n13.3  Geometric Sequences 1259\nbelow is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any\nterm of the sequence by the common ratio 6 generates the subsequent term.\nDefinition of a Geometric Sequence"
	},
	{
		"chunk_id" : "00003764",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Definition of a Geometric Sequence\nAgeometric sequenceis one in which any term divided by the previous term is a constant. This constant is called the\ncommon ratioof the sequence. The common ratio can be found by dividing any term in the sequence by the\nprevious term. If is the initial term of a geometric sequence and is the common ratio, the sequence will be\n...\nHOW TO\nGiven a set of numbers, determine if they represent a geometric sequence.\n1. Divide each term by the previous term."
	},
	{
		"chunk_id" : "00003765",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Divide each term by the previous term.\n2. Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.\nEXAMPLE1\nFinding Common Ratios\nIs the sequence geometric? If so, find the common ratio.\n \nSolution\nDivide each term by the previous term to determine whether a common ratio exists.\n\nThe sequence is geometric because there is a common ratio. The common ratio is 2.\n\nThe sequence is not geometric because there is not a common ratio.\nAnalysis"
	},
	{
		"chunk_id" : "00003766",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nThe graph of each sequence is shown inFigure 1. It seems from the graphs that both (a) and (b) appear have the form of\nthe graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this\ninterpretation holds, but (b) is not.\nFigure1\n1260 13  Sequences, Probability, and Counting Theory\nQ&A If you are told that a sequence is geometric, do you have to divide every term by the previous term\nto find the common ratio?"
	},
	{
		"chunk_id" : "00003767",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to find the common ratio?\nNo. If you know that the sequence is geometric, you can choose any one term in the sequence and divide\nit by the previous term to find the common ratio.\nTRY IT #1 Is the sequence geometric? If so, find the common ratio.\nTRY IT #2 Is the sequence geometric? If so, find the common ratio.\nWriting Terms of Geometric Sequences\nNow that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are"
	},
	{
		"chunk_id" : "00003768",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first\nterm and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is\nand the common ratio is we can find subsequent terms by multiplying to get then multiplying the result\nto get and so on.\nThe first four terms are\n...\nHOW TO\nGiven the first term and the common factor, find the first four terms of a geometric sequence."
	},
	{
		"chunk_id" : "00003769",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Multiply the initial term, by the common ratio to find the next term,\n2. Repeat the process, using to find and then to find until all four terms have been identified.\n3. Write the terms separated by commons within brackets.\nEXAMPLE2\nWriting the Terms of a Geometric Sequence\nList the first four terms of the geometric sequence with and\nSolution\nMultiply by to find Repeat the process, using to find and so on.\nThe first four terms are\nTRY IT #3 List the first five terms of the geometric sequence with and"
	},
	{
		"chunk_id" : "00003770",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n13.3  Geometric Sequences 1261\nUsing Recursive Formulas for Geometric Sequences\nArecursive formulaallows us to find any term of a geometric sequence by using the previous term. Each term is the\nproduct of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is\nnine times the previous term. As with any recursive formula, the initial term must be given.\nRecursive Formula for a Geometric Sequence"
	},
	{
		"chunk_id" : "00003771",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Recursive Formula for a Geometric Sequence\nThe recursive formula for a geometric sequence with common ratio and first term is\n...\nHOW TO\nGiven the first several terms of a geometric sequence, write its recursive formula.\n1. State the initial term.\n2. Find the common ratio by dividing any term by the preceding term.\n3. Substitute the common ratio into the recursive formula for a geometric sequence.\nEXAMPLE3\nUsing Recursive Formulas for Geometric Sequences"
	},
	{
		"chunk_id" : "00003772",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Recursive Formulas for Geometric Sequences\nWrite a recursive formula for the following geometric sequence.\nSolution\nThe first term is given as 6. The common ratio can be found by dividing the second term by the first term.\nSubstitute the common ratio into the recursive formula for geometric sequences and define\nAnalysis\nThe sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential\nfunction as shown inFigure 2\nFigure2"
	},
	{
		"chunk_id" : "00003773",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "function as shown inFigure 2\nFigure2\nQ&A Do we have to divide the second term by the first term to find the common ratio?\nNo. We can divide any term in the sequence by the previous term. It is, however, most common to divide\n1262 13  Sequences, Probability, and Counting Theory\nthe second term by the first term because it is often the easiest method of finding the common ratio.\nTRY IT #4 Write a recursive formula for the following geometric sequence.\nUsing Explicit Formulas for Geometric Sequences"
	},
	{
		"chunk_id" : "00003774",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Explicit Formulas for Geometric Sequences\nBecause a geometric sequence is an exponential function whose domain is the set of positive integers, and the common\nratio is the base of the function, we can write explicit formulas that allow us to find particular terms.\nLets take a look at the sequence This is a geometric sequence with a common ratio of 2 and\nan exponential function with a base of 2. An explicit formula for this sequence is\nThe graph of the sequence is shown inFigure 3.\nFigure3"
	},
	{
		"chunk_id" : "00003775",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure3\nExplicit Formula for a Geometric Sequence\nThe th term of a geometric sequence is given by theexplicit formula:\nEXAMPLE4\nWriting Terms of Geometric Sequences Using the Explicit Formula\nGiven a geometric sequence with and find\nSolution\nThe sequence can be written in terms of the initial term and the common ratio\nFind the common ratio using the given fourth term.\nAccess for free at openstax.org\n13.3  Geometric Sequences 1263\nFind the second term by multiplying the first term by the common ratio."
	},
	{
		"chunk_id" : "00003776",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nThe common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to\nfind the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine\ntimes or by multiplying by the common ratio raised to the ninth power.\nTRY IT #5 Given a geometric sequence with and , find\nEXAMPLE5\nWriting an Explicit Formula for the th Term of a Geometric Sequence"
	},
	{
		"chunk_id" : "00003777",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Write an explicit formula for the term of the following geometric sequence.\nSolution\nThe first term is 2. The common ratio can be found by dividing the second term by the first term.\nThe common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.\nThe graph of this sequence inFigure 4shows an exponential pattern.\nFigure4\nTRY IT #6 Write an explicit formula for the following geometric sequence.\n1264 13  Sequences, Probability, and Counting Theory"
	},
	{
		"chunk_id" : "00003778",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solving Application Problems with Geometric Sequences\nIn real-world scenarios involving geometric sequences, we may need to use an initial term of instead of In these\nproblems, we can alter the explicit formula slightly by using the following formula:\nEXAMPLE6\nSolving Application Problems with Geometric Sequences\nIn 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4%\neach year."
	},
	{
		"chunk_id" : "00003779",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each year.\n Write a formula for the student population.  Estimate the student population in 2020.\nSolution\n\nThe situation can be modeled by a geometric sequence with an initial term of 284. The student population will be\n104% of the prior year, so the common ratio is 1.04.\nLet be the student population and be the number of years after 2013. Using the explicit formula for a geometric\nsequence we get\n\nWe can find the number of years since 2013 by subtracting."
	},
	{
		"chunk_id" : "00003780",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We are looking for the population after 7 years. We can substitute 7 for to estimate the population in 2020.\nThe student population will be about 374 in 2020.\nTRY IT #7 A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The\nbusiness estimates the number of hits will increase by 2.6% per week.\n Write a formula for the number of hits.  Estimate the number of hits in 5 weeks.\nMEDIA"
	},
	{
		"chunk_id" : "00003781",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with geometric sequences.\nGeometric Sequences(http://openstax.org/l/geometricseq)\nDetermine the Type of Sequence(http://openstax.org/l/sequencetype)\nFind the Formula for a Sequence(http://openstax.org/l/sequenceformula)\n13.3 SECTION EXERCISES\nVerbal\n1. What is a geometric 2. How is the common ratio of 3. What is the procedure for\nsequence? a geometric sequence determining whether a\nfound? sequence is geometric?"
	},
	{
		"chunk_id" : "00003782",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "found? sequence is geometric?\n4. What is the difference 5. Describe how exponential\nbetween an arithmetic functions and geometric\nsequence and a geometric sequences are similar. How\nsequence? are they different?\nAccess for free at openstax.org\n13.3  Geometric Sequences 1265\nAlgebraic\nFor the following exercises, find the common ratio for the geometric sequence.\n6. 7. 8.\nFor the following exercises, determine whether the sequence is geometric. If so, find the common ratio.\n9. 10. 11.\n12. 13."
	},
	{
		"chunk_id" : "00003783",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9. 10. 11.\n12. 13.\nFor the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio.\n14. 15.\nFor the following exercises, write the first five terms of the geometric sequence, given any two terms.\n16. 17.\nFor the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.\n18. The first term is and the 19. The first term is 16 and the\ncommon ratio is Find the common ratio is Find\n5thterm. the 4thterm."
	},
	{
		"chunk_id" : "00003784",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "5thterm. the 4thterm.\nFor the following exercises, find the specified term for the geometric sequence, given the first four terms.\n20. 21.\nFind Find\nFor the following exercises, write the first five terms of the geometric sequence.\n22. 23.\nFor the following exercises, write a recursive formula for each geometric sequence.\n24. 25. 26.\n27. 28. 29.\n30. 31.\nFor the following exercises, write the first five terms of the geometric sequence.\n32. 33.\n1266 13  Sequences, Probability, and Counting Theory"
	},
	{
		"chunk_id" : "00003785",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "For the following exercises, write an explicit formula for each geometric sequence.\n34. 35. 36.\n37. 38. 39.\n40. 41.\nFor the following exercises, find the specified term for the geometric sequence given.\n42. Let 43. Let Find\nFind\nFor the following exercises, find the number of terms in the given finite geometric sequence.\n44. 45.\nGraphical\nFor the following exercises, determine whether the graph shown represents a geometric sequence.\n46. 47."
	},
	{
		"chunk_id" : "00003786",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "46. 47.\nFor the following exercises, use the information provided to graph the first five terms of the geometric sequence.\n48. 49. 50.\nExtensions\n51. Use recursive formulas to 52. Use explicit formulas to 53. Find the 5thterm of the\ngive two examples of give two examples of geometric sequence\ngeometric sequences geometric sequences\nwhose 3rdterms are whose 7thterms are\nAccess for free at openstax.org\n13.4  Series and Their Notations 1267"
	},
	{
		"chunk_id" : "00003787",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.4  Series and Their Notations 1267\n54. Find the 7thterm of the geometric 55. At which term does the 56. At which term does the\nsequence sequence sequence\nexceed begin to have integer\nvalues?\n57. For which term does the 58. Use the recursive formula 59. Use the explicit formula to\ngeometric sequence to write a geometric write a geometric\nfirst have sequence whose common sequence whose common\nratio is an integer. Show ratio is a decimal number\na non-integer value?"
	},
	{
		"chunk_id" : "00003788",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a non-integer value?\nthe first four terms, and between 0 and 1. Show the\nthen find the 10thterm. first 4 terms, and then find\nthe 8thterm.\n60. Is it possible for a\nsequence to be both\narithmetic and geometric?\nIf so, give an example.\n13.4 Series and Their Notations\nLearning Objectives\nIn this section, you will:\nUse summation notation.\nUse the formula for the sum of the first n terms of an arithmetic series.\nUse the formula for the sum of the first n terms of a geometric series."
	},
	{
		"chunk_id" : "00003789",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Use the formula for the sum of an infinite geometric series.\nSolve annuity problems.\nA parent decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund\npays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to\nstart college in 6 years? In this section, we will learn how to answer this question. To do so, we need to consider the\namount of money invested and the amount of interest earned."
	},
	{
		"chunk_id" : "00003790",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using Summation Notation\nTo find the total amount of money in the college fund and the sum of the amounts deposited, we need to add the\namounts deposited each month and the amounts earned monthly. The sum of the terms of a sequence is called aseries.\nConsider, for example, the following series.\nThenth partial sumof a series is the sum of a finite number of consecutive terms beginning with the first term. The\nnotation represents the partial sum."
	},
	{
		"chunk_id" : "00003791",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "notation represents the partial sum.\nSummation notationis used to represent series. Summation notation is often known as sigma notation because it uses\nthe Greek capital lettersigma, to represent the sum. Summation notation includes an explicit formula and specifies\nthe first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A\nvariable called theindex of summationis written below the sigma. The index of summation is set equal to thelower"
	},
	{
		"chunk_id" : "00003792",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "limit of summation, which is the number used to generate the first term in the series. The number above the sigma,\ncalled theupper limit of summation, is the number used to generate the last term in a series.\n1268 13  Sequences, Probability, and Counting Theory\nIf we interpret the given notation, we see that it asks us to find the sum of the terms in the series for\nthrough We can begin by substituting the terms for and listing out the terms of this series."
	},
	{
		"chunk_id" : "00003793",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can find the sum of the series by adding the terms:\nSummation Notation\nThe sum of the first terms of aseriescan be expressed insummation notationas follows:\nThis notation tells us to find the sum of from to\nis called theindex of summation, 1 is thelower limit of summation, and is theupper limit of summation.\nQ&A Does the lower limit of summation have to be 1?\nNo. The lower limit of summation can be any number, but 1 is frequently used. We will look at examples"
	},
	{
		"chunk_id" : "00003794",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "with lower limits of summation other than 1.\n...\nHOW TO\nGiven summation notation for a series, evaluate the value.\n1. Identify the lower limit of summation.\n2. Identify the upper limit of summation.\n3. Substitute each value of from the lower limit to the upper limit into the formula.\n4. Add to find the sum.\nEXAMPLE1\nUsing Summation Notation\nEvaluate\nSolution\nAccording to the notation, the lower limit of summation is 3 and the upper limit is 7. So we need to find the sum of"
	},
	{
		"chunk_id" : "00003795",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from to We find the terms of the series by substituting and into the function We add the\nterms to find the sum.\nAccess for free at openstax.org\n13.4  Series and Their Notations 1269\nTRY IT #1 Evaluate\nUsing the Formula for Arithmetic Series\nJust as we studied special types of sequences, we will look at special types of series. Recall that anarithmetic sequenceis\na sequence in which the difference between any two consecutive terms is thecommon difference, The sum of the"
	},
	{
		"chunk_id" : "00003796",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms of an arithmetic sequence is called anarithmetic series. We can write the sum of the first terms of an\narithmetic series as:\nWe can also reverse the order of the terms and write the sum as\nIf we add these two expressions for the sum of the first terms of an arithmetic series, we can derive a formula for the\nsum of the first terms of any arithmetic series.\nBecause there are terms in the series, we can simplify this sum to"
	},
	{
		"chunk_id" : "00003797",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We divide by 2 to find the formula for the sum of the first terms of an arithmetic series.\nFormula for the Sum of the FirstnTerms of an Arithmetic Series\nAnarithmetic seriesis the sum of the terms of an arithmetic sequence. The formula for the sum of the first terms\nof an arithmetic sequence is\n...\nHOW TO\nGiven terms of an arithmetic series, find the sum of the first terms.\n1. Identify and\n2. Determine\n3. Substitute values for and into the formula\n4. Simplify to find"
	},
	{
		"chunk_id" : "00003798",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Simplify to find\n1270 13  Sequences, Probability, and Counting Theory\nEXAMPLE2\nFinding the FirstnTerms of an Arithmetic Series\nFind the sum of each arithmetic series.\n  \nSolution\n\nWe are given and\nCount the number of terms in the sequence to find\nSubstitute values for and into the formula and simplify.\n\nWe are given and\nUse the formula for the general term of an arithmetic sequence to find\nSubstitute values for into the formula and simplify.\n\nTo find substitute into the given explicit formula."
	},
	{
		"chunk_id" : "00003799",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We are given that To find substitute into the given explicit formula.\nSubstitute values for and into the formula and simplify.\nUse the formula to find the sum of each arithmetic series.\nTRY IT #2\nTRY IT #3\nAccess for free at openstax.org\n13.4  Series and Their Notations 1271\nTRY IT #4\nEXAMPLE3\nSolving Application Problems with Arithmetic Series\nOn the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-"
	},
	{
		"chunk_id" : "00003800",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "mile. After 8 weeks, what will be the total number of miles she has walked?\nSolution\nThis problem can be modeled by an arithmetic series with and We are looking for the total number of\nmiles walked after 8 weeks, so we know that and we are looking for To find we can use the explicit formula\nfor an arithmetic sequence.\nWe can now use the formula for arithmetic series.\nShe will have walked a total of 11 miles."
	},
	{
		"chunk_id" : "00003801",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "She will have walked a total of 11 miles.\nTRY IT #5 A man earns $100 in the first week of June. Each week, he earns $12.50 more than the previous\nweek. After 12 weeks, how much has he earned?\nUsing the Formula for Geometric Series\nJust as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric\nsequence is called ageometric series. Recall that ageometric sequenceis a sequence in which the ratio of any two"
	},
	{
		"chunk_id" : "00003802",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "consecutive terms is thecommon ratio, We can write the sum of the first terms of a geometric series as\nJust as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first terms\nof a geometric series. We will begin by multiplying both sides of the equation by\nNext, we subtract this equation from the original equation.\nNotice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel"
	},
	{
		"chunk_id" : "00003803",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "out. To obtain a formula for divide both sides by\nFormula for the Sum of the FirstnTerms of a Geometric Series\nAgeometric seriesis the sum of the terms in a geometric sequence. The formula for the sum of the first terms of a\n1272 13  Sequences, Probability, and Counting Theory\ngeometric sequence is represented as\n...\nHOW TO\nGiven a geometric series, find the sum of the firstnterms.\n1. Identify\n2. Substitute values for and into the formula\n3. Simplify to find\nEXAMPLE4"
	},
	{
		"chunk_id" : "00003804",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3. Simplify to find\nEXAMPLE4\nFinding the FirstnTerms of a Geometric Series\nUse the formula to find the indicated partial sum of each geometric series.\n for the series \nSolution\n\nand we are given that\nWe can find by dividing the second term of the series by the first.\nSubstitute values for into the formula and simplify.\n\nFind by substituting into the given explicit formula.\nWe can see from the given explicit formula that The upper limit of summation is 6, so"
	},
	{
		"chunk_id" : "00003805",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Substitute values for and into the formula, and simplify.\nUse the formula to find the indicated partial sum of each geometric series.\nTRY IT #6 for the series\nTRY IT #7\nAccess for free at openstax.org\n13.4  Series and Their Notations 1273\nEXAMPLE5\nSolving an Application Problem with a Geometric Series\nAt a new job, an employees starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end\nof 5 years.\nSolution"
	},
	{
		"chunk_id" : "00003806",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of 5 years.\nSolution\nThe problem can be represented by a geometric series with and Substitute values for\nand into the formula and simplify to find the total amount earned at the end of 5 years.\nHe will have earned a total of $138,099.03 by the end of 5 years.\nTRY IT #8 At a new job, an employees starting salary is $32,100. She receives a 2% annual raise. How much\nwill she have earned by the end of 8 years?\nUsing the Formula for the Sum of an Infinite Geometric Series"
	},
	{
		"chunk_id" : "00003807",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an\ninfinite sequence rather than the sum of only the first terms. Aninfinite seriesis the sum of the terms of an infinite\nsequence. An example of an infinite series is\n\nThis series can also be written in summation notation as where the upper limit of summation is infinity. Because"
	},
	{
		"chunk_id" : "00003808",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the\nsum of this infinite series is not defined. When the sum is not a real number, we say the seriesdiverges.\nDetermining Whether the Sum of an Infinite Geometric Series is Defined\nIf the terms of aninfinite geometric sequenceapproach 0, the sum of an infinite geometric series can be defined. The\nterms in this series approach 0:"
	},
	{
		"chunk_id" : "00003809",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "terms in this series approach 0:\nThe common ratio As gets very large, the values of get very small and approach 0. Each successive term\naffects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms\napproaches a finite value. The terms of any infinite geometric series with approach 0; the sum of a geometric\nseries is defined when\nDetermining Whether the Sum of an Infinite Geometric Series is Defined"
	},
	{
		"chunk_id" : "00003810",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The sum of an infinite series is defined if the series is geometric and\n...\nHOW TO\nGiven the first several terms of an infinite series, determine if the sum of the series exists.\n1. Find the ratio of the second term to the first term.\n2. Find the ratio of the third term to the second term.\n3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series\nis geometric."
	},
	{
		"chunk_id" : "00003811",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is geometric.\n4. If a common ratio, was found in step 3, check to see if . If so, the sum is defined. If not, the sum is\nnot defined.\n1274 13  Sequences, Probability, and Counting Theory\nEXAMPLE6\nDetermining Whether the Sum of an Infinite Series is Defined\nDetermine whether the sum of each infinite series is defined.\n\n\n   \nSolution\n The ratio of the second term to the first is which is not the same as the ratio of the third term to the second,\nThe series is not geometric."
	},
	{
		"chunk_id" : "00003812",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The series is not geometric.\n The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is\ngeometric with a common ratio of The sum of the infinite series is defined.\n The given formula is exponential with a base of the series is geometric with a common ratio of The sum of\nthe infinite series is defined.\n The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot\nyield a finite sum."
	},
	{
		"chunk_id" : "00003813",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "yield a finite sum.\nDetermine whether the sum of the infinite series is defined.\nTRY IT #9\nTRY IT #10\n\nTRY IT #11\nFinding Sums of Infinite Series\nWhen the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite\nseries is related to the formula for the sum of the first terms of a geometric series.\nWe will examine an infinite series with What happens to as increases?\nThe value of decreases rapidly. What happens for greater values of"
	},
	{
		"chunk_id" : "00003814",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "As gets very large, gets very small. We say that, as increases without bound, approaches 0. As approaches 0,\napproaches 1. When this happens, the numerator approaches This give us a formula for the sum of an infinite\ngeometric series.\nAccess for free at openstax.org\n13.4  Series and Their Notations 1275\nFormula for the Sum of an Infinite Geometric Series\nThe formula for the sum of an infinite geometric series with is\n...\nHOW TO\nGiven an infinite geometric series, find its sum.\n1. Identify and"
	},
	{
		"chunk_id" : "00003815",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify and\n2. Confirm that\n3. Substitute values for and into the formula,\n4. Simplify to find\nEXAMPLE7\nFinding the Sum of an Infinite Geometric Series\nFind the sum, if it exists, for the following:\n\n  \n\n\nSolution\n There is not a constant ratio; the series is not geometric.\n\nThere is a constant ratio; the series is geometric. and so the sum exists. Substitute\nand into the formula and simplify to find the sum:\n"
	},
	{
		"chunk_id" : "00003816",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "\nThe formula is exponential, so the series is geometric with Find by substituting into the given explicit\nformula:\nSubstitute and into the formula, and simplify to find the sum:\n The formula is exponential, so the series is geometric, but The sum does not exist.\nEXAMPLE8\nFinding an Equivalent Fraction for a Repeating Decimal\nFind an equivalent fraction for the repeating decimal\n1276 13  Sequences, Probability, and Counting Theory\nSolution"
	},
	{
		"chunk_id" : "00003817",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nWe notice the repeating decimal so we can rewrite the repeating decimal as a sum of terms.\nLooking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and\nthe second term multiplied to 0.1 in the third term.\nNotice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So,\nsubstituting into our formula for an infinite geometric sum, we have\nFind the sum, if it exists.\nTRY IT #12\n"
	},
	{
		"chunk_id" : "00003818",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Find the sum, if it exists.\nTRY IT #12\n\nTRY IT #13\n\nTRY IT #14\nSolving Annuity Problems\nAt the beginning of the section, we looked at a problem in which a parent invested a set amount of money each month\ninto a college fund for six years. Anannuityis an investment in which the purchaser makes a sequence of periodic,\nequal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned."
	},
	{
		"chunk_id" : "00003819",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "In the example, the parent invests $50 each month. This is the value of the initial deposit. The account paid 6%annual\ninterest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual\npercentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account\neach month by 100.5% to find the value of the account after interest has been added."
	},
	{
		"chunk_id" : "00003820",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We can find the value of the annuity right after the last deposit by using a geometric series with and\nAfter the first deposit, the value of the annuity will be $50. Let us see if we can determine the\namount in the college fund and the interest earned.\nWe can find the value of the annuity after deposits using the formula for the sum of the first terms of a geometric\nseries. In 6 years, there are 72 months, so We can substitute into the formula,\nand simplify to find the value of the annuity after 6 years."
	},
	{
		"chunk_id" : "00003821",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "After the last deposit, the parent will have a total of $4,320.44 in the account. Notice, the parent made 72 payments of\n$50 each for a total of This means that because of the annuity, the parent earned $720.44 interest in\ntheir college fund.\n...\nHOW TO\nGiven an initial deposit and an interest rate, find the value of an annuity.\nAccess for free at openstax.org\n13.4  Series and Their Notations 1277\n1. Determine the value of the initial deposit.\n2. Determine the number of deposits.\n3. Determine"
	},
	{
		"chunk_id" : "00003822",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Determine the number of deposits.\n3. Determine\na. Divide the annual interest rate by the number of times per year that interest is compounded.\nb. Add 1 to this amount to find\n4. Substitute values for into the formula for the sum of the first terms of a geometric series,\n5. Simplify to find the value of the annuity after deposits.\nEXAMPLE9\nSolving an Annuity Problem\nA deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual"
	},
	{
		"chunk_id" : "00003823",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last\ndeposit?\nSolution\nThe value of the initial deposit is $100, so A total of 120 monthly deposits are made in the 10 years, so\nTo find divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly\ndeposit.\nSubstitute into the formula for the sum of the first terms of a geometric series,\nand simplify to find the value of the annuity."
	},
	{
		"chunk_id" : "00003824",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and simplify to find the value of the annuity.\nSo the account has $19,351.43 after the last deposit is made.\nTRY IT #15 At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6%\nannual interest, compounded monthly, and paid into the account at the end of the month. How\nmuch is in the account if deposits are made for 10 years?\nMEDIA\nAccess these online resources for additional instruction and practice with series.\nArithmetic Series(http://openstax.org/l/arithmeticser)"
	},
	{
		"chunk_id" : "00003825",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Geometric Series(http://openstax.org/l/geometricser)\nSummation Notation(http://openstax.org/l/sumnotation)\n13.4 SECTION EXERCISES\nVerbal\n1. What is an partial sum? 2. What is the difference 3. What is a geometric series?\nbetween an arithmetic\nsequence and an arithmetic\nseries?\n1278 13  Sequences, Probability, and Counting Theory\n4. How is finding the sum of an 5. What is an annuity?\ninfinite geometric series\ndifferent from finding the\npartial sum?\nAlgebraic"
	},
	{
		"chunk_id" : "00003826",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "different from finding the\npartial sum?\nAlgebraic\nFor the following exercises, express each description of a sum using summation notation.\n6. The sum of terms 7. The sum from of to 8. The sum of from\nfrom to of to\n9. The sum that results from\nadding the number 4 five\ntimes\nFor the following exercises, express each arithmetic sum using summation notation.\n10. 11.\n12.\nFor the following exercises, use the formula for the sum of the first terms of each arithmetic sequence.\n13. 14. 15."
	},
	{
		"chunk_id" : "00003827",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13. 14. 15.\nFor the following exercises, express each geometric sum using summation notation.\n16. 17.\n18.\nFor the following exercises, use the formula for the sum of the first terms of each geometric sequence, and then state\nthe indicated sum.\n19. 20. 21.\nFor the following exercises, determine whether the infinite series has a sum. If so, write the formula for the sum. If not,\nstate the reason.\n\n22. 23. 24.\n25.\n\nAccess for free at openstax.org\n13.4  Series and Their Notations 1279\nGraphical"
	},
	{
		"chunk_id" : "00003828",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.4  Series and Their Notations 1279\nGraphical\nFor the following exercises, use the following scenario. Javier makes monthly deposits into a savings account. He opened\nthe account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20.\n26. Graph the arithmetic 27. Graph the arithmetic series\nsequence showing one showing the monthly sums\nyear of Javiers deposits. of one year of Javiers\ndeposits.\n\nFor the following exercises, use the geometric series"
	},
	{
		"chunk_id" : "00003829",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "28. Graph the first 7 partial 29. What number does\nsums of the series. seem to be approaching in\nthe graph? Find the sum to\nexplain why this makes\nsense.\nNumeric\nFor the following exercises, find the indicated sum.\n30. 31. 32.\n33.\nFor the following exercises, use the formula for the sum of the first terms of an arithmetic series to find the sum.\n34. 35.\n36. 37.\nFor the following exercises, use the formula for the sum of the first terms of a geometric series to find the partial sum."
	},
	{
		"chunk_id" : "00003830",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "38. for the series 39. for the series 40.\n41.\n1280 13  Sequences, Probability, and Counting Theory\nFor the following exercises, find the sum of the infinite geometric series.\n42. 43. 44.\n\n\n45.\nFor the following exercises, determine the value of the annuity for the indicated monthly deposit amount, the number of\ndeposits, and the interest rate.\n46. Deposit amount: total 47. Deposit amount: 48. Deposit amount:\ndeposits: interest rate: total deposits: interest total deposits: interest"
	},
	{
		"chunk_id" : "00003831",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "compounded monthly rate: compounded rate: compounded\nmonthly quarterly\n49. Deposit amount:\ntotal deposits: interest\nrate: compounded\nsemi-annually\nExtensions\n50. The sum of terms 51. Write an explicit formula 52. Find the smallest value ofn\nfrom through is for such that such that\nWhat isx?\nAssume this\nis an arithmetic series.\n53. How many terms must be 54. Write as an infinite 55. The sum of an infinite\nadded before the series geometric series using geometric series is five"
	},
	{
		"chunk_id" : "00003832",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "has a summation notation. Then times the value of the first\nsum less than use the formula for finding term. What is the common\nthe sum of an infinite ratio of the series?\ngeometric series to convert\nto a fraction.\nAccess for free at openstax.org\n13.5  Counting Principles 1281\n56. To get the best loan rates 57. Karl has two years to save\navailable, the Coleman to buy a used car\nfamily want to save enough when he graduates. To the\nmoney to place 20% down nearest dollar, what would"
	},
	{
		"chunk_id" : "00003833",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "on a $160,000 home. They his monthly deposits need\nplan to make monthly to be if he invests in an\ndeposits of $125 in an account offering a 4.2%\ninvestment account that annual interest rate that\noffers 8.5% annual interest compounds monthly?\ncompounded semi-\nannually. Will the Colemans\nhave enough for a 20%\ndown payment after five\nyears of saving? How much\nmoney will they have\nsaved?\nReal-World Applications\n58. Keisha devised a week-long 59. A boulder rolled down a 60. A scientist places 50 cells in"
	},
	{
		"chunk_id" : "00003834",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "study plan to prepare for mountain, traveling 6 feet a petri dish. Every hour, the\nfinals. On the first day, she in the first second. Each population increases by\nplans to study for hour, successive second, its 1.5%. What will the cell\nand each successive day distance increased by 8 count be after 1 day?\nshe will increase her study feet. How far did the\ntime by minutes. How boulder travel after 10\nmany hours will Keisha seconds?\nhave studied after one\nweek?"
	},
	{
		"chunk_id" : "00003835",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "have studied after one\nweek?\n61. A pendulum travels a 62. Rachael deposits $1,500\ndistance of 3 feet on its into a retirement fund each\nfirst swing. On each year. The fund earns 8.2%\nsuccessive swing, it travels annual interest,\nthe distance of the compounded monthly. If\nprevious swing. What is the she opened her account\ntotal distance traveled by when she was 19 years old,\nthe pendulum when it how much will she have by\nstops swinging? the time she is 55? How\nmuch of that amount will\nbe interest earned?"
	},
	{
		"chunk_id" : "00003836",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "much of that amount will\nbe interest earned?\n13.5 Counting Principles\nLearning Objectives\nIn this section, you will:\nSolve counting problems using the Addition Principle.\nSolve counting problems using the Multiplication Principle.\nSolve counting problems using permutations involving n distinct objects.\nSolve counting problems using combinations.\nFind the number of subsets of a given set.\nSolve counting problems using permutations involving n non-distinct objects."
	},
	{
		"chunk_id" : "00003837",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be\npersonalized for an additional fee with images or a monogram. A customer can choose not to personalize or could\n1282 13  Sequences, Probability, and Counting Theory\nchoose to have one, two, or three images or a monogram. The customer can choose the order of the images and the\nletters in the monogram. The company is working with an agency to develop a marketing campaign with a focus on the"
	},
	{
		"chunk_id" : "00003838",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "huge number of options they offer. Counting the possibilities is challenging!\nWe encounter a wide variety of counting problems every day. There is a branch of mathematics devoted to the study of\ncounting problems such as this one. Other applications of counting include secure passwords, horse racing outcomes,\nand college scheduling choices. We will examine this type of mathematics in this section.\nUsing the Addition Principle"
	},
	{
		"chunk_id" : "00003839",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Using the Addition Principle\nThe company that sells customizable cases offers cases for tablets and smartphones. There are 3 supported tablet\nmodels and 5 supported smartphone models. TheAddition Principletells us that we can add the number of tablet\noptions to the number of smartphone options to find the total number of options. By the Addition Principle, there are 8\ntotal options, as we can see inFigure 1.\nFigure1\nThe Addition Principle"
	},
	{
		"chunk_id" : "00003840",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Figure1\nThe Addition Principle\nAccording to theAddition Principle, if one event can occur in ways and a second event with no common outcomes\ncan occur in ways, then the firstorsecond event can occur in ways.\nEXAMPLE1\nUsing the Addition Principle\nThere are 2 vegetarian entre options and 5 meat entre options on a dinner menu. What is the total number of entre\noptions?\nSolution\nWe can add the number of vegetarian options to the number of meat options to find the total number of entre options."
	},
	{
		"chunk_id" : "00003841",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "There are 7 total options.\nTRY IT #1 A student is shopping for a new computer. He is deciding among 3 desktop computers and 4\nlaptop computers. What is the total number of computer options?\nUsing the Multiplication Principle\nTheMultiplication Principleapplies when we are making more than one selection. Suppose we are choosing an\nAccess for free at openstax.org\n13.5  Counting Principles 1283"
	},
	{
		"chunk_id" : "00003842",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.5  Counting Principles 1283\nappetizer, an entre, and a dessert. If there are 2 appetizer options, 3 entre options, and 2 dessert options on a fixed-\nprice dinner menu, there are a total of 12 possible choices of one each as shown in the tree diagram inFigure 2.\nFigure2\nThe possible choices are:\n1. soup, chicken, cake\n2. soup, chicken, pudding\n3. soup, fish, cake\n4. soup, fish, pudding\n5. soup, steak, cake\n6. soup, steak, pudding\n7. salad, chicken, cake\n8. salad, chicken, pudding\n9. salad, fish, cake"
	},
	{
		"chunk_id" : "00003843",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "8. salad, chicken, pudding\n9. salad, fish, cake\n10. salad, fish, pudding\n11. salad, steak, cake\n12. salad, steak, pudding\nWe can also find the total number of possible dinners by multiplying.\nWe could also conclude that there are 12 possible dinner choices simply by applying the Multiplication Principle.\nThe Multiplication Principle\nAccording to theMultiplication Principle, if one event can occur in ways and a second event can occur in ways"
	},
	{
		"chunk_id" : "00003844",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "after the first event has occurred, then the two events can occur in ways. This is also known as the\nFundamental Counting Principle.\nEXAMPLE2\nUsing the Multiplication Principle\nDiane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt and a blouse for\neach outfit and decide whether to wear the sweater. Use the Multiplication Principle to find the total number of possible\noutfits.\nSolution"
	},
	{
		"chunk_id" : "00003845",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "outfits.\nSolution\nTo find the total number of outfits, find the product of the number of skirt options, the number of blouse options, and\nthe number of sweater options.\n1284 13  Sequences, Probability, and Counting Theory\nThere are 16 possible outfits.\nTRY IT #2 A restaurant offers a breakfast special that includes a breakfast sandwich, a side dish, and a\nbeverage. There are 3 types of breakfast sandwiches, 4 side dish options, and 5 beverage choices.\nFind the total number of possible breakfast specials."
	},
	{
		"chunk_id" : "00003846",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Number of Permutations ofnDistinct Objects\nThe Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects\nin order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An\nordering of objects is called apermutation.\nFinding the Number of Permutations ofnDistinct Objects Using the Multiplication Principle"
	},
	{
		"chunk_id" : "00003847",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine\nthe number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the\nnumber of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three\nplaces on the wall.\nThere are four options for the first place, so we write a 4 on the first line."
	},
	{
		"chunk_id" : "00003848",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "After the first place has been filled, there are three options for the second place so we write a 3 on the second line.\nAfter the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally,\nwe find the product.\nThere are 24 possible permutations of the paintings.\n...\nHOW TO\nGiven distinct options, determine how many permutations there are.\n1. Determine how many options there are for the first situation."
	},
	{
		"chunk_id" : "00003849",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Determine how many options are left for the second situation.\n3. Continue until all of the spots are filled.\n4. Multiply the numbers together.\nEXAMPLE3\nFinding the Number of Permutations Using the Multiplication Principle\nAt a swimming competition, nine swimmers compete in a race.\n How many ways can they place first, second, and third?\n How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there\nis only one contestant named Ariel.)"
	},
	{
		"chunk_id" : "00003850",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is only one contestant named Ariel.)\n How many ways can all nine swimmers line up for a photo?\nAccess for free at openstax.org\n13.5  Counting Principles 1285\nSolution\n Draw lines for each place.\nThere are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place.\nOnce first and second place have been won, there are 7 remaining options for third place.\nMultiply to find that there are 504 ways for the swimmers to place."
	},
	{
		"chunk_id" : "00003851",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Draw lines for describing each place.\nWe know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second\nplace, and then 7 remaining options for third place.\nMultiply to find that there are 56 ways for the swimmers to place if Ariel wins first.\n\nDraw lines for describing each place in the photo.\nThere are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person\nremains for the last spot."
	},
	{
		"chunk_id" : "00003852",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "remains for the last spot.\nThere are 362,880 possible permutations for the swimmers to line up.\nAnalysis\nNote that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of distinct objects\ncan always be found by\nA family of five is having portraits taken. Use the Multiplication Principle to find the following.\nTRY IT #3 How many ways can the family line up for the portrait?\nTRY IT #4 How many ways can the photographer line up 3 family members?"
	},
	{
		"chunk_id" : "00003853",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #5 How many ways can the family line up for the portrait if the parents are required to stand on each\nend?\nFinding the Number of Permutations ofnDistinct Objects Using a Formula\nFor some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many\nnumbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, lets look"
	},
	{
		"chunk_id" : "00003854",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "at two common notations for permutations. If we have a set of objects and we want to choose objects from the set in\norder, we write Another way to write this is a notation commonly seen on computers and calculators. To\ncalculate we begin by finding the number of ways to line up all objects. We then divide by to cancel\nout the items that we do not wish to line up.\nLets see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice"
	},
	{
		"chunk_id" : "00003855",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice\npresident, and any of the remaining four people could be elected treasurer. The number of ways this may be done is\nUsing factorials, we get the same result.\n1286 13  Sequences, Probability, and Counting Theory\nThere are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken\n3 at a time. The general formula is as follows."
	},
	{
		"chunk_id" : "00003856",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "3 at a time. The general formula is as follows.\nNote that the formula stills works if we are choosingall objects and placing them in order. In that case we would be\ndividing by or which we said earlier is equal to 1. So the number of permutations of objects taken at a\ntime is or just\nFormula for Permutations ofnDistinct Objects\nGiven distinct objects, the number of ways to select objects from the set in order is\n...\nHOW TO\nGiven a word problem, evaluate the possible permutations."
	},
	{
		"chunk_id" : "00003857",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1. Identify from the given information.\n2. Identify from the given information.\n3. Replace and in the formula with the given values.\n4. Evaluate.\nEXAMPLE4\nFinding the Number of Permutations Using the Formula\nA professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and\narrange the questions?\nSolution\nSubstitute and into the permutation formula and simplify.\nThere are 79,833,600 possible permutations of exam questions!\nAnalysis"
	},
	{
		"chunk_id" : "00003858",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Analysis\nWe can also use a calculator to find permutations. For this problem, we would enter 12, press the function, enter 9,\nand then press the equal sign. The function may be located under the MATH menu with probability commands.\nQ&A Could we have solvedExample 4using the Multiplication Principle?\nYes. We could have multiplied to find the same answer.\nA play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following."
	},
	{
		"chunk_id" : "00003859",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "TRY IT #6 How many ways can the 7 actors line up?\nTRY IT #7 How many ways can 5 of the 7 actors be chosen to line up?\nAccess for free at openstax.org\n13.5  Counting Principles 1287\nFind the Number of Combinations Using the Formula\nSo far, we have looked at problems asking us to put objects in order. There are many problems in which we want to\nselect a few objects from a group of objects, but we do not care about the order. When we are selecting objects and the"
	},
	{
		"chunk_id" : "00003860",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "order does not matter, we are dealing withcombinations. A selection of objects from a set of objects where the\norder does not matter can be written as Just as with permutations, can also be written as In this\ncase, the general formula is as follows.\nAn earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to\nselect 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number"
	},
	{
		"chunk_id" : "00003861",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "because selecting paintings 1, 2, 3 would be the same as selecting paintings 2, 3, 1. To find the number of ways to select\n3 of the 4 paintings, disregarding the order of the paintings, divide the number of permutations by the number of ways\nto order 3 paintings. There are ways to order 3 paintings. There are or 4 ways to select 3 of the 4\npaintings. This number makes sense because every time we are selecting 3 paintings, we arenotselecting 1 painting."
	},
	{
		"chunk_id" : "00003862",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "There are 4 paintings we could choosenotto select, so there are 4 ways to select 3 of the 4 paintings.\nFormula for Combinations ofnDistinct Objects\nGiven distinct objects, the number of ways to select objects from the set is\n...\nHOW TO\nGiven a number of options, determine the possible number of combinations.\n1. Identify from the given information.\n2. Identify from the given information.\n3. Replace and in the formula with the given values.\n4. Evaluate.\nEXAMPLE5"
	},
	{
		"chunk_id" : "00003863",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "4. Evaluate.\nEXAMPLE5\nFinding the Number of Combinations Using the Formula\nA fast food restaurant offers five side dish options. Your meal comes with two side dishes.\n How many ways can you select your side dishes?  How many ways can you select 3 side dishes?\nSolution\n We want to choose 2 side dishes from 5 options.  We want to choose 3 side dishes from 5 options.\nAnalysis\nWe can also use a graphing calculator to find combinations. Enter 5, then press enter 3, and then press the equal"
	},
	{
		"chunk_id" : "00003864",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sign. The function may be located under the MATH menu with probability commands.\nQ&A Is it a coincidence that parts (a) and (b) inExample 5have the same answers?\nNo. When we choose r objects from n objects, we arenotchoosing objects. Therefore,\n1288 13  Sequences, Probability, and Counting Theory\nTRY IT #8 An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for\na banana split?\nFinding the Number of Subsets of a Set"
	},
	{
		"chunk_id" : "00003865",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Finding the Number of Subsets of a Set\nWe have looked only at combination problems in which we chose exactly objects. In some problems, we want to\nconsider choosing every possible number of objects. Consider, for example, a pizza restaurant that offers 5 toppings.\nAny number of toppings can be ordered. How many different pizzas are possible?\nTo answer this question, we need to consider pizzas with any number of toppings. There is way to order a"
	},
	{
		"chunk_id" : "00003866",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "pizza with no toppings. There are ways to order a pizza with exactly one topping. If we continue this process,\nwe get\nThere are 32 possible pizzas. This result is equal to\nWe are presented with a sequence of choices. For each of the objects we have two choices: include it in the subset or\nnot. So for the whole subset we have made choices, each with two options. So there are a total of\npossible resulting subsets, all the way from the empty subset, which we obtain when we say no each time, to the"
	},
	{
		"chunk_id" : "00003867",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "original set itself, which we obtain when we say yes each time.\nFormula for the Number of Subsets of a Set\nA set containingndistinct objects has subsets.\nEXAMPLE6\nFinding the Number of Subsets of a Set\nA restaurant offers butter, cheese, chives, and sour cream as toppings for a baked potato. How many different ways are\nthere to order a potato?\nSolution\nWe are looking for the number of subsets of a set with 4 objects. Substitute into the formula.\nThere are 16 possible ways to order a potato."
	},
	{
		"chunk_id" : "00003868",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "There are 16 possible ways to order a potato.\nTRY IT #9 A sundae bar at a wedding has 6 toppings to choose from. Any number of toppings can be\nchosen. How many different sundaes are possible?\nFinding the Number of Permutations ofnNon-Distinct Objects\nWe have studied permutations where all of the objects involved were distinct. What happens if some of the objects are\nindistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be"
	},
	{
		"chunk_id" : "00003869",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ways to order the stickers. However, 4 of the stickers are identical stars, and 3 are identical moons. Because all of the\nobjects are not distinct, many of the permutations we counted are duplicates. The general formula for this situation\nis as follows.\nIn this example, we need to divide by the number of ways to order the 4 stars and the ways to order the 3 moons to find\nthe number of unique permutations of the stickers. There are ways to order the stars and ways to order the moon."
	},
	{
		"chunk_id" : "00003870",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "There are 3,326,400 ways to order the sheet of stickers.\nAccess for free at openstax.org\n13.5  Counting Principles 1289\nFormula for Finding the Number of Permutations ofnNon-Distinct Objects\nIf there are elements in a set and are alike, are alike, are alike, and so on through the number of\npermutations can be found by\nEXAMPLE7\nFinding the Number of Permutations ofnNon-Distinct Objects\nFind the number of rearrangements of the letters in the word DISTINCT.\nSolution"
	},
	{
		"chunk_id" : "00003871",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Solution\nThere are 8 letters. Both I and T are repeated 2 times. Substitute and into the formula.\nThere are 10,080 arrangements.\nTRY IT #10 Find the number of rearrangements of the letters in the word CARRIER.\nMEDIA\nAccess these online resources for additional instruction and practice with combinations and permutations.\nCombinations(http://openstax.org/l/combinations)\nPermutations(http://openstax.org/l/permutations)\n13.5 SECTION EXERCISES\nVerbal"
	},
	{
		"chunk_id" : "00003872",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.5 SECTION EXERCISES\nVerbal\nFor the following exercises, assume that there are ways an event can happen, ways an event can happen, and\nthat are non-overlapping.\n1. Use the Addition Principle of 2. Use the Multiplication\ncounting to explain how Principle of counting to\nmany ways event explain how many ways\ncan occur. event can occur.\nAnswer the following questions.\n3. When given two separate 4. Describe how the 5. What is the term for the"
	},
	{
		"chunk_id" : "00003873",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "events, how do we know permutation of objects arrangement that selects\nwhether to apply the differs from the objects from a set of\nAddition Principle or the permutation of choosing objects when the order of\nMultiplication Principle objects from a set of the objects is not\nwhen calculating possible objects. Include how each is important? What is the\noutcomes? What calculated. formula for calculating the\nconjunctions may help to number of possible\ndetermine which operations outcomes for this type of"
	},
	{
		"chunk_id" : "00003874",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to use? arrangement?\n1290 13  Sequences, Probability, and Counting Theory\nNumeric\nFor the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform\nthe calculations.\n6. Let the set 7. Let the set\nHow many ways are there to choose a positive or\nHow many ways are there to an odd number from\nchoose a negative or an even\nnumber from\n8. How many ways are there to 9. How many ways are there to 10. How many outcomes are"
	},
	{
		"chunk_id" : "00003875",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "pick a red ace or a club from pick a paint color from 5 possible from tossing a\na standard card playing shades of green, 4 shades of pair of coins?\ndeck? blue, or 7 shades of yellow?\n11. How many outcomes are 12. How many two-letter 13. How many ways are there\npossible from tossing a stringsthe first letter to construct a string of 3\ncoin and rolling a 6-sided from and the second digits if numbers can be\ndie? letter from can be repeated?\nformed from the sets\nand\n14. How many ways are there"
	},
	{
		"chunk_id" : "00003876",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "and\n14. How many ways are there\nto construct a string of 3\ndigits if numbers cannot be\nrepeated?\nFor the following exercises, compute the value of the expression.\n15. 16. 17.\n18. 19. 20.\n21. 22. 23.\n24.\nFor the following exercises, find the number of subsets in each given set.\n25. 26. 27. A set containing 5 distinct\nnumbers, 4 distinct letters,\nand 3 distinct symbols\n28. The set of even numbers 29. The set of two-digit\nfrom 2 to 28 numbers between 1 and\n100 containing the digit 0"
	},
	{
		"chunk_id" : "00003877",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "100 containing the digit 0\nAccess for free at openstax.org\n13.5  Counting Principles 1291\nFor the following exercises, find the distinct number of arrangements.\n30. The letters in the word 31. The letters in the word 32. The letters in the word\njuggernaut academia academia that begin and\nend in a\n33. The symbols in the string 34. The symbols in the string\n#,#,#,@,@,$,$,$,%,%,%,% #,#,#,@,@,$,$,$,%,%,%,%\nthat begin and end with\n%\nExtensions"
	},
	{
		"chunk_id" : "00003878",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "that begin and end with\n%\nExtensions\n35. The set, consists of 36. The number of 5-element 37. Can ever equal\nwhole subsets from a set Explain.\nnumbers, each being the containing elements is\nsame number of digits equal to the number of\nlong. How many digits long 6-element subsets from\nis a number from (Hint: the same set. What is the\nuse the fact that a whole value of (Hint:the order\nnumber cannot start with in which the elements for\nthe digit 0.) the subsets are chosen is\nnot important.)"
	},
	{
		"chunk_id" : "00003879",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "not important.)\n38. Suppose a set has 2,048 39. How many arrangements\nsubsets. How many distinct can be made from the\nobjects are contained in letters of the word\nmountains if all the\nvowels must form a string?\nReal-World Applications\n40. A family consisting of 2 41. A cell phone company 42. In horse racing, a trifecta\nparents and 3 children is to offers 6 different voice occurs when a bettor wins\npose for a picture with 2 packages and 8 different by selecting the first three"
	},
	{
		"chunk_id" : "00003880",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "family members in the data packages. Of those, 3 finishers in the exact order\nfront and 3 in the back. packages include both (1st place, 2nd place, and\nvoice and data. How many 3rd place). How many\n How many\nways are there to choose different trifectas are\narrangements are possible\neither voice or data, but possible if there are 14\nwith no restrictions?\nnot both? horses in a race?\n How many\narrangements are possible\nif the parents must sit in\nthe front?\n How many\narrangements are possible"
	},
	{
		"chunk_id" : "00003881",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the front?\n How many\narrangements are possible\nif the parents must be next\nto each other?\n1292 13  Sequences, Probability, and Counting Theory\n43. A wholesale T-shirt 44. Hector wants to place 45. An art store has 4 brands\ncompany offers sizes small, billboard advertisements of paint pens in 12\nmedium, large, and extra- throughout the county for different colors and 3 types\nlarge in organic or non- his new business. How of ink. How many paint"
	},
	{
		"chunk_id" : "00003882",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "organic cotton and colors many ways can Hector pens are there to choose\nwhite, black, gray, blue, choose 15 neighborhoods from?\nand red. How many to advertise in if there are\ndifferent T-shirts are there 30 neighborhoods in the\nto choose from? county?\n46. How many ways can a 47. How many ways can a 48. A conductor needs 5\ncommittee of 3 freshmen baseball coach arrange the cellists and 5 violinists to\nand 4 juniors be formed order of 9 batters if there play at a diplomatic event."
	},
	{
		"chunk_id" : "00003883",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from a group of are 15 players on the To do this, he ranks the\nfreshmen and juniors? team? orchestras 10 cellists and\n16 violinists in order of\nmusical proficiency. What\nis the ratio of the total\ncellist rankings possible to\nthe total violinist rankings\npossible?\n49. A motorcycle shop has 10 50. A skateboard shop stocks 51. Just-For-Kicks Sneaker\nchoppers, 6 bobbers, and 5 10 types of board decks, 3 Company offers an online\ncaf racersdifferent types of trucks, and 4 types customizing service. How"
	},
	{
		"chunk_id" : "00003884",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "types of vintage of wheels. How many many ways are there to\nmotorcycles. How many different skateboards can design a custom pair of\nways can the shop choose be constructed? Just-For-Kicks sneakers if a\n3 choppers, 5 bobbers, and customer can choose from\n2 caf racers for a weekend a basic shoe up to 11\nshowcase? customizable options?\n52. A car wash offers the 53. Suni bought 20 plants to 54. How many unique ways\nfollowing optional services arrange along the border can a string of Christmas"
	},
	{
		"chunk_id" : "00003885",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to the basic wash: clear of her garden. How many lights be arranged from 9\ncoat wax, triple foam distinct arrangements can red, 10 green, 6 white, and\npolish, undercarriage she make if the plants are 12 gold color bulbs?\nwash, rust inhibitor, wheel comprised of 6 tulips, 6\nbrightener, air freshener, roses, and 8 daisies?\nand interior shampoo. How\nmany washes are possible\nif any number of options\ncan be added to the basic\nwash?\n13.6 Binomial Theorem\nLearning Objectives\nIn this section, you will:"
	},
	{
		"chunk_id" : "00003886",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Learning Objectives\nIn this section, you will:\nApply the Binomial Theorem.\nA polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials\nto powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a\nshortcut that will allow us to find without multiplying the binomial by itself times.\nAccess for free at openstax.org\n13.6  Binomial Theorem 1293\nIdentifying Binomial Coefficients"
	},
	{
		"chunk_id" : "00003887",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Identifying Binomial Coefficients\nInCounting Principles, we studiedcombinations. In the shortcut to finding we will need to use combinations to\nfind the coefficients that will appear in the expansion of the binomial. In this case, we use the notation instead of\nbut it can be calculated in the same way. So\nThe combination is called abinomial coefficient. An example of a binomial coefficient is\nBinomial Coefficients\nIf and are integers greater than or equal to 0 with then thebinomial coefficientis"
	},
	{
		"chunk_id" : "00003888",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Q&A Is a binomial coefficient always a whole number?\nYes. Just as the number of combinations must always be a whole number, a binomial coefficient will\nalways be a whole number.\nEXAMPLE1\nFinding Binomial Coefficients\nFind each binomial coefficient.\n  \nSolution\nUse the formula to calculate each binomial coefficient. You can also use the function on your calculator.\n  \nAnalysis\nNotice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you"
	},
	{
		"chunk_id" : "00003889",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with\ncombinations.\nTRY IT #1 Find each binomial coefficient.\n \nUsing the Binomial Theorem\nWhen we expand by multiplying, the result is called abinomial expansion, and it includes binomial coefficients.\nIf we wanted to expand we might multiply by itself fifty-two times. This could take hours! If we"
	},
	{
		"chunk_id" : "00003890",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more\ncomplicated binomial expansions.\n1294 13  Sequences, Probability, and Counting Theory\nFirst, lets examine the exponents. With each successive term, the exponent for decreases and the exponent for\nincreases. The sum of the two exponents is for each term.\nNext, lets examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The"
	},
	{
		"chunk_id" : "00003891",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "coefficients follow a pattern:\nThese patterns lead us to theBinomial Theorem, which can be used to expand any binomial.\nAnother way to see the coefficients is to examine the expansion of a binomial in general form, to successive\npowers 1, 2, 3, and 4.\nCan you guess the next expansion for the binomial\nFigure1\nSeeFigure 1, which illustrates the following:\n There are terms in the expansion of\n The degree (or sum of the exponents) for each term is\n The powers on begin with and decrease to 0."
	},
	{
		"chunk_id" : "00003892",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The powers on begin with and decrease to 0.\n The powers on begin with 0 and increase to\n The coefficients are symmetric.\nTo determine the expansion on we see thus, there will be 5+1 = 6 terms. Each term has a combined\ndegree of 5. In descending order for powers of the pattern is as follows:\n Introduce and then for each successive term reduce the exponent on by 1 until is reached.\n Introduce and then increase the exponent on by 1 until is reached.\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003893",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n13.6  Binomial Theorem 1295\nThe next expansion would be\nBut where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an\narray known asPascal's Triangle, shown inFigure 2. Pascal didn't invent the triangle. The underlying principles had been\ndeveloped and written about for over 1500 years, first by the Indian mathematician (and poet) Pingala in the second"
	},
	{
		"chunk_id" : "00003894",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "century BCE. Others throughout Asia and Europe worked with the concepts throughout, and the triangle was first\npublished in its graphical form by Omar Khayyam, an Iranian mathematician and astronomer, for whom the triangle is\nnamed in Iran. French mathematician Blaise Pascal repopularized it when he republished it and used it to solve a\nnumber of probability problems.\nFigure2\nTo generate Pascals Triangle, we start by writing a 1. In the row below, row 2, we write two 1s. In the 3rdrow, flank the"
	},
	{
		"chunk_id" : "00003895",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "ends of the rows with 1s, and add to find the middle number, 2. In the row, flank the ends of the row with 1s.\nEach element in the triangle is the sum of the two elements immediately above it.\nTo see the connection between Pascals Triangle and binomial coefficients, let us revisit the expansion of the binomials\nin general form.\nThe Binomial Theorem\nTheBinomial Theoremis a formula that can be used to expand any binomial.\n...\nHOW TO\nGiven a binomial, write it in expanded form."
	},
	{
		"chunk_id" : "00003896",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Given a binomial, write it in expanded form.\n1. Determine the value of according to the exponent.\n2. Evaluate the through using the Binomial Theorem formula.\n3. Simplify.\n1296 13  Sequences, Probability, and Counting Theory\nEXAMPLE2\nExpanding a Binomial\nWrite in expanded form.\n \nSolution\n Substitute into the formula. Evaluate the through terms. Simplify.\n Substitute into the formula. Evaluate the through terms. Notice that is in the place that was"
	},
	{
		"chunk_id" : "00003897",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "occupied by and that is in the place that was occupied by So we substitute them. Simplify.\nAnalysis\nNotice the alternating signs in part b. This happens because raised to odd powers is negative, but raised to\neven powers is positive. This will occur whenever the binomial contains a subtraction sign.\nTRY IT #2 Write in expanded form.\n \nUsing the Binomial Theorem to Find a Single Term\nExpanding a binomial with a high exponent such as can be a lengthy process."
	},
	{
		"chunk_id" : "00003898",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial\nto find a single specific term.\nNote the pattern of coefficients in the expansion of\nThe second term is The third term is We can generalize this result.\nThe (r+1)th Term of a Binomial Expansion\nThe term of thebinomial expansionof is:\n...\nHOW TO\nGiven a binomial, write a specific term without fully expanding.\n1. Determine the value of according to the exponent."
	},
	{
		"chunk_id" : "00003899",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n13.6  Binomial Theorem 1297\n2. Determine\n3. Determine\n4. Replace in the formula for the term of the binomial expansion.\nEXAMPLE3\nWriting a Given Term of a Binomial Expansion\nFind the tenth term of without fully expanding the binomial.\nSolution\nBecause we are looking for the tenth term, we will use in our calculations.\nTRY IT #3 Find the sixth term of without fully expanding the binomial.\nMEDIA"
	},
	{
		"chunk_id" : "00003900",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with binomial expansion.\nThe Binomial Theorem(http://openstax.org/l/binomialtheorem)\nBinomial Theorem Example(http://openstax.org/l/btexample)\n13.6 SECTION EXERCISES\nVerbal\n1. What is a binomial 2. What role do binomial 3. What is the Binomial\ncoefficient, and how it is coefficients play in a Theorem and what is its\ncalculated? binomial expansion? Are use?\nthey restricted to any type of\nnumber?\n4. When is it an advantage to"
	},
	{
		"chunk_id" : "00003901",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "number?\n4. When is it an advantage to\nuse the Binomial Theorem?\nExplain.\nAlgebraic\nFor the following exercises, evaluate the binomial coefficient.\n5. 6. 7.\n8. 9. 10.\n1298 13  Sequences, Probability, and Counting Theory\n11. 12.\nFor the following exercises, use the Binomial Theorem to expand each binomial.\n13. 14. 15.\n16. 17. 18.\n19. 20. 21.\n22.\nFor the following exercises, use the Binomial Theorem to write the first three terms of each binomial.\n23. 24. 25.\n26. 27. 28.\n29."
	},
	{
		"chunk_id" : "00003902",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "23. 24. 25.\n26. 27. 28.\n29.\nFor the following exercises, find the indicated term of each binomial without fully expanding the binomial.\n30. The fourth term of 31. The fourth term of 32. The third term of\n33. The eighth term of 34. The seventh term of 35. The fifth term of\n36. The tenth term of 37. The ninth term of 38. The fourth term of\n39. The eighth term of\nGraphical\nFor the following exercises, use the Binomial Theorem to expand the binomial Then find and graph\neach indicated sum on one set of axes."
	},
	{
		"chunk_id" : "00003903",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "each indicated sum on one set of axes.\n40. Find and graph such 41. Find and graph such 42. Find and graph such\nthat is the first term that is the sum of the that is the sum of the\nof the expansion. first two terms of the first three terms of the\nexpansion. expansion.\nAccess for free at openstax.org\n13.7  Probability 1299\n43. Find and graph such 44. Find and graph such\nthat is the sum of the that is the sum of the\nfirst four terms of the first five terms of the\nexpansion. expansion.\nExtensions"
	},
	{
		"chunk_id" : "00003904",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "expansion. expansion.\nExtensions\n45. In the expansion of each term has the 46. In the expansion of\nthe coefficient of\nform , where successively takes on\nis the same as the\ncoefficient of which other\nthe value If what is\nterm?\nthe corresponding term?\n47. Consider the expansion of 48. Find 49. Which expression cannot\nWhat is the be expanded using the\nand write the answer as a\nexponent of in the Binomial Theorem?\nbinomial coefficient in the\nterm? Explain.\nform Prove it.\n\nHint:Use the fact that, for "
	},
	{
		"chunk_id" : "00003905",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "form Prove it.\n\nHint:Use the fact that, for \nany integer such that\n\n\n13.7 Probability\nLearning Objectives\nIn this section, you will:\nConstruct probability models.\nCompute probabilities of equally likely outcomes.\nCompute probabilities of the union of two events.\nUse the complement rule to find probabilities.\nCompute probability using counting theory.\nFigure1 An example of a spaghetti model, which can be used to predict possible paths of a tropical storm.1"
	},
	{
		"chunk_id" : "00003906",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1300 13  Sequences, Probability, and Counting Theory\nResidents of the Southeastern United States are all too familiar with charts, known as spaghetti models, such as the one\ninFigure 1. They combine a collection of weather data to predict the most likely path of a hurricane. Each colored line\nrepresents one possible path. The group of squiggly lines can begin to resemble strands of spaghetti, hence the name.\nIn this section, we will investigate methods for making these types of predictions."
	},
	{
		"chunk_id" : "00003907",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Constructing Probability Models\nSuppose we roll a six-sided number cube. Rolling a number cube is an example of anexperiment, or an activity with an\nobservable result. The numbers on the cube are possible results, oroutcomes, of this experiment. The set of all possible\noutcomes of an experiment is called thesample spaceof the experiment. The sample space for this experiment is\nAneventis any subset of a sample space."
	},
	{
		"chunk_id" : "00003908",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Aneventis any subset of a sample space.\nThe likelihood of an event is known asprobability. The probability of an event is a number that always satisfies\nwhere 0 indicates an impossible event and 1 indicates a certain event. Aprobability modelis a mathematical\ndescription of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a 1%\nchance of winning a raffle and a 99% chance of losing the raffle, a probability model would look much likeTable 1."
	},
	{
		"chunk_id" : "00003909",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Outcome Probability\nWinning the raffle 1%\nLosing the raffle 99%\nTable1\nThe sum of the probabilities listed in a probability model must equal 1, or 100%.\n...\nHOW TO\nGiven a probability event where each event is equally likely, construct a probability model.\n1. Identify every outcome.\n2. Determine the total number of possible outcomes.\n3. Compare each outcome to the total number of possible outcomes.\nEXAMPLE1\nConstructing a Probability Model"
	},
	{
		"chunk_id" : "00003910",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "EXAMPLE1\nConstructing a Probability Model\nConstruct a probability model for rolling a single, fair die, with the event being the number shown on the die.\nSolution\nBegin by making a list of all possible outcomes for the experiment. The possible outcomes are the numbers that can be\nrolled: 1, 2, 3, 4, 5, and 6. There are six possible outcomes that make up the sample space.\nAssign probabilities to each outcome in the sample space by determining a ratio of the outcome to the number of"
	},
	{
		"chunk_id" : "00003911",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "possible outcomes. There is one of each of the six numbers on the cube, and there is no reason to think that any\nparticular face is more likely to show up than any other one, so the probability of rolling any number is\nOutcome Roll of 1 Roll of 2 Roll of 3 Roll of 4 Roll of 5 Roll of 6\nProbability\nTable2\n1 The figure is for illustrative purposes only and does not model any particular storm.\nAccess for free at openstax.org\n13.7  Probability 1301"
	},
	{
		"chunk_id" : "00003912",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13.7  Probability 1301\nQ&A Do probabilities always have to be expressed as fractions?\nNo. Probabilities can be expressed as fractions, decimals, or percents. Probability must always be a\nnumber between 0 and 1, inclusive of 0 and 1.\nTRY IT #1 Construct a probability model for tossing a fair coin.\nComputing Probabilities of Equally Likely Outcomes\nLet be a sample space for an experiment. When investigating probability, an event is any subset of When the"
	},
	{
		"chunk_id" : "00003913",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of\noutcomes in the event by the total number of outcomes in Suppose a number cube is rolled, and we are interested in\nfinding the probability of the event rolling a number less than or equal to 4. There are 4 possible outcomes in the\nevent and 6 possible outcomes in so the probability of the event is\nComputing the Probability of an Event with Equally Likely Outcomes"
	},
	{
		"chunk_id" : "00003914",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The probability of an event in an experiment with sample space with equally likely outcomes is given by\nis a subset of so it is always true that\nEXAMPLE2\nComputing the Probability of an Event with Equally Likely Outcomes\nA six-sided number cube is rolled. Find the probability of rolling an odd number.\nSolution\nThe event rolling an odd number contains three outcomes. There are 6 equally likely outcomes in the sample space.\nDivide to find the probability of the event."
	},
	{
		"chunk_id" : "00003915",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Divide to find the probability of the event.\nTRY IT #2 A number cube is rolled. Find the probability of rolling a number greater than 2.\nComputing the Probability of the Union of Two Events\nWe are often interested in finding the probability that one of multiple events occurs. Suppose we are playing a card\ngame, and we will win if the next card drawn is either a heart or a king. We would be interested in finding the probability"
	},
	{
		"chunk_id" : "00003916",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "of the next card being a heart or a king. Theunion of two events is the event that occurs if\neither or both events occur.\nSuppose the spinner inFigure 2is spun. We want to find the probability of spinning orange or spinning a\n1302 13  Sequences, Probability, and Counting Theory\nFigure2\nThere are a total of 6 sections, and 3 of them are orange. So the probability of spinning orange is There are a\ntotal of 6 sections, and 2 of them have a So the probability of spinning a is If we added these two"
	},
	{
		"chunk_id" : "00003917",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "probabilities, we would be counting the sector that is both orange and a twice. To find the probability of spinning an\norange or a we need to subtract the probability that the sector is both orange and has a\nThe probability of spinning orange or a is\nProbability of the Union of Two Events\nThe probability of the union of two events and (written ) equals the sum of the probability of and the\nprobability of minus the probability of and occurring together which is called theintersectionof and and"
	},
	{
		"chunk_id" : "00003918",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is written as ).\nEXAMPLE3\nComputing the Probability of the Union of Two Events\nA card is drawn from a standard deck. Find the probability of drawing a heart or a 7.\nSolution\nA standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability of drawing a heart\nis There are four 7s in a standard deck, and there are a total of 52 cards. So the probability of drawing a 7 is"
	},
	{
		"chunk_id" : "00003919",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "The only card in the deck that is both a heart and a 7 is the 7 of hearts, so the probability of drawing both a heart and a 7\nis Substitute into the formula.\nThe probability of drawing a heart or a 7 is\nTRY IT #3 A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.\nComputing the Probability of Mutually Exclusive Events\nSuppose the spinner inFigure 2is spun again, but this time we are interested in the probability of spinning an orange or"
	},
	{
		"chunk_id" : "00003920",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a There are no sectors that are both orange and contain a so these two events have no outcomes in common.\nEvents are said to bemutually exclusive eventswhen they have no outcomes in common. Because there is no overlap,\nthere is nothing to subtract, so the general formula is\nAccess for free at openstax.org\n13.7  Probability 1303\nNotice that with mutually exclusive events, the intersection of and is the empty set. The probability of spinning an"
	},
	{
		"chunk_id" : "00003921",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "orange is and the probability of spinning a is We can find the probability of spinning an orange or a simply\nby adding the two probabilities.\nThe probability of spinning an orange or a is\nProbability of the Union of Mutually Exclusive Events\nThe probability of the union of twomutually exclusiveevents is given by\n...\nHOW TO\nGiven a set of events, compute the probability of the union of mutually exclusive events.\n1. Determine the total number of outcomes for the first event."
	},
	{
		"chunk_id" : "00003922",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "2. Find the probability of the first event.\n3. Determine the total number of outcomes for the second event.\n4. Find the probability of the second event.\n5. Add the probabilities.\nEXAMPLE4\nComputing the Probability of the Union of Mutually Exclusive Events\nA card is drawn from a standard deck. Find the probability of drawing a heart or a spade.\nSolution\nThe events drawing a heart and drawing a spade are mutually exclusive because they cannot occur at the same"
	},
	{
		"chunk_id" : "00003923",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "time. The probability of drawing a heart is and the probability of drawing a spade is also so the probability of\ndrawing a heart or a spade is\nTRY IT #4 A card is drawn from a standard deck. Find the probability of drawing an ace or a king.\nUsing the Complement Rule to Compute Probabilities\nWe have discussed how to calculate the probability that an event will happen. Sometimes, we are interested in finding"
	},
	{
		"chunk_id" : "00003924",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the probability that an event willnothappen. Thecomplement of an event denoted is the set of outcomes in the\nsample space that are not in For example, suppose we are interested in the probability that a horse will lose a race. If\nevent is the horse winning the race, then the complement of event is the horse losing the race.\nTo find the probability that the horse loses the race, we need to use the fact that the sum of all probabilities in a\nprobability model must be 1."
	},
	{
		"chunk_id" : "00003925",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "probability model must be 1.\nThe probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the\nprobability of the horse winning the race is the probability of the horse losing the race is simply\n1304 13  Sequences, Probability, and Counting Theory\nThe Complement Rule\nThe probability that thecomplement of an eventwill occur is given by\nEXAMPLE5\nUsing the Complement Rule to Calculate Probabilities\nTwo six-sided number cubes are rolled."
	},
	{
		"chunk_id" : "00003926",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Two six-sided number cubes are rolled.\n Find the probability that the sum of the numbers rolled is less than or equal to 3.\n Find the probability that the sum of the numbers rolled is greater than 3.\nSolution\nThe first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes,\nand each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are or"
	},
	{
		"chunk_id" : "00003927",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.\nTable3\n We need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1,\n1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is\n Rather than listing all the possibilities, we can use the Complement Rule. Because we have already found the"
	},
	{
		"chunk_id" : "00003928",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that\nthe sum of the numbers rolled is greater than 3.\nTRY IT #5 Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is\nless than 10.\nComputing Probability Using Counting Theory\nMany interesting probability problems involve counting principles, permutations, and combinations. In these problems,"
	},
	{
		"chunk_id" : "00003929",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "we will use permutations and combinations to find the number of elements in events and sample spaces. These\nAccess for free at openstax.org\n13.7  Probability 1305\nproblems can be complicated, but they can be made easier by breaking them down into smaller counting problems.\nAssume, for example, that a store has 8 cellular phones and that 3 of those are defective. We might want to find the\nprobability that a couple purchasing 2 phones receives 2 phones that are not defective. To solve this problem, we need"
	},
	{
		"chunk_id" : "00003930",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to calculate all of the ways to select 2 phones that are not defective as well as all of the ways to select 2 phones. There\nare 5 phones that are not defective, so there are ways to select 2 phones that are not defective. There are 8\nphones, so there are ways to select 2 phones. The probability of selecting 2 phones that are not defective is:\nEXAMPLE6\nComputing Probability Using Counting Theory\nA child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears."
	},
	{
		"chunk_id" : "00003931",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Find the probability that only bears are chosen.  Find the probability that 2 bears and 3 dogs are chosen.\n Find the probability that at least 2 dogs are chosen.\nSolution\n We need to count the number of ways to choose only bears and the total number of possible ways to select 5\ntoys. There are 6 bears, so there are ways to choose 5 bears. There are 14 toys, so there are ways to\nchoose any 5 toys."
	},
	{
		"chunk_id" : "00003932",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "choose any 5 toys.\n We need to count the number of ways to choose 2 bears and 3 dogs and the total number of possible ways to\nselect 5 toys. There are 6 bears, so there are ways to choose 2 bears. There are 5 dogs, so there are\nways to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication\nPrinciple. There are ways to choose 2 bears and 3 dogs. We can use this result to find the probability."
	},
	{
		"chunk_id" : "00003933",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " It is often easiest to solve at least problems using the Complement Rule. We will begin by finding the\nprobability that fewer than 2 dogs are chosen. If less than 2 dogs are chosen, then either no dogs could be chosen, or\n1 dog could be chosen.\nWhen no dogs are chosen, all 5 toys come from the 9 toys that are not dogs. There are ways to choose toys\nfrom the 9 toys that are not dogs. Since there are 14 toys, there are ways to choose the 5 toys from all of the\ntoys."
	},
	{
		"chunk_id" : "00003934",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "toys.\nIf there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs.\nSince we are choosing both dogs and other toys at the same time, we will use the Multiplication Principle. There are\nways to choose 1 dog and 1 other toy.\nBecause these events would not occur together and are therefore mutually exclusive, we add the probabilities to find\nthe probability that fewer than 2 dogs are chosen."
	},
	{
		"chunk_id" : "00003935",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "We then subtract that probability from 1 to find the probability that at least 2 dogs are chosen.\n1306 13  Sequences, Probability, and Counting Theory\nTRY IT #6 A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow\ngumballs, and 2 green gumballs.\n Find the probability that all 3 gumballs selected are purple.\n Find the probability that no yellow gumballs are selected.\n Find the probability that at least 1 yellow gumball is selected.\nMEDIA"
	},
	{
		"chunk_id" : "00003936",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "MEDIA\nAccess these online resources for additional instruction and practice with probability.\nIntroduction to Probability(http://openstax.org/l/introprob)\nDetermining Probability(http://openstax.org/l/determineprob)\n13.7 SECTION EXERCISES\nVerbal\n1. What term is used to 2. What is a sample space? 3. What is an experiment?\nexpress the likelihood of an\nevent occurring? Are there\nrestrictions on its values? If\nso, what are they? If not,\nexplain.\n4. What is the difference 5. Theunion of two setsis"
	},
	{
		"chunk_id" : "00003937",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "between events and defined as a set of elements\noutcomes? Give an example that are present in at least\nof both using the sample one of the sets. How is this\nspace of tossing a coin 50 similar to the definition used\ntimes. for theunion of two events\nfrom a probability model?\nHow is it different?\nNumeric\nFor the following exercises, use the spinner shown inFigure 3to find the probabilities indicated.\nFigure3\n6. Landing on red 7. Landing on a vowel 8. Not landing on blue"
	},
	{
		"chunk_id" : "00003938",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "9. Landing on purple or a 10. Landing on blue or a vowel 11. Landing on green or blue\nvowel\nAccess for free at openstax.org\n13.7  Probability 1307\n12. Landing on yellow or a 13. Not landing on yellow or a\nconsonant consonant\nFor the following exercises, two coins are tossed.\n14. What is the sample space? 15. Find the probability of 16. Find the probability of\ntossing two heads. tossing exactly one tail.\n17. Find the probability of\ntossing at least one tail."
	},
	{
		"chunk_id" : "00003939",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "tossing at least one tail.\nFor the following exercises, four coins are tossed.\n18. What is the sample space? 19. Find the probability of 20. Find the probability of\ntossing exactly two heads. tossing exactly three\nheads.\n21. Find the probability of 22. Find the probability of 23. Find the probability of\ntossing four heads or four tossing all tails. tossing not all tails.\ntails.\n24. Find the probability of 25. Find the probability of\ntossing exactly two heads tossing either two heads or"
	},
	{
		"chunk_id" : "00003940",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or at least two tails. three heads.\nFor the following exercises, one card is drawn from a standard deck of cards. Find the probability of drawing the\nfollowing:\n26. A club 27. A two 28. Six or seven\n29. Red six 30. An ace or a diamond 31. A non-ace\n32. A heart or a non-jack\nFor the following exercises, two dice are rolled, and the results are summed.\n33. Construct a table showing the sample space of 34. Find the probability of rolling a sum of\noutcomes and sums."
	},
	{
		"chunk_id" : "00003941",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "outcomes and sums.\n35. Find the probability of rolling at least one four or a 36. Find the probability of rolling an odd sum less\nsum of than\n37. Find the probability of rolling a sum greater than 38. Find the probability of\nor equal to rolling a sum less than\n39. Find the probability of 40. Find the probability of 41. Find the probability of\nrolling a sum less than or rolling a sum between rolling a sum of or\ngreater than and inclusive.\n1308 13  Sequences, Probability, and Counting Theory"
	},
	{
		"chunk_id" : "00003942",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "42. Find the probability of\nrolling any sum other than\nor\nFor the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the\nfollowing:\n43. A head on the coin or a 44. A tail on the coin or red ace 45. A head on the coin or a\nclub face card\n46. No aces\nFor the following exercises, use this scenario: a bag of M&Ms contains blue, brown, orange, yellow, red, and\ngreen M&Ms. Reaching into the bag, a person grabs 5 M&Ms."
	},
	{
		"chunk_id" : "00003943",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "47. What is the probability of 48. What is the probability of 49. What is the probability of\ngetting all blue M&Ms? getting blue M&Ms? getting blue M&Ms?\n50. What is the probability of\ngetting no brown M&Ms?\nExtensions\nUse the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting numbers\nfrom the numbers to After the player makes his selections, winning numbers are randomly selected from"
	},
	{
		"chunk_id" : "00003944",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "numbers to A win occurs if the player has correctly selected or of the winning numbers. (Round all\nanswers to the nearest hundredth of a percent.)\n51. What is the percent chance 52. What is the percent chance 53. What is the percent chance\nthat a player selects exactly that a player selects exactly that a player selects all 5\n3 winning numbers? 4 winning numbers? winning numbers?\n54. What is the percent chance 55. How much less is a players\nof winning? chance of selecting 3\nwinning numbers than the"
	},
	{
		"chunk_id" : "00003945",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "winning numbers than the\nchance of selecting either 4\nor 5 winning numbers?\nReal-World Applications\nUse this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and\nabout 40 million were elderly (aged 65 and over).2\n56. If you meet a U.S. citizen, 57. If you meet five U.S. 58. If you meet five U.S.\nwhat is the percent chance citizens, what is the citizens, what is the\nthat the person is elderly? percent chance that exactly percent chance that three"
	},
	{
		"chunk_id" : "00003946",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "(Round to the nearest one is elderly? (Round to are elderly? (Round to the\ntenth of a percent.) the nearest tenth of a nearest tenth of a percent.)\npercent.)\n2 United States Census Bureau. http://www.census.gov\nAccess for free at openstax.org\n13.7  Probability 1309\n59. If you meet five U.S. 60. It is predicted that by 2030,\ncitizens, what is the one in five U.S. citizens will\npercent chance that four be elderly. How much\nare elderly? (Round to the greater will the chances of"
	},
	{
		"chunk_id" : "00003947",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nearest thousandth of a meeting an elderly person\npercent.) be at that time? What\npolicy changes do you\nforesee if these statistics\nhold true?\n1310 13  Chapter Review\nChapter Review\nKey Terms\nAddition Principle if one event can occur in ways and a second event with no common outcomes can occur in\nways, then the first or second event can occur in ways\nannuity an investment in which the purchaser makes a sequence of periodic, equal payments"
	},
	{
		"chunk_id" : "00003948",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "arithmetic sequence a sequence in which the difference between any two consecutive terms is a constant\narithmetic series the sum of the terms in an arithmetic sequence\nbinomial coefficient the number of ways to chooserobjects fromnobjects where order does not matter; equivalent\nto denoted\nbinomial expansion the result of expanding by multiplying\nBinomial Theorem a formula that can be used to expand any binomial\ncombination a selection of objects in which order does not matter"
	},
	{
		"chunk_id" : "00003949",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common difference the difference between any two consecutive terms in an arithmetic sequence\ncommon ratio the ratio between any two consecutive terms in a geometric sequence\ncomplement of an event the set of outcomes in the sample space that are not in the event\ndiverge a series is said to diverge if the sum is not a real number\nevent any subset of a sample space\nexperiment an activity with an observable result"
	},
	{
		"chunk_id" : "00003950",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "experiment an activity with an observable result\nexplicit formula a formula that defines each term of a sequence in terms of its position in the sequence\nfinite sequence a function whose domain consists of a finite subset of the positive integers for some\npositive integer\nFundamental Counting Principle if one event can occur in ways and a second event can occur in ways after the\nfirst event has occurred, then the two events can occur in ways; also known as the Multiplication Principle"
	},
	{
		"chunk_id" : "00003951",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "geometric sequence a sequence in which the ratio of a term to a previous term is a constant\ngeometric series the sum of the terms in a geometric sequence\nindex of summation in summation notation, the variable used in the explicit formula for the terms of a series and\nwritten below the sigma with the lower limit of summation\ninfinite sequence a function whose domain is the set of positive integers\ninfinite series the sum of the terms in an infinite sequence"
	},
	{
		"chunk_id" : "00003952",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "lower limit of summation the number used in the explicit formula to find the first term in a series\nMultiplication Principle if one event can occur in ways and a second event can occur in ways after the first event\nhas occurred, then the two events can occur in ways; also known as the Fundamental Counting Principle\nmutually exclusive events events that have no outcomes in common\nn factorial the product of all the positive integers from 1 to\nnth partial sum the sum of the first terms of a sequence"
	},
	{
		"chunk_id" : "00003953",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "nth term of a sequence a formula for the general term of a sequence\noutcomes the possible results of an experiment\npermutation a selection of objects in which order matters\nprobability a number from 0 to 1 indicating the likelihood of an event\nprobability model a mathematical description of an experiment listing all possible outcomes and their associated\nprobabilities\nrecursive formula a formula that defines each term of a sequence using previous term(s)"
	},
	{
		"chunk_id" : "00003954",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sample space the set of all possible outcomes of an experiment\nsequence a function whose domain is a subset of the positive integers\nseries the sum of the terms in a sequence\nsummation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies\nthe first and last terms in the series\nterm a number in a sequence\nunion of two events the event that occurs if either or both events occur"
	},
	{
		"chunk_id" : "00003955",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "upper limit of summation the number used in the explicit formula to find the last term in a series\nAccess for free at openstax.org\n13  Chapter Review 1311\nKey Equations\nFormula for a factorial\nrecursive formula for nth term of an arithmetic sequence\nexplicit formula for nth term of an arithmetic sequence\nrecursive formula for term of a geometric sequence\nexplicit formula for term of a geometric sequence\nsum of the first terms of an arithmetic series\nsum of the first terms of a geometric series"
	},
	{
		"chunk_id" : "00003956",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sum of the first terms of a geometric series\nsum of an infinite geometric series with\nnumber of permutations of distinct objects taken at a time\nnumber of combinations of distinct objects taken at a time\nnumber of permutations of non-distinct objects\nBinomial Theorem\nterm of a binomial expansion\nprobability of an event with equally likely outcomes\nprobability of the union of two events\nprobability of the union of mutually exclusive events\nprobability of the complement of an event\n1312 13  Chapter Review"
	},
	{
		"chunk_id" : "00003957",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1312 13  Chapter Review\nKey Concepts\n13.1Sequences and Their Notations\n A sequence is a list of numbers, called terms, written in a specific order.\n Explicit formulas define each term of a sequence using the position of the term. SeeExample 1,Example 2, and\nExample 3.\n An explicit formula for the term of a sequence can be written by analyzing the pattern of several terms. See\nExample 4.\n Recursive formulas define each term of a sequence using previous terms."
	},
	{
		"chunk_id" : "00003958",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Recursive formulas must state the initial term, or terms, of a sequence.\n A set of terms can be written by using a recursive formula. SeeExample 5andExample 6.\n A factorial is a mathematical operation that can be defined recursively.\n The factorial of is the product of all integers from 1 to SeeExample 7.\n13.2Arithmetic Sequences\n An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant."
	},
	{
		"chunk_id" : "00003959",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The constant between two consecutive terms is called the common difference.\n The common difference is the number added to any one term of an arithmetic sequence that generates the\nsubsequent term. SeeExample 1.\n The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common\ndifference repeatedly. SeeExample 2andExample 3.\n A recursive formula for an arithmetic sequence with common difference is given by See\nExample 4."
	},
	{
		"chunk_id" : "00003960",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 4.\n As with any recursive formula, the initial term of the sequence must be given.\n An explicit formula for an arithmetic sequence with common difference is given by See\nExample 5.\n An explicit formula can be used to find the number of terms in a sequence. SeeExample 6.\n In application problems, we sometimes alter the explicit formula slightly to SeeExample 7.\n13.3Geometric Sequences\n A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant."
	},
	{
		"chunk_id" : "00003961",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The constant ratio between two consecutive terms is called the common ratio.\n The common ratio can be found by dividing any term in the sequence by the previous term. SeeExample 1.\n The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common\nratio repeatedly. SeeExample 2andExample 4.\n A recursive formula for a geometric sequence with common ratio is given by for ."
	},
	{
		"chunk_id" : "00003962",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " As with any recursive formula, the initial term of the sequence must be given. SeeExample 3.\n An explicit formula for a geometric sequence with common ratio is given by SeeExample 5.\n In application problems, we sometimes alter the explicit formula slightly to SeeExample 6.\n13.4Series and Their Notations\n The sum of the terms in a sequence is called a series.\n A common notation for series is called summation notation, which uses the Greek letter sigma to represent the\nsum. SeeExample 1."
	},
	{
		"chunk_id" : "00003963",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sum. SeeExample 1.\n The sum of the terms in an arithmetic sequence is called an arithmetic series.\n The sum of the first terms of an arithmetic series can be found using a formula. SeeExample 2andExample 3.\n The sum of the terms in a geometric sequence is called a geometric series.\n The sum of the first terms of a geometric series can be found using a formula. SeeExample 4andExample 5.\n The sum of an infinite series exists if the series is geometric with"
	},
	{
		"chunk_id" : "00003964",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " If the sum of an infinite series exists, it can be found using a formula. SeeExample 6,Example 7, andExample 8.\n An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an\nannuity can be found using geometric series. SeeExample 9.\n13.5Counting Principles\n If one event can occur in ways and a second event with no common outcomes can occur in ways, then the first\nor second event can occur in ways. SeeExample 1."
	},
	{
		"chunk_id" : "00003965",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or second event can occur in ways. SeeExample 1.\n If one event can occur in ways and a second event can occur in ways after the first event has occurred, then the\ntwo events can occur in ways. SeeExample 2.\nAccess for free at openstax.org\n13  Exercises 1313\n A permutation is an ordering of objects.\n If we have a set of objects and we want to choose objects from the set in order, we write\n Permutation problems can be solved using the Multiplication Principle or the formula for SeeExample 3and"
	},
	{
		"chunk_id" : "00003966",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Example 4.\n A selection of objects where the order does not matter is a combination.\n Given distinct objects, the number of ways to select objects from the set is and can be found using a\nformula. SeeExample 5.\n A set containing distinct objects has subsets. SeeExample 6.\n For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations.\nSeeExample 7.\n13.6Binomial Theorem\n is called a binomial coefficient and is equal to SeeExample 1."
	},
	{
		"chunk_id" : "00003967",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " The Binomial Theorem allows us to expand binomials without multiplying. SeeExample 2.\n We can find a given term of a binomial expansion without fully expanding the binomial. SeeExample 3.\n13.7Probability\n Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is\ncertain.\n The probabilities in a probability model must sum to 1. SeeExample 1."
	},
	{
		"chunk_id" : "00003968",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the\nnumber of outcomes in the event by the total number of outcomes in the sample space for the experiment. See\nExample 2.\n To find the probability of the union of two events, we add the probabilities of the two events and subtract the\nprobability that both events occur simultaneously. SeeExample 3."
	},
	{
		"chunk_id" : "00003969",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events.\nSeeExample 4.\n The probability of the complement of an event is the difference between 1 and the probability that the event occurs.\nSeeExample 5.\n In some probability problems, we need to use permutations and combinations to find the number of elements in\nevents and sample spaces. SeeExample 6.\nExercises\nReview Exercises\nSequences and Their Notation"
	},
	{
		"chunk_id" : "00003970",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Review Exercises\nSequences and Their Notation\n1. Write the first four terms of 2. Evaluate 3. Write the first four terms of\nthe sequence defined by the the sequence defined by the\nrecursive formula explicit formula\n4. Write the first four terms of\nthe sequence defined by the\nexplicit formula\nArithmetic Sequences\n5. Is the sequence 6. Is the sequence 7. An arithmetic sequence has\narithmetic? If the first term and\narithmetic? If so, find the so, find the common common difference"
	},
	{
		"chunk_id" : "00003971",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common difference. difference. What are the first five\nterms?\n1314 13  Exercises\n8. An arithmetic sequence has 9. Write a recursive formula 10. Write a recursive formula\nterms and for the arithmetic sequence for the arithmetic\nWhat is the first sequence\nterm?\nand then find the 31stterm.\n11. Write an explicit formula 12. How many terms are in the\nfor the arithmetic finite arithmetic sequence\nsequence\nGeometric Sequences"
	},
	{
		"chunk_id" : "00003972",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "sequence\nGeometric Sequences\n13. Find the common ratio for 14. Is the sequence 4, 16, 28, 15. A geometric sequence has\nthe geometric sequence 40  geometric? If so find terms and\nthe common ratio. If not, . What are\nexplain why. the first five terms?\n16. A geometric sequence has 17. What are the first five 18. Write a recursive formula\nthe first term and terms of the geometric for the geometric\ncommon ratio What sequence sequence\nis the 8thterm?"
	},
	{
		"chunk_id" : "00003973",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is the 8thterm?\n19. Write an explicit formula for the 20. How many terms are in the\ngeometric sequence finite geometric sequence\nSeries and Their Notation\n21. Use summation notation to 22. Use summation notation to 23. Use the formula for the\nwrite the sum of terms write the sum that results sum of the first terms of\nfrom to from adding the number an arithmetic series to find\ntwenty times. the sum of the first eleven\nterms of the arithmetic\nseries 2.5, 4, 5.5,  .\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003974",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Access for free at openstax.org\n13  Exercises 1315\n24. A ladder has tapered 25. Use the formula for the 26. The fees for the first three\nrungs, the lengths of which sum of the firstnterms of years of a hunting club\nincrease by a common a geometric series to find membership are given in\ndifference. The first rung is for the series Table 1. If fees continue to\n5 inches long, and the last rise at the same rate, how\nrung is 20 inches long. much will the total cost be"
	},
	{
		"chunk_id" : "00003975",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "What is the sum of the for the first ten years of\nlengths of the rungs? membership?\nMembership\nYear\nFees\n1 $1500\n2 $1950\n3 $2535\nTable1\n27. Find the sum of the infinite 28. A ball has a bounce-back 29. Alejandro deposits $80 of\ngeometric series ratio of the height of the his monthly earnings into\n previous bounce. Write a an annuity that earns\nseries representing the 6.25% annual interest,\ntotal distance traveled by compounded monthly.\nthe ball, assuming it was How much money will he"
	},
	{
		"chunk_id" : "00003976",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the ball, assuming it was How much money will he\ninitially dropped from a have saved after 5 years?\nheight of 5 feet. What is the\ntotal distance? (Hint: the\ntotal distance the ball\ntravels on each bounce is\nthe sum of the heights of\nthe rise and the fall.)\n30. The twins Hoa and Binh\nboth opened retirement\naccounts on their 21st\nbirthday. Hoa deposits\n$4,800.00 each year,\nearning 5.5% annual\ninterest, compounded\nmonthly. Binh deposits\n$3,600.00 each year,\nearning 8.5% annual\ninterest, compounded"
	},
	{
		"chunk_id" : "00003977",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "earning 8.5% annual\ninterest, compounded\nmonthly. Which twin will\nearn the most interest by\nthe time they are years\nold? How much more?\n1316 13  Exercises\nCounting Principles\n31. How many ways are there 32. In a group of musicians, 33. How many ways are there\nto choose a number from play piano, play to construct a 4-digit code\nthe set trumpet, and play both if numbers can be\npiano and trumpet. How repeated?\nthat is divisible by either many musicians play either\nor piano or trumpet?"
	},
	{
		"chunk_id" : "00003978",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "or piano or trumpet?\n34. A palette of water color 35. Calculate 36. In a group of first-year,\npaints has 3 shades of second-year, third-\ngreen, 3 shades of blue, 2 year, and fourth-year\nshades of red, 2 shades of students, how many ways\nyellow, and 1 shade of can a president, vice\nblack. How many ways are president, and treasurer be\nthere to choose one shade elected?\nof each color?\n37. Calculate 38. A coffee shop has 7 39. How many subsets does\nGuatemalan roasts, 4 the set\nCuban roasts, and 10 Costa have?"
	},
	{
		"chunk_id" : "00003979",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Cuban roasts, and 10 Costa have?\nRican roasts. How many\nways can the shop choose\n2 Guatemalan, 2 Cuban,\nand 3 Costa Rican roasts\nfor a coffee tasting event?\n40. A day spa charges a basic 41. How many distinct ways 42. How many distinct\nday rate that includes use can the word DEADWOOD rearrangements of the\nof a sauna, pool, and be arranged? letters of the word\nshowers. For an extra DEADWOOD are there if\ncharge, guests can choose the arrangement must\nfrom the following begin and end with the"
	},
	{
		"chunk_id" : "00003980",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "from the following begin and end with the\nadditional services: letter D?\nmassage, body scrub,\nmanicure, pedicure, facial,\nand straight-razor shave.\nHow many ways are there\nto order additional services\nat the day spa?\nBinomial Theorem\n43. Evaluate the binomial 44. Use the Binomial Theorem 45. Use the Binomial Theorem\nto expand to write the first three\ncoefficient\nterms of\n46. Find the fourth term of\nwithout fully\nexpanding the binomial.\nAccess for free at openstax.org\n13  Exercises 1317\nProbability"
	},
	{
		"chunk_id" : "00003981",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "13  Exercises 1317\nProbability\nFor the following exercises, assume two die are rolled.\n47. Construct a table showing the sample space. 48. What is the probability that a roll includes a\n49. What is the probability of 50. What is the probability that a roll includes a 2 or\nrolling a pair? results in a pair?\n51. What is the probability that 52. What is the probability of rolling a 5 or a 6?\na roll doesnt include a 2 or\nresult in a pair?\n53. What is the probability that\na roll includes neither a 5\nnor a 6?"
	},
	{
		"chunk_id" : "00003982",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a roll includes neither a 5\nnor a 6?\nFor the following exercises, use the following data: An elementary school survey found that 350 of the 500 students\npreferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round\nto the nearest tenth of a percent.)\n54. What is the percent chance 55. What is the percent chance 56. What is the percent chance\nthat all the children that at least one of the that exactly 3 of the"
	},
	{
		"chunk_id" : "00003983",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "attending the party prefer children attending the children attending the\nsoda? party prefers milk? party prefer soda?\n57. What is the percent chance\nthat exactly 3 of the\nchildren attending the\nparty prefer milk?\nPractice Test\n1. Write the first four terms of 2. Write the first four terms of 3. Is the sequence\nthe sequence defined by the the sequence defined by the\nrecursive formula explicit formula arithmetic? If so find the\ncommon difference."
	},
	{
		"chunk_id" : "00003984",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "common difference.\n4. An arithmetic sequence has 5. Write a recursive formula 6. Write an explicit formula for\nthe first term and for the arithmetic sequence the arithmetic sequence\ncommon difference\nWhat is the 6thterm? and then find the 22ndterm. and then find the 32ndterm.\n7. Is the sequence 8. What is the 11thterm of the 9. Write a recursive formula\ngeometric sequence for the geometric sequence\ngeometric? If so find the\ncommon ratio. If not,\nexplain why.\n1318 13  Exercises"
	},
	{
		"chunk_id" : "00003985",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "explain why.\n1318 13  Exercises\n10. Write an explicit formula 11. Use summation notation to 12. A community baseball\nfor the geometric write the sum of terms stadium has 10 seats in the\nsequence from to first row, 13 seats in the\nsecond row, 16 seats in the\nthird row, and so on. There\nare 56 rows in all. What is\nthe seating capacity of the\nstadium?\n13. Use the formula for the 14. Find the sum of the infinite 15. Ramla deposits $3,600 into\nsum of the first terms of geometric series a retirement fund each"
	},
	{
		"chunk_id" : "00003986",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "a geometric series to find  year. The fund earns 7.5%\nannual interest,\ncompounded monthly. If\nshe opened her account\nwhen she was 20 years old,\nhow much will she have by\nthe time shes 55? How\nmuch of that amount was\ninterest earned?\n16. In a competition of 50 17. A buyer of a new sedan can 18. To allocate annual\nprofessional ballroom custom order the car by bonuses, a manager must\ndancers, 22 compete in the choosing from 5 different choose his top four"
	},
	{
		"chunk_id" : "00003987",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fox-trot competition, 18 exterior colors, 3 different employees and rank them\ncompete in the tango interior colors, 2 sound first to fourth. In how many\ncompetition, and 6 systems, 3 motor designs, ways can he create the\ncompete in both the fox- and either manual or Top-Four list out of the\ntrot and tango automatic transmission. 32 employees?\ncompetitions. How many How many choices does\ndancers compete in the the buyer have?\nfox-trot or tango\ncompetitions?"
	},
	{
		"chunk_id" : "00003988",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "fox-trot or tango\ncompetitions?\n19. A music group needs to 20. A self-serve frozen yogurt 21. How many distinct ways\nchoose 3 songs to play at shop has 8 candy toppings can the word\nthe annual Battle of the and 4 fruit toppings to EVANESCENCE be arranged\nBands. How many ways choose from. How many if the anagram must end\ncan they choose their set if ways are there to top a with the letter E?\nhave 15 songs to pick frozen yogurt?\nfrom?\n22. Use the Binomial Theorem 23. Find the seventh term of"
	},
	{
		"chunk_id" : "00003989",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to expand without fully\nexpanding the binomial.\nAccess for free at openstax.org\n13  Exercises 1319\nFor the following exercises, use the spinner inFigure 1.\nFigure1\n24. Construct a probability 25. What is the probability of 26. What is the probability of\nmodel showing each landing on an odd landing on blue?\npossible outcome and its number?\nassociated probability. (Use\nthe first letter for colors.)\n27. What is the probability of 28. What is the probability of 29. A bowl of candy holds 16"
	},
	{
		"chunk_id" : "00003990",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "landing on blue or an odd landing on anything other peppermint, 14\nnumber? than blue or an odd butterscotch, and 10\nnumber? strawberry flavored\ncandies. Suppose a person\ngrabs a handful of 7\ncandies. What is the\npercent chance that exactly\n3 are butterscotch? (Show\ncalculations and round to\nthe nearest tenth of a\npercent.)\n1320 13  Exercises\nAccess for free at openstax.org"
	},
	{
		"chunk_id" : "00003991",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Illustrative\nMathematics\nA-SSE Triangle Series\nAlignments to Content Standards: A-SSE.B.4\nTask\nConsider the picture below, consisting of a nested sequence of five equilateral\ntriangles, colored in black. Each of the black triangles is made by connecting the three\nmidpoints of the sides of the immediately larger white triangle.\nFind and evaluate a sum to compute the total area of the black region (that is, the sum\nof the areas of the five black triangles) given that the largest triangle in the diagram has"
	},
	{
		"chunk_id" : "00003992",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "area 1.\nIM Commentary\nThe purpose of this task is to emphasize the adjective \"geometric\"\" in the \"\"geometric\"\""
	},
	{
		"chunk_id" : "00003993",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "introduce the geometric series as a worthy object of study, or as a geometric\napplication of its use.\nEdit this solution\nSolution\nThe geometric key to the task is to note that of the various equilateral triangle sizes in\nthe problem, each has 1 the area of the immediately larger triangle of which it is a part\n4\n(since each smaller such triangle has half its base and half its height).\nIn particular, the area of the largest black triangle (and each of the white triangles of"
	},
	{
		"chunk_id" : "00003994",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the same size) is 1 of the area of the large triangle, i.e., 1 (1) = 1. Similarly, the second\n4 4 4\nlargest black triangle has 1 the area of one of these white triangles, so has area\n4\n1  1 = 1 . Continuing in this way, the third, fourth, and fifth black triangles have\n4 4 42\nrespective areas 1 , 1 , and 1 . The sum of the areas of the black triangles can now be\n43 44 45\nevaluated by the formula for a finite geometric series (with common ratio 1):\n4\n1 1 1 1 1\nBlack Area = + + + +\n4 4 2 4 3 4 4 4 5"
	},
	{
		"chunk_id" : "00003995",
		"source" : "Algebra-and-Trigonometry-2e-WEB-17-1330.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "1 1 1 1 1\nBlack Area = + + + +\n4 4 2 4 3 4 4 4 5\n1 1 1 1 1\n= 1 + + + +\n( )\n4 4 4 2 4 3 4 4\n5\n1\n1 1 \n4\n= \n4 1  1\n4\n341\n= .\n1024\nA-SSE Triangle Series\nTypeset May 4, 2016 at 20:05:24. Licensed by Illustrative Mathematics under a\nCreative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .\n2"
	},
	{
		"chunk_id" : "00003996",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nBacteria Populations\nNo Tags\nAlignments to Content Standards: F-LE.A.4\nStudent View\nTask\nA hospital is conducting a study to see how different environmental\nconditions influence the growth of streptococcus pneumonia, one of\nthe bacteria which causes pneumonia. Three different populations are\nstudied giving rise to the following equations:\np (t) = 1000et/3,\n1\np (t) = 1500e3t/8,\n2\np (t) = 5000et/4.\n3\nt"
	},
	{
		"chunk_id" : "00003997",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "1\np (t) = 1500e3t/8,\n2\np (t) = 5000et/4.\n3\nt\nHere represents the number of hours since the beginning of the\np (t)\nexperiment which lasts for 24 hours and i represents the size of the\nith\nbacteria population.\np (t)\na. Explain, in terms of the structure of the expressions defining 1\np (t)\nand 2 , why these two populations never share the same value at\nany time during the experiment.\np (t)\nb. Explain, in terms of the structure of the expressions defining 1\np (t)"
	},
	{
		"chunk_id" : "00003998",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "p (t)\nand 3 , why these two populations will be equal at exactly one time\nduring the experiment. Determine this time.\nIM Commentary\nThis task provides a real world context for interpreting and solving\nexponential equations. There are two solutions provided for part (a).\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nThe first solution demonstrates how to deduce the conclusion by\nthinking in terms of the functions and their rates of change. The second"
	},
	{
		"chunk_id" : "00003999",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "approach illustrates a rigorous algebraic demonstration that the two\npopulations can never be equal.\nSolutions\nSolution: 1\n\np (0) = 1000\na. The first bacteria population has a value of 1 at the\nbeginning of the experiment while the second population has a value of\np (0) = 1500\n2 . The growth rate for the first population is determined by\n1 p (t) 1\nthe constant 3 in the exponent of 1 . Because 3 is positive, the\np (t)\nfunction 1 grows as time passes. The growth rate for the second\n3"
	},
	{
		"chunk_id" : "00004000",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "3\npopulation, , is larger than than the growth rate for the first\n8\npopulation. Since the second population begins at a higher value than\nthe first and it grows at a faster rate, the second population will always\nbe larger than the first.\np (0) = 5000\nb. The third bacterial population has an initial value of 3 ,\nfive times larger than the first bacteria population. Because the rate of\n1\ngrowth for the first population, , is larger than the growth rate of the\n3\n1"
	},
	{
		"chunk_id" : "00004001",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "3\n1\nthird population, , this means that eventually the first population will\n4\nbe larger than the third provided these growth rates continue for a\nsufficiently long period. Because its rate of growth continues to be\nfaster than that of the third, the first population will be larger than the\nthird population once passing the instant when they are equal.\n\nTo find out when the first and third populations are equal, we solve the\np (t) = p (t)\nequation 1 3 :\n1000et/3 = 5000et/4.\net/4"
	},
	{
		"chunk_id" : "00004002",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "equation 1 3 :\n1000et/3 = 5000et/4.\net/4\nDividing both sides of the equation by 1000 and then by gives\net/12 = 5.\nRewriting the equation using the definition of natural log gives\nt\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\n= ln5.\n12\nSo the first and third populations are equal when\nt = 12ln5.\nln5\nUsing a calculator to find , the first and third populations are equal\nafter approximately 19 hours and 19 minutes.\n\nSolution: 2\n"
	},
	{
		"chunk_id" : "00004003",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "\nSolution: 2\n\na. It is interesting to see what goes wrong algebraically if we try to set\nt\nthe first population equal to the second population and solve for :\n1000et/3 = 1500e3t/8.\n1000 e3t/8\nDividing both sides by and by gives\n3\net/24 = .\n2\nt\nSince only positive values of make sense within the context of the\net/24 3\nproblem, is always less than one and so can never be equal to .\n2\n\nAlternatively, we may observe that\n1500e3t/8 3et/24\n= .\n1000et/3 2\n3 > 1 et/24 1 t"
	},
	{
		"chunk_id" : "00004004",
		"source" : "Bacteria Populations-1-3.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "1500e3t/8 3et/24\n= .\n1000et/3 2\n3 > 1 et/24 1 t\nSince and is always larger than for positive , it follows\n2\nthat\n1500e3t/8 3\n>\n1000et/3 2"
	},
	{
		"chunk_id" : "00004005",
		"source" : "Crude Oil and Gas Mileage-1-2.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nCrude Oil and Gas\nMileage\nNo Tags\nAlignments to Content Standards: F-BF.A.1.c\nStudent View\nTask\nAccording to the U.S. Energy Information Administration, a barrel of\ncrude oil produces approximately 20 gallons of gasoline. EPA mileage\nestimates indicate a 2011 Ford Focus averages 28 miles per gallon of\ngasoline.\nG(x)\na. Write an expression for , the number of gallons of gasoline\nx\nproduced by barrels of crude oil.\nM(g)"
	},
	{
		"chunk_id" : "00004006",
		"source" : "Crude Oil and Gas Mileage-1-2.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "x\nproduced by barrels of crude oil.\nM(g)\nb. Write an expression for , the number of miles on average that a\ng\n2011 Ford Focus can drive on gallons of gasoline.\nM(G(x)) M(G(x))\nc. Write an expression for . What does represent in\nterms of the context?\nd. One estimate (from www.oilvoice.com) claimed that the 2010\nDeepwater Horizon disaster in the Gulf of Mexico spilled 4.9 million\nbarrels of crude oil. How many miles of Ford Focus driving would this\nspilled oil fuel?\nIM Commentary"
	},
	{
		"chunk_id" : "00004007",
		"source" : "Crude Oil and Gas Mileage-1-2.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "spilled oil fuel?\nIM Commentary\nIn reference to the solution for part d), in 2010, Ford sold just over\n170,000 Focus models. The oil spilled from the Deepwater Horizon\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\ndisaster would allow EACH Ford Focus sold in 2010 to drive over 15,000\nmiles.\nNote that F.BF.1c does not require student facility with the notation\nf g\n.\nBased on a problem by Hilton Russel.\nSolution\nx 20x"
	},
	{
		"chunk_id" : "00004008",
		"source" : "Crude Oil and Gas Mileage-1-2.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Solution\nx 20x\na. At 20 gallons per barrel, barrels produces gallons so\nG(x) = 20x\n.\n28 g\nb. At miles per gallon, gallons of gasoline will allow the Ford Focus\n28g M(g) = 28g\nto drive miles so .\nc. We have\nM(G(x)) = M(20x) = 28(20x) = 560x.\nM(G(x))\nThe composition represents the number of miles that may be\nx\ndriven on gasoline refined from barrels of crude oil.\nM(G(4.9 million)) = 560(4.9 million) = 2.744 billion miles\nd. .\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a"
	},
	{
		"chunk_id" : "00004009",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "A Comprehensive Institutional Guide to\nDisability Accommodations in\nPostsecondary Education\nI. The Legal Mandate for Equal Access in Higher\nEducation\nThe foundation of disability rights in American higher education rests upon a framework of\nfederal civil rights laws designed to prohibit discrimination and ensure equal opportunity. For\npostsecondary institutions, understanding this legal architecture is not merely a matter of"
	},
	{
		"chunk_id" : "00004010",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "compliance but a prerequisite for fostering an equitable and inclusive academic environment.\nThese laws represent a paradigm shift from the K-12 educational system, moving from a model\nof entitlement and guaranteed success to one of access and individual responsibility. This\ntransition requires a sophisticated understanding of the distinct, yet overlapping, mandates of\nSection 504 of the Rehabilitation Act, the Americans with Disabilities Act, and the Fair Housing\nAct."
	},
	{
		"chunk_id" : "00004011",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Act.\nA. Dissecting Section 504 of the Rehabilitation Act of 1973\nSection 504 of the Rehabilitation Act of 1973 stands as the seminal piece of legislation\nprotecting the rights of individuals with disabilities in programs and activities receiving federal\n1\nfinancial assistance. Its core principle, articulated with powerful simplicity, establishes a broad\nprohibition against discrimination. The statute declares: \"No otherwise qualified individual with a\""
	},
	{
		"chunk_id" : "00004012",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "disability in the United States... shall, solely by reason of her or his disability, be excluded from\nthe participation in, be denied the benefits of, or be subjected to discrimination under any\n1\nprogram or activity receiving Federal financial assistance\". This language is critical"
	},
	{
		"chunk_id" : "00004013",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "The scope of Section 504 is exceptionally broad within the context of higher education. Its\napplicability is triggered by the receipt of federal financial assistance, a criterion met by the vast\n2\nmajority of postsecondary institutions in the United States. This assistance can take many\nforms, including federal grants used to pay student tuition, research funding, or other federal aid\n6\nprograms. Consequently, nearly every public and private college, university, and vocational\n2"
	},
	{
		"chunk_id" : "00004014",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "2\nschool is bound by its requirements. The U.S. Department of Education's Office for Civil Rights\n(OCR) is the primary body charged with enforcing Section 504 regulations in educational\n1\nsettings.\nThe law imposes an affirmative duty on institutions to ensure non-discrimination. It is not enough\nfor a university to simply refrain from overt discriminatory acts. Instead, the institution must \"take"
	},
	{
		"chunk_id" : "00004016",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "4\nact under the law.\nB. Understanding the Americans with Disabilities Act (ADA) of 1990 and its\nAmendments\nEnacted in 1990, the Americans with Disabilities Act (ADA) significantly expanded the\nlandscape of disability rights, extending protections to nearly all areas of public life. Modeled\nafter the Civil Rights Act of 1964, the ADA is a comprehensive \"equal opportunity\"\" law for"
	},
	{
		"chunk_id" : "00004017",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "of the ADA prohibits discrimination by state and local government entities, which includes all\n6\npublic colleges and universities, regardless of whether they receive federal funding. Title III\nprohibits discrimination by public accommodations, which includes private colleges, universities,\n5\nand trade schools. Together, these titles ensure that virtually every higher education institution\n7\nin the country is covered by a federal mandate to prevent disability-based discrimination."
	},
	{
		"chunk_id" : "00004018",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "A pivotal moment in the evolution of disability law was the passage of the ADA Amendments Act\nof 2008 (ADAAA). This legislation was a direct congressional response to a series of Supreme\nCourt decisions that had narrowly interpreted the definition of \"disability"
	},
	{
		"chunk_id" : "00004019",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "definition \"shall be construed in favor of broad coverage of individuals... to the maximum extent"
	},
	{
		"chunk_id" : "00004020",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "academic adjustments. It mandates that all of an institution's services, programs, and activities,\n\"when viewed in its entirety"
	},
	{
		"chunk_id" : "00004021",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "be in an accessible format, videos must be captioned, and digital documents must be\n11\ncompatible with screen-reading software used by students with vision loss. This holistic view\nof access is central to the ADA's goal of ensuring full and equal participation.\nC. The Fair Housing Act (FHA) and its Application to Student Housing\nWhile Section 504 and the ADA govern broad programmatic and academic access, the Fair\nHousing Act (FHA), as amended in 1988, provides specific and crucial protections related to\n11"
	},
	{
		"chunk_id" : "00004022",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "11\nstudent housing. The FHA prohibits discrimination in housing on the basis of disability and\n11\napplies directly to university-managed residence halls, dormitories, and apartments. This\nmeans that a university cannot deny housing to a student because of their disability or the\n7\ndisability of a person associated with them.\nThe FHA imposes two key obligations on housing providers, including universities. First,"
	},
	{
		"chunk_id" : "00004023",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "institutions must make \"reasonable exceptions\"\" in their policies and operations to afford people"
	},
	{
		"chunk_id" : "00004024",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "11\ndisability.\nSecond, the FHA requires landlords to permit tenants with disabilities to make \"reasonable"
	},
	{
		"chunk_id" : "00004025",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "A critical point of clarification for university administrators is the distinction between \"service"
	},
	{
		"chunk_id" : "00004026",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "definition of an \"assistance animal\"\" is much broader. It includes animals that provide emotional"
	},
	{
		"chunk_id" : "00004027",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "11\nareas of campus, such as classrooms, libraries, or dining halls.\nD. From Entitlement to Access: Critical Distinctions Between IDEA and\nADA/Section 504\nPerhaps the most significant challenge facing students with disabilities, their families, and\ninstitutional staff is navigating the profound legal and philosophical shift that occurs upon\ngraduation from high school. The K-12 public school system operates primarily under the\n10"
	},
	{
		"chunk_id" : "00004028",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "10\nIndividuals with Disabilities Education Act (IDEA), an \"educational entitlement act\"\". IDEA"
	},
	{
		"chunk_id" : "00004029",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "5\nIndividualized Education Plan (IEP) or 504 Plan. The focus is on ensuring the student's\nacademic success, which may involve fundamentally modifying the curriculum or altering\n5\nacademic standards.\nUpon entering postsecondary education, this entire framework disappears. IDEA no longer\n10\napplies. Students are now covered by the ADA and Section 504, which are not entitlement\nstatutes but civil rights laws designed to prohibit discrimination and ensure equal\n10"
	},
	{
		"chunk_id" : "00004030",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "10\naccess. This creates a fundamental shift in both the nature of the services provided and the\nlocus of responsibility. The responsibility for initiating the process transfers entirely from the\ninstitution to the individual. Students in college must self-identify to the disability services office,\n4\nprovide appropriate documentation of their disability, and formally request accommodations.\nThe university has no obligation to seek out students with disabilities or to provide"
	},
	{
		"chunk_id" : "00004031",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodations that have not been requested.\nFurthermore, the goal of the legal protection changes from ensuring \"success\"\" to ensuring"
	},
	{
		"chunk_id" : "00004032",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodations serving as the tools to provide an equal opportunity to do so.\nThis abrupt change in legal paradigms creates what can be described as a \"transition chasm.\"\""
	},
	{
		"chunk_id" : "00004033",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "university will proactively manage their needs as their high school did. This mismatch in\nunderstanding is a primary source of confusion and frustration, often leading to significant\ndelays in students receiving the accommodations they need to access their education. The\nfailure of a student to connect with the disability services office is often not a personal failing but\na systemic one, stemming from an inadequate appreciation of this profound transition."
	},
	{
		"chunk_id" : "00004034",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Therefore, an institution's responsibility extends beyond simply providing accommodations upon\nrequest; it includes a duty to actively bridge this chasm through proactive outreach during the\nadmissions process, clear communication in orientation materials, and robust transition planning\nsupport for incoming students and their families.\nFeature K-12 System (IDEA / Postsecondary System (ADA /\nSection 504) Section 504)\nApplicable Laws Individuals with Disabilities Americans with Disabilities Act"
	},
	{
		"chunk_id" : "00004035",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Education Act (IDEA); (ADA); Section 504\nSection 504\nCore Principle Educational Entitlement Civil Rights (Non-discrimination\n(FAPE - Free Appropriate and Equal Access)\nPublic Education)\nResponsibility School district is Student is responsible for\nresponsible for identifying, self-identifying, providing\nevaluating, and providing documentation, and requesting\nservices. accommodations.\nDocumentation School develops an Student must provide current\nIndividualized Education documentation of disability and"
	},
	{
		"chunk_id" : "00004036",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Plan (IEP) or 504 Plan. its functional impact. An IEP is\nnot sufficient.\nScope of Service Focus on ensuring Focus on ensuring equal access.\nacademic success. May Accommodations cannot\ninvolve modifying fundamentally alter essential\ncurriculum and altering course/program requirements.\nstandards.\nAdvocacy Parents and teachers are Student is expected to be their\noften the primary own primary advocate.\nadvocates.\nFunding School district pays for all Student may be responsible for"
	},
	{
		"chunk_id" : "00004037",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "required evaluations and paying for evaluations. Institution\nservices. pays for approved\naccommodations.\nII. Defining Disability and Determining Eligibility\nThe process of providing reasonable accommodations begins with a crucial first step:\ndetermining whether a student is a qualified individual with a disability under the law. The\nADAAA of 2008 significantly reshaped this determination by mandating a broad and inclusive"
	},
	{
		"chunk_id" : "00004038",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "interpretation. For postsecondary institutions, this requires a nuanced understanding of the legal\ndefinition of disability, the standards for what constitutes a \"substantial limitation"
	},
	{
		"chunk_id" : "00004039",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "1. Actual Disability: This is the most common and straightforward prong. It refers to a\nperson who has a physical or mental impairment that substantially limits one or more\n1\nmajor life activities. A physical impairment is defined as any physiological disorder or\ncondition, cosmetic disfigurement, or anatomical loss affecting one of the body's\n2\nsystems, such as neurological, musculoskeletal, or respiratory systems. A mental"
	},
	{
		"chunk_id" : "00004040",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "impairment includes conditions such as emotional or mental illness and specific learning\ndisabilities. The regulations intentionally do not provide an exhaustive list of conditions,\n2\nrecognizing the vast range of potential impairments.\n2. Record of Disability: This prong protects individuals who may not currently have a\n1\nsubstantially limiting impairment but have a history or record of one. For example, a\nstudent who had cancer that is now in remission would be covered under this prong."
	},
	{
		"chunk_id" : "00004041",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "This provision ensures that individuals are not discriminated against based on a past\nmedical history.\n3. Regarded As Disabled: This prong protects individuals who are perceived or treated by\nothers as having an impairment, whether or not they actually have one or whether the\n1\nimpairment limits a major life activity. This part of the definition is primarily aimed at\ncombating discrimination based on stereotypes, myths, or fears about disability. While it"
	},
	{
		"chunk_id" : "00004042",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "is a crucial anti-discrimination tool, it is less frequently the basis for providing academic\naccommodations, which are typically tied to the functional limitations of an actual\nimpairment.\nB. \"Substantial Limitation\"\" and \"\"Major Life Activities\"\": An ADAAA-Informed"
	},
	{
		"chunk_id" : "00004043",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "an impairment \"severely\"\" or \"\"significantly\"\" restrict a major life activity"
	},
	{
		"chunk_id" : "00004044",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "people in the general population.\nThe amendments also expanded and clarified the non-exhaustive list of \"major life activities.\"\""
	},
	{
		"chunk_id" : "such as caring for oneself",
		"source" : " performing manual tasks",
		"MyUnknownColumn" : " seeing",
		"subject" : " hearing",
		"content" : " eating"
	},
	{
		"chunk_id" : "standing",
		"source" : " lifting",
		"MyUnknownColumn" : " bending",
		"subject" : " speaking",
		"content" : " breathing"
	},
	{
		"chunk_id" : "00004045",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "operation of \"major bodily functions.\"\" This encompasses the functions of the immune system"
	},
	{
		"chunk_id" : "normal cell growth",
		"source" : " and the digestive",
		"MyUnknownColumn" : " bowel",
		"subject" : " bladder",
		"content" : " neurological"
	},
	{
		"chunk_id" : "00004046",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "C. Evaluating Eligibility: Considerations for Episodic, Temporary, and\nMitigated Conditions\nThe ADAAA provided crucial guidance on how to evaluate eligibility for individuals with\nconditions that are not constant or are managed with treatment. This guidance requires\ndisability services professionals to look beyond a student's current presentation and consider\nthe underlying impairment.\nA cornerstone of the ADAAA is its treatment of mitigating measures. The law mandates that"
	},
	{
		"chunk_id" : "00004047",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "the determination of whether an impairment substantially limits a major life activity must be\n8\nmade without regard to the ameliorative effects of such measures. Mitigating measures include\nmedication, medical supplies, equipment, hearing aids, prosthetics, and \"learned behavioral or"
	},
	{
		"chunk_id" : "00004048",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "well-controlled by medication is still considered an individual with a disability because, without\nthe medication, their ability to concentrate and think would be substantially limited. Similarly, a\nstudent with a hearing impairment who functions well with a cochlear implant is evaluated based\non their hearing ability without the device.\nThe law also explicitly covers conditions that are episodic or in remission. An impairment that"
	},
	{
		"chunk_id" : "00004049",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "is episodic (e.g., epilepsy, multiple sclerosis, certain psychiatric conditions) or in remission (e.g.,\ncancer) is defined as a disability if it would substantially limit a major life activity when it is\n10\nactive. This provision ensures that students are not denied protection simply because their\nsymptoms are not manifest at the moment of evaluation.\nThe status of temporary impairments is more nuanced. While the ADA is intended to cover"
	},
	{
		"chunk_id" : "00004050",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "long-term or permanent conditions, a temporary impairment, such as a broken leg or a severe\nconcussion, may be covered if it is sufficiently severe and substantially limits a major life activity\n6\nfor an extended period. However, impairments with an actual or expected duration of six\n10\nmonths or less are generally not considered disabilities under the ADA. Even if a temporary\ncondition does not meet the legal definition of a disability, many institutions have policies to"
	},
	{
		"chunk_id" : "00004051",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "provide temporary support services, such as note-taking assistance for a student with a broken\n6\nhand, as a matter of good practice.\nD. The Role and Standards of Disability Documentation: A Modern\nApproach\nThe process of reviewing a student's disability documentation often presents a significant\nchallenge for institutions. A tension exists between the legal mandate for a flexible,\nnon-burdensome process and an institutional desire for clear, prescriptive standards to ensure"
	},
	{
		"chunk_id" : "00004052",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "compliance and prevent misuse of resources. Historically, many colleges and universities\nadopted rigid documentation policies, often requiring recent (e.g., within three to five years),\n14\ncomprehensive psycho-educational evaluations that can be prohibitively expensive. This\npractice creates a significant barrier to access, disproportionately affecting students from less\n17\naffluent backgrounds who may not have had access to such costly private testing. This"
	},
	{
		"chunk_id" : "00004053",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "\"documentation dilemma\"\" forces institutions to confront a fundamental question: is the"
	},
	{
		"chunk_id" : "00004054",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "goal is not to re-diagnose the student but to understand the current impact of the disability on\n9\ntheir academic experience and to support the need for specific accommodations. Under this\nframework, documentation is gathered from multiple sources, with professional judgment being\n9\nthe key to its interpretation.\nThe AHEAD framework identifies three types of documentation:\n1. Primary Documentation: Student Self-Report. The student's own narrative of their"
	},
	{
		"chunk_id" : "00004055",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "experience with their disability, the barriers they encounter, and the accommodations\n9\nthat have been effective or ineffective is considered a \"vital source of information\"\"."
	},
	{
		"chunk_id" : "00004056",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "interactions of the disability services professional during the intake interview are a valid\n9\nand important form of documentation. An experienced professional can use their\nobservations of a student's language, performance, and strategies to help validate the\nstudent's narrative.\n3. Tertiary Documentation: Third-Party Reports. External documents, such as medical\nrecords, psycho-educational evaluations, and school records like an IEP or 504 Plan,\n9"
	},
	{
		"chunk_id" : "00004057",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "9\nserve as supporting information. While an IEP from the K-12 system is not, by itself,\nsufficient documentation for the postsecondary level, it can provide a valuable history of\n6\nthe disability and previously used accommodations.\nAdopting this flexible model requires institutions to abandon rigid, one-size-fits-all policies. For\ninstance, blanket requirements regarding the \"recency\"\" of documentation are inappropriate for"
	},
	{
		"chunk_id" : "00004058",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "9\nstable, lifelong conditions. The documentation process must be non-burdensome and should\nnot discourage students from seeking the support they need. By shifting the focus from\nprescriptive paperwork to a holistic review centered on the student's lived experience,\ninstitutions can create a documentation process that is not only legally compliant with the\nADAAA but also fundamentally more equitable and accessible.\nIII. The Accommodation Process: A Collaborative\nFramework"
	},
	{
		"chunk_id" : "00004059",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Framework\nOnce a student's eligibility is established, the institution and the student engage in a structured,\ncollaborative process to determine and implement reasonable accommodations. This\n\"interactive process\"\" is the cornerstone of providing effective support. It is not a one-time"
	},
	{
		"chunk_id" : "00004060",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "responsibilities of each party is essential to ensure that accommodations are provided in a\ntimely, effective, and legally compliant manner.\nA. Initiating the Process: Student Self-Identification and Disclosure\n13\nAs established, the accommodation process in higher education begins with the student.\nUnlike the K-12 system, the university has no affirmative duty to identify students with\ndisabilities. The student must take the first step by self-identifying and disclosing their disability"
	},
	{
		"chunk_id" : "00004061",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "to the designated office responsible for coordinating accommodationstypically known as the\n6\nDisability Services Office (DSO), Student Disability Resources, or a similar title.\nBecause a student may initially disclose their need for support to any number of university\nemployeessuch as an admissions counselor, a faculty advisor, or a specific course\n14\ninstructorit is imperative that all student-facing staff are trained on the proper protocol. This"
	},
	{
		"chunk_id" : "00004062",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "protocol is simple but critical: upon a student's disclosure of a disability or request for an\naccommodation, the employee must immediately and confidentially direct the student to the\n14\nDSO. This ensures that the student connects with the office that is equipped to handle\nsensitive medical information and is authorized to determine and approve accommodations.\nThe initial disclosure is a highly personal and often difficult step for students. University"
	},
	{
		"chunk_id" : "00004063",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "personnel must handle this interaction with sensitivity and a firm commitment to\n18\nconfidentiality. Faculty and staff should reassure the student that their information will be kept\n18\nprivate and should never ask for a diagnosis or request to see medical documentation. The\nrole of a faculty member or general staff person is not to vet the disability but to serve as a\nbridge, connecting the student to the expert resources available in the DSO.\nB. The Interactive Process: A Step-by-Step Procedural Guide"
	},
	{
		"chunk_id" : "00004064",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "The interactive process is a deliberative and collaborative dialogue designed to identify the\nspecific barriers a student faces and determine the accommodations necessary to provide equal\n9\naccess. While the specific procedures may vary slightly between institutions, the core steps\nare consistent.\n1. Formal Request: After the initial contact, the student is typically asked to complete a\n14\nformal Request for Accommodations form. This form gathers basic information and"
	},
	{
		"chunk_id" : "00004065",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "asks the student to identify the nature of their impairment and the specific\naccommodations they are requesting.\n2. Documentation Review: The DSO professional reviews the documentation provided by\nthe student. As discussed, this includes the student's self-report and may also include\n9\nthird-party medical or educational records. The purpose of this review is to establish\neligibility and gain a clear understanding of the student's functional limitations in an\nacademic setting."
	},
	{
		"chunk_id" : "00004066",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "academic setting.\n3. Intake and Needs Assessment: The heart of the interactive process is the intake\n9\nmeeting between the student and a DSO professional. During this conversation, the\nspecialist explores the student's educational history, their past use of accommodations,\nthe specific barriers they anticipate in the college environment, and the rationale for their\nrequested accommodations. This is a crucial diagnostic step.\n4. Determination of Accommodations: Based on the documentation and the intake"
	},
	{
		"chunk_id" : "00004067",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "discussion, the DSO determines which accommodations are reasonable and appropriate\n13\nfor the student. This is a matter of professional judgment. If a student's requested\naccommodation is deemed unreasonable, would fundamentally alter a program, or is not\nsupported by the documentation, the DSO is obligated to discuss the concerns with the\n14\nstudent and explore equally effective alternative accommodations. The student is not\n14\nentitled to their preferred accommodation, only to an effective one."
	},
	{
		"chunk_id" : "00004068",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "5. Formal Notification to Faculty: Once accommodations are approved, the DSO issues\na formal document, often called an \"Accommodation Letter\"\" or \"\"Letter of"
	},
	{
		"chunk_id" : "00004069",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "6. Implementation and Collaboration: The student is responsible for delivering the\n13\nAccommodation Letter to each of their instructors. Upon receiving the letter, the\nfaculty member is responsible for implementing the listed accommodations. This may\nrequire a brief, private conversation between the student and instructor to discuss\n19\nlogistics, such as how to arrange for extended time on an upcoming exam. It is\nimportant to note that accommodations are not retroactive; they are effective only from\n18"
	},
	{
		"chunk_id" : "00004070",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "18\nthe point at which the instructor receives the official letter.\nC. Delineating Roles and Responsibilities: The Student, The DSO, and The\nFaculty\nThe success of the accommodation process hinges on each party understanding and fulfilling\ntheir distinct but interconnected responsibilities. Ambiguity in these roles is a common source of\nbreakdown, conflict, and failure to provide access. The DSO does not simply process"
	},
	{
		"chunk_id" : "00004071",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "paperwork; it serves as a central hub of expertise, requiring a unique and sophisticated blend of\nskills. The DSO professional must act as a legal expert, interpreting complex medical\ndocumentation and applying the standards of the ADA and Section 504. They must also\npossess pedagogical knowledge, understanding course design and academic standards to\nproperly assess whether an accommodation would constitute a \"fundamental alteration.\"\" Finally"
	},
	{
		"chunk_id" : "00004073",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "be positioned and supported as a hub of \"diplomatic expertise"
	},
	{
		"chunk_id" : "00004074",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Responsibilities\n1. Disclosure & Self-identify to the Provide a clear and Refer any student\nRequest DSO; complete accessible intake disclosing a\nrequired request process; maintain disability or\nforms; articulate confidentiality; requesting\nneeds and barriers. serve as the central accommodation\npoint of contact. directly to the DSO;\nmaintain\nconfidentiality.\n2. Documentation Provide Review Do not request or\ndocumentation documentation to accept medical\n(self-report, determine documentation"
	},
	{
		"chunk_id" : "00004075",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "(self-report, determine documentation\nthird-party reports) eligibility; engage in from the student.\nthat establishes professional\ndisability and judgment; request\nsupports additional\naccommodation information only\nrequests. when necessary;\nmaintain secure\nrecords.\n3. Determination Participate actively Conduct a Consult with the\n(Interactive in the intake thorough, DSO if there are\nProcess) meeting; discuss individualized questions about\nfunctional needs assessment; how an"
	},
	{
		"chunk_id" : "00004076",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "functional needs assessment; how an\nlimitations and past determine accommodation\naccommodations; reasonable and might interact with\nconsider alternative appropriate essential course\naccommodations if accommodations; requirements.\nproposed. deny requests that\nare unreasonable\nor a fundamental\nalteration.\n4. Notification & Deliver the official Issue the official Receive and\nCommunication Accommodation Accommodation acknowledge the\nLetter to each Letter listing Accommodation"
	},
	{
		"chunk_id" : "00004077",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Letter to each Letter listing Accommodation\ninstructor in a approved Letter; review the\ntimely manner. accommodations approved\n(without diagnosis); accommodations.\nbe available for\nconsultation.\n5. Communicate with Provide direct Implement all\nImplementation faculty regarding services (e.g., accommodations\nthe logistics of interpreters, listed in the letter in\nimplementing alternative media); good faith and in a\naccommodations act as a resource timely manner;"
	},
	{
		"chunk_id" : "00004078",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodations act as a resource timely manner;\n(e.g., scheduling and mediator if maintain student's\nexams); follow implementation confidentiality.\nDSO procedures issues arise.\nfor services like\ntesting centers.\n6. Ongoing Notify the DSO Intervene to Consult with the\nIssues/Review immediately if an resolve DSO if an\napproved implementation approved\naccommodation is problems; accommodation\nnot being provided re-engage in the appears to be\nor is ineffective; interactive process fundamentally"
	},
	{
		"chunk_id" : "00004079",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "request reviews or if a student's needs altering the course;\nchanges to change or do not unilaterally\naccommodations accommodations deny an approved\nas needed. are ineffective. accommodation.\nIV. A Comprehensive Guide to Reasonable\nAccommodations\nThe core of disability services in higher education is the provision of \"reasonable"
	},
	{
		"chunk_id" : "00004080",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "disability, thereby providing a student with an equal opportunity to access and participate in all\naspects of university life. Understanding the spectrum of available accommodations, and the\nlegal principles that govern their provision, is essential for faculty, staff, and students.\nAccommodations are not intended to lower academic standards or guarantee success, but\nrather to \"level the playing field\"\" so that a student's performance reflects their abilities"
	},
	{
		"chunk_id" : "00004081",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "24\ndisability.\nA. The Principle of \"Reasonable Accommodation\"\" vs. \"\"Fundamental"
	},
	{
		"chunk_id" : "00004082",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "25\nequal opportunity. The institution has the flexibility to choose among equally effective\naccommodations and is not required to provide the specific accommodation requested by the\n14\nstudent, so long as the alternative is effective.\nHowever, this obligation is not limitless. An institution is not required to provide an\naccommodation that would fundamentally alter an essential requirement of a course or program\n6\nof study. For example, a request to waive a required clinical rotation for a nursing student"
	},
	{
		"chunk_id" : "00004083",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "would likely be considered a fundamental alteration because the rotation is essential to the\nprogram's learning objectives and licensure requirements. The determination of what is\n\"essential\"\" is made on a case-by-case basis through a deliberative process involving the faculty"
	},
	{
		"chunk_id" : "00004084",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "requires the institution to consider its overall resources, not just the budget of a single\n4\ndepartment, and is rarely successfully invoked. Second, institutions are not responsible for\n14\nproviding services or devices of a \"personal nature\"\". This includes personal care attendants"
	},
	{
		"chunk_id" : "00004085",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "26\n\"invisible disabilities"
	},
	{
		"chunk_id" : "00004086",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "these students, such as deadline flexibility and attendance modifications, are frequently\nperceived by faculty not as simple access tools like a ramp, but as fundamental challenges to\ntheir pedagogical methods and course management. This perception is a major source of\nfriction and resistance. This reality suggests that a purely reactive, accommodation-based\nmodel is becoming insufficient. A more effective and equitable approach involves a proactive"
	},
	{
		"chunk_id" : "00004087",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "pedagogical shift toward principles of Universal Design for Learning (UDL), which encourages\nfaculty to build flexibility and multiple means of engagement, representation, and expression into\ntheir courses from the outset. This proactive approach can reduce the need for many\nretroactive, individualized accommodations and create a more inclusive learning environment\nfor all students.\nB. Academic Adjustments and Modifications"
	},
	{
		"chunk_id" : "00004088",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "B. Academic Adjustments and Modifications\nThese accommodations modify how a student learns material or demonstrates their knowledge.\nThey are among the most frequently requested and provided supports.\n Testing Accommodations: These are designed to address barriers in the traditional\ntesting environment. Common examples include extended time (typically 1.5x or 2x the\nstandard time), testing in a reduced-distraction environment, the use of assistive"
	},
	{
		"chunk_id" : "00004089",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "technology such as text-to-speech or dictation software, the assistance of a human\nreader or scribe, and the use of alternative test formats, such as substituting an oral\n25\nexam for a written one or vice versa.\n Note-Taking Assistance: For students whose disabilities impact their ability to take\nnotes effectively, several options are available. These include receiving copies of notes\nfrom a peer volunteer, permission to audio record lectures for later review, the use of"
	},
	{
		"chunk_id" : "00004090",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "smartpens that sync audio with written notes, and receiving copies of the instructor's\n25\nlecture slides or notes in advance.\n Classroom & Assignment Modifications: These accommodations provide flexibility\nwithin the structure of a course. They can include preferential seating to minimize\ndistractions or improve visibility/audibility, flexibility with deadlines for assignments when\na disability-related flare-up occurs, and flexibility with attendance policies for students\n25"
	},
	{
		"chunk_id" : "00004091",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "25\nwith chronic, episodic conditions. This category also includes alternative methods for\nclass participation for a student with a speech impairment or severe social anxiety.\nC. Auxiliary Aids and Services\nThis category includes devices and services necessary to ensure effective communication and\n4\naccess to information for students with sensory, physical, or other disabilities.\n For Deaf and Hard of Hearing Students: To ensure access to auditory information,"
	},
	{
		"chunk_id" : "00004092",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "institutions may provide American Sign Language (ASL) interpreters for classes and\nacademic activities, real-time captioning services (also known as CART), assistive\nlistening devices (such as FM systems that amplify the instructor's voice directly to the\nstudent), and ensuring all video and multimedia content used in a course is accurately\n25\ncaptioned.\n For Blind and Low Vision Students: Access to visual information is provided through"
	},
	{
		"chunk_id" : "00004093",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "various means. This includes converting required texts and course materials into\nalternate formats like Braille, large print, or accessible electronic text that can be used\n25\nwith screen-reading software. Other aids include human readers for in-class\nmaterials, tactile graphics for charts and diagrams, and specialized assistants or\n4\nadaptive equipment for laboratory settings.\n Assistive Technology: This is a broad and rapidly evolving category of tools. It includes"
	},
	{
		"chunk_id" : "00004094",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "text-to-speech software (e.g., Kurzweil 3000) that reads digital text aloud, voice-to-text or\ndictation software that converts spoken words to text, talking calculators, television\nenlargers (CCTVs), and ensuring that the university's online learning management\n25\nsystem is fully accessible and compatible with these technologies.\nD. Environmental and Non-Academic Accommodations\nEqual access extends beyond the classroom to all facets of campus life, including housing,\ndining, and extracurricular activities."
	},
	{
		"chunk_id" : "00004095",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "dining, and extracurricular activities.\n Housing & Dining: Under the FHA and ADA, university housing must be accessible.\nThis can include providing wheelchair-accessible rooms and bathrooms, single rooms for\nstudents whose disability prevents them from having a roommate, visual (strobe) fire\nalarms for deaf students, and individual climate control for students with certain medical\n25\nconditions. As noted previously, an exception to a \"no pets\"\" policy for an Emotional"
	},
	{
		"chunk_id" : "00004096",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "25\nSupport Animal (ESA) is a common housing accommodation. Dining accommodations\nmay involve modifications to a required meal plan for students with severe dietary\n25\nrestrictions related to a disability.\n Physical Access: Institutions must ensure that all programs are accessible, which may\n25\nrequire relocating a class from an inaccessible classroom to an accessible one. It also\nincludes providing accessible lab tables or workstations and maintaining clear paths of\n25\ntravel throughout the campus."
	},
	{
		"chunk_id" : "00004097",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "25\ntravel throughout the campus.\n Service Animals: Institutions must permit trained service animals (dogs or miniature\nhorses under the ADA) to accompany their handlers in all areas of the facility where\n11\nstudents are normally allowed to go. Staff may only ask two questions if it is not\nobvious what service the animal provides: (1) Is the animal required because of a\n11\ndisability? and (2) What work or task has the animal been trained to perform?."
	},
	{
		"chunk_id" : "00004098",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : " Access to Non-Academic Programs: The mandate for equal access applies to all\nuniversity-sponsored programs and activities. This includes ensuring that career\nservices, counseling centers, athletic programs, student clubs, and libraries are fully\n4\naccessible to students with disabilities. For example, career services may need to\nprovide an ASL interpreter for a meeting with a deaf student or ensure its printed\n11\nmaterials are available in alternative formats."
	},
	{
		"chunk_id" : "00004099",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "materials are available in alternative formats.\nThe following table provides a practical reference connecting common disability categories to\nthe functional limitations students may experience and the types of accommodations that may\nbe appropriate to address those limitations. This table is for informational purposes and is not a\nsubstitute for the individualized, interactive process.\nDisability Common Functional Examples of Potentially\nCategory Limitations in an Academic Appropriate Accommodations"
	},
	{
		"chunk_id" : "00004100",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Setting\nLearning Slower reading or Extended time on exams;\nDisabilities information processing text-to-speech software;\n(e.g., Dyslexia, speed; difficulty with reading audio-recorded lectures;\nDysgraphia, decoding, comprehension, or note-taking assistance; use of a\nDyscalculia) fluency; difficulty with written computer with spell/grammar\nexpression (spelling, check for exams; use of a\ngrammar, organization); calculator.\ndifficulty with mathematical\ncalculation or reasoning."
	},
	{
		"chunk_id" : "00004101",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "calculation or reasoning.\nADHD Difficulty with sustaining Extended time on exams;\nattention and concentration; testing in a reduced-distraction\ndistractibility; challenges with environment; permission for\ntime management, breaks during long exams;\norganization, and initiating note-taking assistance; deadline\ntasks (executive functions). flexibility.\nPsychiatric Difficulty with concentration Extended time on exams;\nDisabilities and memory due to anxiety testing in a reduced-distraction"
	},
	{
		"chunk_id" : "00004102",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "(e.g., Anxiety, or depressive symptoms; or private setting; attendance\nDepression, variable energy levels and flexibility; deadline flexibility;\nPTSD) attendance due to episodic alternative ways to complete\nnature of the condition; participation requirements.\ndifficulty in high-stress\nsituations (e.g., timed\nexams, public speaking).\nAutism Difficulty with social Clear, explicit, and written\nSpectrum communication and \"reading instructions; advance notice of\""
	},
	{
		"chunk_id" : "00004103",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Disorder between the lines\"; changes to syllabus or"
	},
	{
		"chunk_id" : "00004104",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "standard classroom seating flexibility with\nor lab equipment; physical attendance/tardiness; lab\nbarriers in buildings. assistant; extended time for\nexams to manage fatigue.\nChronic Unpredictable or frequent Attendance flexibility; deadline\nMedical absences due to medical flexibility; permission for breaks\nConditions flare-ups; fatigue; need for during class and exams;\n(e.g., Crohn's breaks to manage medical adjustments to food/drink\ndisease, needs (e.g., check blood policies in the classroom."
	},
	{
		"chunk_id" : "00004105",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Diabetes, sugar, take medication).\nChronic Fatigue)\nDeaf / Hard of Inability or difficulty ASL interpreter; real-time\nHearing accessing auditory captioning (CART); assistive\ninformation from lectures, listening device (FM system);\ndiscussions, and multimedia. preferential seating; captioned\nvideos; note-taking assistance.\nBlind / Low Inability or difficulty Materials in alternate formats\nVision accessing visual information (Braille, large print, accessible"
	},
	{
		"chunk_id" : "00004106",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "from textbooks, whiteboards, electronic text); screen-reading\npresentations, and online software; audio-recorded\nmaterials. lectures; reader for exams; lab\nassistant; preferential seating.\nV. Upholding Confidentiality: Navigating FERPA and Best\nPractices\nMaintaining the confidentiality of a student's disability information is a paramount legal and\nethical obligation for all members of the university community. The protocols governing privacy"
	},
	{
		"chunk_id" : "00004107",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "are not merely bureaucratic rules; they form the bedrock of the trust that is essential for the\nentire accommodation system to function. A student's decision to self-disclose a disability is a\nvoluntary act that requires a profound sense of safety and confidence that their sensitive\npersonal information will be handled with the utmost discretion. Any breach of this trust, whether\nintentional or accidental, can have a chilling effect, deterring other students from seeking the"
	},
	{
		"chunk_id" : "00004108",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "support they need and are legally entitled to receive.\nA. Disability Records as Protected Educational Records\nThe primary federal law governing the privacy of student records is the Family Educational\n23\nRights and Privacy Act (FERPA). FERPA applies to all educational institutions that receive\nfunds from the U.S. Department of Education and is designed to protect the privacy of\n30\n\"personally identifiable information\"\" contained within a student's \"\"education record\"\". All\""
	},
	{
		"chunk_id" : "00004109",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "disability-related information maintained by the DSOincluding diagnostic documentation,\ncorrespondence, consultation notes, and accommodation lettersis considered part of the\n24\nstudent's education record and is therefore protected by FERPA. These records are typically\nkept in a secure location within the DSO, separate from the student's general academic file to\n23\nensure a higher level of privacy.\nA key provision of FERPA is that it gives students who are 18 or older the right to control the\n30"
	},
	{
		"chunk_id" : "00004110",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "30\ndisclosure of their education records. This means that an institution may not release a\nstudent's disability information to a third partyincluding the student's parentswithout the\n23\nstudent's prior written consent. This is a frequent point of confusion for parents accustomed\nto being involved in their child's education under the K-12 system. Even if a student has signed\na general university FERPA waiver allowing the registrar or financial aid office to speak with"
	},
	{
		"chunk_id" : "00004111",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "their parents, a separate, specific release must be on file with the DSO before its staff can\n31\ndiscuss the student's accommodations or disability status with a parent or guardian.\nB. Information Sharing Protocols: The \"Legitimate Educational Interest\"\""
	},
	{
		"chunk_id" : "00004112",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "share information from a student's education record with other officials within the institution who\n22\nhave a \"legitimate educational interest\"\" in the information. This \"\"need-to-know\"\" principle is the"
	},
	{
		"chunk_id" : "00004113",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "in their course. Therefore, the DSO is permitted to provide the faculty member with an official\n23\nAccommodation Letter that lists the approved accommodations. However, a faculty member\ndoes\nnot have a legitimate educational interest in knowing the student's specific diagnosis or the\ndetails of their medical history. For this reason, the Accommodation Letter must not contain any\n22\ndiagnostic information. The letter itself, issued by the authorized university office, is all the\n19"
	},
	{
		"chunk_id" : "00004114",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "19\njustification a faculty member needs to provide the listed accommodations.\nFaculty and staff must understand their role within this confidential framework. They are\npartners in implementing accommodations, but they are not involved in the diagnostic or\neligibility determination process. Therefore, it is inappropriate for a faculty member to ask a\nstudent to disclose their disability or to question their need for an accommodation that has been\n18"
	},
	{
		"chunk_id" : "00004115",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "18\napproved by the DSO. If a student voluntarily chooses to disclose the nature of their disability\nto an instructor, that information is also considered confidential and must not be shared with\n19\nothers.\nC. Practical Guidance for Maintaining Student Privacy\nAdhering to confidentiality requires constant vigilance and the adoption of specific best practices\nby all faculty and staff. These practices protect the student's privacy, foster a climate of trust,"
	},
	{
		"chunk_id" : "00004116",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "and shield both the individual employee and the institution from legal liability.\n Conduct Conversations in Private: All discussions with a student about their\naccommodations must take place in a private and confidential setting, such as during\noffice hours or a scheduled appointment. These conversations should never occur in the\n19\nclassroom or hallway where they can be overheard by other students.\n Secure Documents and Communications: Physical and digital documents related to"
	},
	{
		"chunk_id" : "00004117",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "student accommodations must be kept secure. Do not leave an Accommodation Letter\nor related notes visible on a desk or computer screen where others might see them. At\n19\nthe end of the semester, these documents should be disposed of securely. When\nsending an email to a group of students registered with the DSO (e.g., to coordinate an\nexam), use the Blind Carbon Copy (BCC) function to prevent students from seeing each\n19\nother's names and email addresses."
	},
	{
		"chunk_id" : "00004118",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "19\nother's names and email addresses.\n Avoid Public Identification: Never single out or publicly identify students with\ndisabilities in any way. A common violation is an instructor announcing to the class, \"All"
	},
	{
		"chunk_id" : "00004119",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : " Refer to the Designated Office: Faculty should be trained to act as a referral point, not\nan intake point. If a student attempts to provide a faculty member with their primary\nmedical or diagnostic documentation, the instructor should politely refuse to accept or\n19\nread it and immediately refer the student to the DSO. Similarly, if a faculty member\nhas questions or concerns about the appropriateness of an accommodation, their\n19\nconversation should be with the DSO, not with the student."
	},
	{
		"chunk_id" : "00004120",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Upholding these standards is not simply a matter of following rules. It is a fundamental\nexpression of respect for the student and a critical component of building an inclusive campus\nculture where students feel safe enough to come forward and advocate for their needs.\nVI. Addressing Disputes: Grievance and Appeals\nProcedures\nDespite the best efforts of all parties, disagreements can arise during the accommodation\nprocess. A student may believe an accommodation was unfairly denied, or that an approved"
	},
	{
		"chunk_id" : "00004121",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodation is not being implemented effectively. To ensure fairness and due process,\nfederal law requires institutions to establish and publish a clear, prompt, and equitable grievance\n11\nprocedure for resolving such disability-related complaints. A well-structured grievance\nprocess not only provides a mechanism for individual dispute resolution but also serves as a\nvital feedback loop, allowing the institution to identify and correct systemic problems."
	},
	{
		"chunk_id" : "00004122",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "A. Establishing a Prompt and Equitable Internal Grievance Process\nBoth Section 504 of the Rehabilitation Act and the ADA mandate that covered institutions adopt\n11\nan internal grievance procedure to address complaints of disability discrimination. The\npurpose of this process is to provide a formal channel for students to raise concerns about any\naction prohibited by these laws, including the denial of a requested accommodation, a failure to\n33"
	},
	{
		"chunk_id" : "00004123",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "33\nprovide an approved accommodation, or other forms of discrimination.\nTo oversee compliance and manage these procedures, institutions must designate at least one\n11\nemployee as the ADA/Section 504 Coordinator. This individual is the central point of contact\nfor disability-related issues and typically plays a key role in the grievance process. The policy\nmust be widely published and easily accessible to all students, for example, in the student\n14\nhandbook and on the university website."
	},
	{
		"chunk_id" : "00004124",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "14\nhandbook and on the university website.\nA critical component of any grievance policy is a strong and explicit prohibition against\nretaliation. The law forbids an institution from intimidating, threatening, coercing, or\ndiscriminating against any individual because they have filed a grievance or otherwise opposed\n2\na discriminatory practice. Students must be able to raise concerns without fear of reprisal.\nB. Informal Resolution vs. Formal Grievance"
	},
	{
		"chunk_id" : "00004125",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "B. Informal Resolution vs. Formal Grievance\nBest practices in dispute resolution strongly encourage attempting to resolve conflicts at the\n33\nlowest and most informal level possible before escalating to a formal process. A robust\ngrievance procedure should therefore outline an informal resolution pathway.\nIf a student's concern relates to the implementation of an approved accommodation, the first\n33\nstep should be to discuss the issue directly with the faculty or staff member involved. Often,"
	},
	{
		"chunk_id" : "00004126",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "implementation problems arise from simple miscommunication that can be quickly corrected. If\nthe student is uncomfortable having this conversation alone, or if the direct conversation does\nnot resolve the issue, they should contact the DSO for assistance. The DSO can act as a\n33\nmediator and clarify the faculty member's obligations.\nIf a student's concern is with a decision made by the DSOfor example, the denial of a specific"
	},
	{
		"chunk_id" : "00004127",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodation requestthe informal step is to discuss their disagreement with the Director of\n37\nthe DSO. This allows for a review of the decision and provides an opportunity to present any\nnew information that might affect the outcome.\nC. The Formal Grievance and Appeals Pathway\nWhen informal measures fail or are inappropriate, the student has the right to initiate a formal\ngrievance. The process should follow a clear, multi-step pathway."
	},
	{
		"chunk_id" : "00004128",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "1. Filing the Formal Complaint: The student submits a formal, written grievance to the\n33\ndesignated ADA/Section 504 Coordinator. The complaint should be filed within a\nreasonable timeframe (e.g., within 60 calendar days of the alleged violation) and should\ninclude specific details: the nature of the complaint, the date of the incident, the\nindividuals involved, the steps already taken to resolve the issue, and the desired\n33\nremedy."
	},
	{
		"chunk_id" : "00004129",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "33\nremedy.\n2. Investigation: Upon receipt of the complaint, the ADA/Section 504 Coordinator (or their\ndesignee, who must not be the person who made the initial decision) will conduct a\n37\nthorough and impartial investigation. This investigation may include reviewing the\nstudent's file and all relevant documentation, interviewing the complainant, the\n37\nfaculty/staff involved, and any other relevant witnesses.\n3. Written Decision: The Coordinator will analyze the findings of the investigation and"
	},
	{
		"chunk_id" : "00004130",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "issue a formal written decision to the student and other relevant parties. This decision\nshould be rendered within a specified timeframe, such as 20 or 30 business days, and\n33\nshould explain the rationale for the finding.\n4. Internal Appeal: The institutional policy should provide for at least one level of appeal\n35\nfor a student who is not satisfied with the Coordinator's decision. The appeal is\ntypically made in writing to a senior-level administrator who was not involved in the"
	},
	{
		"chunk_id" : "00004131",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "original decision, such as a Dean of Students, Provost, or College President. This\n35\nindividual's decision is generally considered final at the institutional level.\nViewing the grievance process solely as a mechanism for resolving individual disputes is a\nmissed opportunity. Each grievance, whether informal or formal, is a valuable data point that\nsignals a point of friction or failure within the institution's accessibility framework. By"
	},
	{
		"chunk_id" : "00004132",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "systematically tracking and analyzing the nature, frequency, and location of these grievances,\nthe ADA/Section 504 Coordinator can identify systemic patterns. For instance, a cluster of\ncomplaints originating from a particular academic department may indicate a need for targeted\nfaculty training. A recurring theme of disputes over a specific accommodation, such as\nattendance flexibility, might reveal that institutional policies in that area are unclear or poorly"
	},
	{
		"chunk_id" : "00004133",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "understood. An institution that uses this data for continuous quality improvement transforms its\ngrievance process from a reactive, legally defensive mechanism into a proactive diagnostic tool\nfor enhancing campus-wide accessibility and reducing future conflicts.\nD. External Remedies: The Office for Civil Rights (OCR) and Department of\nJustice (DOJ)\nIt is essential that students are informed that they have the right to file a complaint with an"
	},
	{
		"chunk_id" : "00004134",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "external federal agency at any time, regardless of whether they have initiated or completed the\n35\ninstitution's internal grievance process.\n U.S. Department of Education, Office for Civil Rights (OCR): The OCR is the primary\n1\nenforcement agency for Section 504 and Title II of the ADA in educational settings. A\nstudent can file a complaint with the regional OCR office. A complaint must generally be\n34\nfiled within 180 days of the last act of alleged discrimination. OCR may investigate the"
	},
	{
		"chunk_id" : "00004135",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "complaint, facilitate a resolution between the student and the institution, or issue findings\nand mandate corrective actions.\n U.S. Department of Justice (DOJ): The Civil Rights Division of the DOJ also has\n38\nenforcement authority for the ADA. Students can file a complaint with the DOJ online\nor by mail. The DOJ may investigate the complaint, refer it to its ADA Mediation\n38\nProgram, or, in some cases, file a lawsuit to correct violations.\nVII. Recommendations for Institutional Best Practices"
	},
	{
		"chunk_id" : "00004136",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Moving beyond the minimum requirements of legal compliance to foster a truly inclusive and\naccessible campus environment requires a proactive, holistic, and institution-wide commitment.\nLegal mandates provide the floor, but best practices build the framework for excellence. The\nfollowing recommendations synthesize the core principles discussed in this report into an\nactionable agenda for institutional leaders dedicated to creating a culture where students with\ndisabilities can thrive."
	},
	{
		"chunk_id" : "00004137",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "disabilities can thrive.\nA. Developing Clear, Accessible, and Student-Centered Policies\nEffective institutional practice begins with clear, well-articulated policies that are easily\nunderstood by all members of the campus community.\n Centralize and Simplify: All disability-related policies, procedures, and forms should be\nconsolidated into a single, prominent, and easily navigable section of the university\n14\nwebsite. This \"one-stop shop\"\" approach eliminates confusion and ensures students"
	},
	{
		"chunk_id" : "00004139",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "revise their disability documentation guidelines to align with the flexible,\n9\nnon-burdensome framework promoted by AHEAD and the spirit of the ADAAA. This\nmeans moving away from rigid recency requirements for lifelong conditions and\nembracing student self-report as a primary source of information, thereby removing\nunnecessary barriers to access.\n Mandate a Welcoming Syllabus Statement: Every course syllabus across the\nuniversity should include a standard, welcoming statement about disability access. This"
	},
	{
		"chunk_id" : "00004140",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "statement should affirm the institution's commitment to accessibility, provide the contact\ninformation for the Disability Services Office, and encourage students who need\n21\naccommodations to connect with that office early in the semester. This simple act\nnormalizes the accommodation process and creates a more inviting climate for\ndisclosure.\nB. Essential Components of Faculty and Staff Training Programs\nThe effective implementation of accommodations depends on knowledgeable and supportive"
	},
	{
		"chunk_id" : "00004141",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "faculty and staff. Comprehensive, ongoing training is not an option but a necessity.\n Mandate Foundational Training: All faculty (including adjuncts and teaching assistants)\nand student-facing staff should be required to complete regular training on their legal\n14\nobligations under the ADA and Section 504. This training must cover the\naccommodation process, the critical importance of confidentiality, and the specific roles\nand responsibilities of faculty versus the DSO."
	},
	{
		"chunk_id" : "00004142",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "and responsibilities of faculty versus the DSO.\n Focus on Practical, Scenario-Based Learning: Training should be practical and\nengaging. Instead of simply reciting the law, sessions should use case studies and\nscenarios to walk faculty through common situations: how to respond when a student\nfirst discloses a disability, how to have a productive and private conversation about\nimplementing an accommodation letter, and how to handle a concern that an"
	},
	{
		"chunk_id" : "00004143",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "accommodation might be altering an essential course requirement.\n Promote Universal Design for Learning (UDL): Institutions should invest in robust\nfaculty development programs focused on UDL. This pedagogical framework\nencourages faculty to proactively design their courses with flexibility and multiple means\nof engagement, representation, and assessment. By building accessibility into a course\nfrom the outset, faculty can reduce the need for many retroactive accommodations and"
	},
	{
		"chunk_id" : "00004144",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "create a more inclusive learning environment for all students.\n Reinforce the \"Referral\"\" Duty: All university employees must be trained on their"
	},
	{
		"chunk_id" : "00004145",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "Finally, policies and training must be embedded within a campus culture that genuinely values\naccessibility and inclusion as core institutional principles.\n Visible Leadership Commitment: The President, Provost, and Deans must\nconsistently and publicly champion the importance of disability access. This \"top-down\"\""
	},
	{
		"chunk_id" : "00004146",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "all major institutional processes. This includes ensuring all new construction and major\nrenovations meet the highest standards of physical accessibility, requiring that all new\ntechnology and software procurements are vetted for accessibility, and incorporating\n39\naccessibility reviews into the curriculum development and approval process. Online\ncourses and materials, in particular, must be designed to be accessible from their\n11\ninception."
	},
	{
		"chunk_id" : "00004147",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "11\ninception.\n Centralize and Empower the Disability Services Office: The DSO should be\nrecognized and structured as a hub of critical expertise. It should be positioned within\nthe academic affairs division to facilitate collaboration with faculty and academic\ndepartments. The office must be adequately staffed with well-trained professionals and\nprovided with the resources necessary to fulfill its complex legal, pedagogical, and\ndiplomatic mission."
	},
	{
		"chunk_id" : "00004148",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "diplomatic mission.\n Commit to Data-Driven Improvement: The institution should formalize the use of its\ngrievance process and other feedback mechanisms (e.g., student surveys) as diagnostic\ntools. The ADA/Section 504 Coordinator should be charged with conducting an annual\nanalysis of grievance data to identify systemic trends and report these findings to senior\nleadership. This commitment to continuous, data-informed improvement is the ultimate"
	},
	{
		"chunk_id" : "00004149",
		"source" : "Disability_Accommodations_in_Postsecondary_Education.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Disability_Accommodations",
		"content" : "hallmark of an institution that has moved beyond mere compliance to a deep-seated\ncommitment to equity and inclusion for all students."
	},
	{
		"chunk_id" : "00004150",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "Federal Educational Aid Eligibility Requirements for 2025-\n2026\nThe 2025-2026 academic year brings significant changes to federal student aid through the\nFAFSA Simplification Act, with streamlined applications, revised eligibility calculations, and\nenhanced access for millions of students. The maximum Pell Grant remains $7,395,\nFederal Student Aid +4 while major upcoming legislation will dramatically reshape loan programs\nstarting in 2026. Students benefit from simplified processes, though must navigate new"
	},
	{
		"chunk_id" : "00004151",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "requirements and upcoming program changes that will affect future borrowers.\nFAFSA Simplification Act transforms application process\nThe most significant change for 2025-2026 is the full implementation of the FAFSA\nSimplification Act, fundamentally altering how students apply for and qualify for federal aid. The\nFree Application for Federal Student Aid now contains approximately 36 questions instead of\nthe previous 108, CollegeData dramatically reducing complexity for families. LendEDU"
	},
	{
		"chunk_id" : "00004152",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "The Student Aid Index (SAI) has completely replaced the Expected Family Contribution (EFC)\nCollegeData in determining aid eligibility. ed +3 Unlike the EFC, the SAI can be negative (as low\nas -1,500), better reflecting families with the highest financial need. U.S. Department of Education\nLendEDU This change particularly benefits low-income students who may now qualify for\nmaximum Pell Grant amounts automatically based on family size and income thresholds tied to\nfederal poverty guidelines."
	},
	{
		"chunk_id" : "00004153",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "federal poverty guidelines.\nFederal Tax Information consent is now mandatory for all applicants and contributors, enabling\nautomatic data retrieval from the IRS. U.S. Department of Education While this streamlines\nverification, it creates new challenges for families with complex tax situations or those who\nhaven't filed returns. The application opened December 1, 2024, after a phased rollout that\nbegan with beta testing in October 2024, ed +3 addressing technical issues from the previous"
	},
	{
		"chunk_id" : "00004154",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "year's problematic launch.\nPell Grant eligibility expanded with new categorical system\nThe Pell Grant program introduces three distinct eligibility categories for 2025-2026: Maximum\nPell, Minimum Pell, and Calculated Pell. ed +4 This replaces the previous payment schedule\nsystem with more predictable, formulaic awards.\nMaximum Pell Grant recipients ($7,395) Federal Student Aid Federal Student Aid include dependent"
	},
	{
		"chunk_id" : "00004155",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "students whose families earn up to 175% of federal poverty guidelines (225% for single parents)\nand independent students meeting similar thresholds. Federal Student Aid +2 Students whose\nfamilies aren't required to file federal tax returns automatically receive an SAI of -1,500,\nqualifying them for maximum awards. U.S. Department of Education\nMinimum Pell Grant recipients ($740) Federal Student Aid Partners are those who don't qualify for"
	},
	{
		"chunk_id" : "00004156",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "maximum awards but have family incomes up to 275% of federal poverty guidelines (325% for\nsingle parents). ed Federal Student Aid Partners This safety net ensures aid reaches more middle-\nincome families than under previous systems.\nStudents maintaining Satisfactory Academic Progress must achieve a 2.0 cumulative GPA,\ncomplete at least 67% of attempted credits, and finish within 150% of their program's\npublished length. SoFi +5 The Lifetime Eligibility Used limit remains 600% (equivalent to 12"
	},
	{
		"chunk_id" : "00004157",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "full-time semesters), Federal Student Aid with LEU tracking all Pell Grants received since program\ninception in 1973. ed +5\nFederal Direct Loan programs face major upcoming changes\nInterest rates for 2025-2026 are 6.39% for undergraduate loans, 7.94% for graduate\nunsubsidized loans, and 8.94% for PLUS loans. ed CNBC These fixed rates apply for the life of\nloans disbursed during this award year, determined by the 10-year Treasury Note rate plus\nstatutory add-ons."
	},
	{
		"chunk_id" : "00004158",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "statutory add-ons.\nAnnual borrowing limits vary significantly by dependency status and academic level.\nDependent undergraduates can borrow $5,500-$7,500 annually (with $3,500-$5,500 in\nsubsidized loans), while independent undergraduates access $9,500-$12,500 annually. Graduate\nstudents can borrow up to $20,500 annually in unsubsidized loans, ed plus unlimited PLUS\nloans up to their cost of attendance.\nThe One Big Beautiful Bill Act will dramatically reshape loan programs starting July 1, 2026."
	},
	{
		"chunk_id" : "00004159",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "Federal Student Aid Partners Graduate PLUS loans will be eliminated for new borrowers, replaced with\na $100,000 lifetime limit for graduate unsubsidized loans (down from current $138,500). Parent\nPLUS loans will face new annual limits of $20,000 and lifetime limits of $65,000 per child.\nProfessional students will have annual limits of $50,000 and lifetime limits of $200,000.\nFederal Student Aid Partners \nThe SAVE repayment plan remains blocked by federal court injunction, with borrowers"
	},
	{
		"chunk_id" : "00004160",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "required to choose alternative repayment options. Interest restarted for SAVE borrowers on\nAugust 1, 2025, U.S. Department of Education and they must transition to new plans by July 1, 2028.\nNPR\nWork-Study and specialized grant programs continue with modifications\nFederal Work-Study provides campus-based employment for students demonstrating\nfinancial need, with no minimum or maximum award limits beyond calculated need."
	},
	{
		"chunk_id" : "00004161",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "U.S. Department of Education Students must earn at least the federal minimum wage ($7.25) and\nwork in positions that complement their educational goals when possible.\nU.S. Department of Education Schools must dedicate 7% of their Work-Study allocation to\ncommunity service positions, including reading tutoring or family literacy projects.\nU.S. Department of Education\nFederal Supplemental Educational Opportunity Grants (FSEOG) offer up to $4,000 annually"
	},
	{
		"chunk_id" : "00004162",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "for undergraduate students with exceptional financial need. Federal Student Aid Partners ed Priority\ngoes to Pell Grant recipients with the lowest SAI scores. U.S. Department of Education ed The\nprogram received $905.5 million in federal funding for 2025-2026, serving approximately 3,356\nparticipating schools. Federal Student Aid Partners\nTEACH Grants provide up to $4,000 annually for students preparing to teach in high-need"
	},
	{
		"chunk_id" : "00004163",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "fields. U.S. Department of Education ed Recipients must maintain a 3.25 GPA, teach four years full-\ntime in high-need schools within eight years of program completion, and meet state\ncertification requirements. U.S. Department of Education ed Grants convert to unsubsidized loans if\nservice obligations aren't met. U.S. Department of Education ed High-need fields include bilingual\neducation, mathematics, science, special education, and areas listed in the annual Teacher"
	},
	{
		"chunk_id" : "00004164",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "Shortage Area Nationwide Listing. U.S. Department of Education U.S. Department of Education\nThe Iraq and Afghanistan Service Grant was retired after the 2023-2024 academic year.\nFormer IASG-eligible students now automatically qualify for maximum Pell Grants regardless of\ntheir SAI under Special Rule provisions. Federal Student Aid Partners\nCitizenship and dependency requirements establish aid eligibility\nU.S. citizens, nationals, and eligible non-citizens qualify for federal aid. Eligible non-citizens"
	},
	{
		"chunk_id" : "00004165",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "include permanent residents, conditional permanent residents, refugees, asylum grantees, T-visa\nholders, Cuban-Haitian entrants, trafficking victims with HHS certification, and VAWA-qualified\nimmigrants. University of Georgia Student DACA recipients and most visa holders remain ineligible\nfor federal aid, though some states provide alternative funding. University of Georgia Student\nIndependent student status requires meeting at least one of nine criteria: being 24 or older,"
	},
	{
		"chunk_id" : "00004166",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "married, a graduate student, military veteran or active duty, having legal dependents, being\norphaned or in foster care since age 13, being an emancipated minor, being in legal\nguardianship, or being an unaccompanied homeless youth. BYU +2 Students not meeting these\ncriteria are considered dependent and must provide parent financial information. Edvisors +2\nProfessional judgment allows financial aid administrators to override dependency"
	},
	{
		"chunk_id" : "00004167",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "determinations in cases of unusual circumstances, such as abandonment, abuse, or other\nextraordinary situations. U.S. Department of Education Students can indicate these circumstances on\ntheir FAFSA, triggering school-specific review processes. U.S. Department of Education\nApplication deadlines and academic progress standards\nThe federal FAFSA deadline remains June 30, 2026, USAGov but students should complete"
	},
	{
		"chunk_id" : "00004168",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "applications as early as possible after the December 1, 2024 availability date. Federal Student Aid +4\nState deadlines vary dramatically, from as early as October 1, 2024, to June 30, 2026, with many\noperating on first-come, first-served funding while state aid lasts. Federal Student Aid studentaid\nSatisfactory Academic Progress requirements apply universally across federal aid programs,\nrequiring students to maintain a cumulative 2.0 GPA (qualitative measure), complete at least"
	},
	{
		"chunk_id" : "00004169",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "67% of attempted credits (quantitative measure), and finish their programs within 150% of\npublished length (maximum timeframe). SoFi +3 Schools must review SAP at least annually, with\nmany reviewing each semester or payment period. Brooklyn College +3\nStudents failing SAP typically receive one warning period before aid suspension, with appeal\nprocesses available for extenuating circumstances like medical emergencies, family deaths, or"
	},
	{
		"chunk_id" : "00004170",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "natural disasters. Kennesaw State University Successful appeals often require academic plans and\nregular progress monitoring. NerdWallet Maricopa Community Colleges\nKey regulatory changes eliminate previous barriers\nSelective Service registration is no longer required for federal student aid, representing a\nsignificant change from previous years. This requirement was eliminated through the FAFSA\nSimplification Act, U.S. Department of Education though many states still require registration for state"
	},
	{
		"chunk_id" : "00004171",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "aid programs. Males aged 18-25 assigned male sex at birth may still need to register for state\nbenefits. U.S. Department of Education\nDrug conviction limitations on aid eligibility have been completely eliminated, removing\nbarriers that previously affected students with substance abuse convictions. This change\nexpands access for students working to rebuild their lives after legal challenges.\nDefault resolution options continue for borrowers seeking to regain aid eligibility, including"
	},
	{
		"chunk_id" : "00004172",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "loan rehabilitation (nine consecutive on-time payments), full repayment, or consolidation.\nNerdWallet +2 The temporary Fresh Start Initiative for pre-pandemic defaults ended September\n30, 2024, U.S. Department of Education but traditional rehabilitation options remain available.\nConclusion\nThe 2025-2026 academic year represents a pivotal transition in federal student aid, with\nsimplified applications and expanded Pell Grant access benefiting millions of students. However,"
	},
	{
		"chunk_id" : "00004173",
		"source" : "Federal Educational Aid Eligibility Requirements for 2025-2026_ FAFSA Simplification and Program Changes Analysis.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "Federal_Educational_Aid_Eligibility",
		"content" : "upcoming loan program changes in 2026 will fundamentally alter borrowing options,\nparticularly for graduate students and parents. Students and families should take advantage of\ncurrent loan availability while preparing for more restrictive future lending limits. The\nstreamlined FAFSA process, despite initial implementation challenges, offers clearer pathways to\naid determination and earlier notification of eligibility, helping students make informed\neducational financing decisions."
	},
	{
		"chunk_id" : "00004174",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nFind the Angle\nTags: GeoGebra\nAlignments to Content Standards: 8.G.A.5\nStudent View\nTask\nABC M\nIn triangle , point is the point of intersection of the bisectors\nBAC ABC ACB ABC 42\nof angles , , and . The measure of is ,\nBAC 64 BMC\nand the measure of is . What is the measure of ?\nThis task adapted from a problem published by the Russian Ministry of\nEducation.\nIM Commentary"
	},
	{
		"chunk_id" : "00004175",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Education.\nIM Commentary\nThe task is an example of a direct but non-trivial problem in which\nstudents have to reason with angles and angle measurements (and in\nparticular, their knowledge of the sum of the angles in a triangle) to\ndeduce information from a picture.\nThis tasEkn ignacgleu ydoeusr astnu deexnptse rwimithe enftfaecl tGiveeo dGisetabnrcae wleoarrnkisnhge reest,o wurictehs .t hACeCESS RESOURCES>>\nintent that instructors might use it to more interactively demonstrate"
	},
	{
		"chunk_id" : "00004176",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the relevant content material. The file should be considered a draft\nversion, and feedback on it in the comment section is highly\nencouraged, both in terms of suggestions for improvement and for\nideas on using it effectively. The file can be run via the free online\napplication GeoGebra, or run locally if GeoGebra has been installed on\na computer.\nSome notes on the use of the GeoGebra file: The file gives the proctor\nof the problem the abilty to set 2 of the 3 angles before showing the"
	},
	{
		"chunk_id" : "00004177",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "problem to the student. It would also allow for a student or group to\nuse this to evaluate a specific instance of the problem. Also, a teacher\ncould load this on multiple computers where groups or students could\nmove from station to station to have a variety of problems. Once the\nangles are set, the problem appears and the student is allowed to enter\nwhat they think the third angle should be. A popup alerts the student if\nthey are correct or incorrect. There is a check box that shows the"
	},
	{
		"chunk_id" : "00004178",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "solution with a graphical explanation. From there, you can reset the\nproblem to its initial state.\nAttached Resources\nTask 59 Geogebra File\nSolution\nAll angle measurements are in degrees.\nThe solution is obtained by applying the Triangle Sum Theorem twice.\nABC ACB\nFirst apply it to the triangle to find the measure of angle .\n180 (64 +42) = 180 (106) = 74\nThis angle has measure :\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nBMC BM"
	},
	{
		"chunk_id" : "00004179",
		"source" : "Find the Angle.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "BMC BM\nNow consider the triangle . Since the segment bisects the\nABC MBC\nangle of the triangle, we have the measure of angle is half\nABC 42 21\nthe measure of angle , which is half of , or . Similarly, the\nMCB ACB 74,\nmeasure of angle is half of angle , which is half of\n37 BMC\nwhich is . Now use the Triangle Sum Theorem on the triangle\nBMC\nto find that the measure of angle is\n180 (37 +21) = 180 58 = 122 :\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a"
	},
	{
		"chunk_id" : "00004180",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Foundations of Academic Success: Words\nof Wisdom\nFoundations of Academic\nSuccess: Words of Wisdom\nThomas Priester\nOpen SUNY Textbooks\nFoundations of Academic Success: Words of Wisdom by Thomas Priester is licensed under a\nCreative Commons Attribution 4.0 International License, except where otherwise noted.\nYou are free to:\n Share  copy and redistribute the material in any medium or format\n Adapt  remix, transform, and build upon the material\n for any purpose, even commercially."
	},
	{
		"chunk_id" : "00004181",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : " for any purpose, even commercially.\nThe licensor cannot revoke these freedoms as long as you follow the license terms.\nUnder the following terms:\n Attribution You must giveappropriate credit, provide a link to the license,\nandindicate if changes were made. You may do so in any reasonable manner, but not\nin any way that suggests the licensor endorses you or your use.\n No additional restrictions  You may not apply legal terms or technological"
	},
	{
		"chunk_id" : "00004182",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "measuresthat legally restrict others from doing anything the license permits.\nThis publication was made possible by a SUNY Innovative Instruction Technology Grant (IITG). IITG is a\ncompetitive grants program open to SUNY faculty and support staf across all disciplines. IITG encourages\ndevelopment of innovations that meet the Power of SUNYs transformative vision.\nPublished by Open SUNY Textbooks, Milne Library (IITG PI)\nState University of New York at Geneseo, Geneseo, NY 14454\nContents\nAbout the Book 1"
	},
	{
		"chunk_id" : "00004183",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Contents\nAbout the Book 1\nAbout the Author 3\nReviewer's Notes 5\nAbout Open SUNY Textbooks 7\nAcknowledgments 9\nPreface 11\nPart One: Your Solid Foundation\nThe Student Experience 17\nPractice, Practice, Practice 21\nWhy So Many Questions? 24\nThese Are the Best Years of Your Life 26\nWith a Little Help from My Friends 27\nPart Two: You are the President and CEO of You\nCan You Listen to Yourself? 31\nFailure Is Not an Option 33\nThinking Critically and Creatively 35\nTime Is on Your Side 37"
	},
	{
		"chunk_id" : "00004184",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Time Is on Your Side 37\nWhat Do You Enjoy Studying? 40\nPart Three: The Future You\nFighting for My Future Now 43\nSomething Was Diferent 45\nTransferable 46\nIts Like Online Dating 47\nLearn What You Dont Want 52\nConclusion 54\nAbout the Book\nFoundations of Academic Success: Words of Wisdom (FAS: WoW) introduces you to the various\naspectsofstudentandacademiclifeoncampusandpreparesyoutothriveasasuccessfulcollege\nstudent (since there is a difference between a college student and a successful college student)."
	},
	{
		"chunk_id" : "00004185",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Each section of FAS: WoW is framed by self-authored, true-to-life short stories from actual\nStateUniversity of New York(SUNY) students, employees,andalumni.Theadvicetheyshare\nincludes a variety of techniques to help you cope with the demands of college. The lessons\nlearned are meant to enlarge your awareness of self with respect to your academic and personal\ngoals and assist you to gain the necessary skills to succeed in college.\n1\nAbout the Author"
	},
	{
		"chunk_id" : "00004186",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "1\nAbout the Author\nA hope-inspired educator dedicated to helping others interact with the future, Thomas C.\nPriester holds a Doctor of Education degree in Executive Leadership from St. John Fisher\nCollege, a Master of Science degree in Student Personnel Administration from SUNY Buffalo\nState, and a Bachelor of Arts degree in Secondary English Education from Fredonia (where\nhe is also a member of the Alumni Board of Directors). Having worked previously in the"
	},
	{
		"chunk_id" : "00004187",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "areas of academic success, student life, student leadership development, orientation, academic\nadvising, and residence life,Dr.Priester currently serves as theDirector ofTransitional Studies/\nAssistant Professor at SUNY Genesee Community College in Batavia, NY where he is also an\nadvisor to the campus chapter of the Phi Theta Kappa Honor Society, the chairperson of both\nthe Academic Assessment and the Transitional Studies Committees, and a member of both the"
	},
	{
		"chunk_id" : "00004188",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Institutional Effectiveness and the Academic Senate Curriculum Committees. Additionally, Dr.\nPriester is a faculty member in the Higher Education Student Affairs Administration graduate\nprogram at SUNY Buffalo State in Buffalo, NY, has taught conversational English at Fatec\nAmericana in Americana, So Paulo, Brazil, and has also taught Academic Success at the Attica\nCorrectional Facility in Attica, NY. Dr. Priester has served as a contributing chapter author"
	},
	{
		"chunk_id" : "00004189",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "for the books: Assessing Student Learning in the Community and Two-Year College (Stylus, 2013)\nand Examining the Impact of Community Colleges on the Global Workforce (IGI Global, 2015) and\nhas most recently published the open access textbook: Foundations of Academic Success: Words of\nWisdom (Open SUNY Textbooks, 2015).\n3\nReviewer's Notes\nFoundations of Academic Success is an engaging, informational, and succinct read that connects"
	},
	{
		"chunk_id" : "00004190",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the reader to personal essays about succeeding in college. The text allows the reader to see\ndifferent perspectives of the shared experience of navigating higher education. As an adjunct\nlecturerteachingacourseentitledLearningtoLearn,Iseetheadviceandlifelessonsdiscussed\nasbothhelpfulandinformativetomanytypesofstudents.Ibelievethatthisbookhasauniversal\nappeal for any classroom that is discussing the college life cycle, and would advocate using"
	},
	{
		"chunk_id" : "00004191",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "this text with students who seek out campus services, such as Career Services, Study Abroad,\nand Academic Advising. I also see this text applicable in the education of student leaders, peer\nmentors, and peer advocates at the undergraduate and graduate levels. This text goes beyond\ntraditional Student Affairs and Student Development theories to connect the reader with real,\nhonest, and understandable life lessons."
	},
	{
		"chunk_id" : "00004192",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "honest, and understandable life lessons.\nAbout the Reviewer: Sherill Anderson is currently the Second-year Experience Coordinator for The\nCollege at Brockport. Ms. Anderson attended Alfred University, graduating in 2006 with her Bachelors\nof Arts in Sociology. She completed her Master of Education degree in Counseling, with a concentration\nin College Student Development, as well as a certificate of advanced study, in 2008; also from Alfred"
	},
	{
		"chunk_id" : "00004193",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "University. She began her doctoral studies in May of 2013 with St. John Fisher College in the\nEd.D program in Executive Leadership. Ms. Anderson pursued her research exploring the employer\nobservations of both negative and positive social behavioral characteristics that impact employer hiring\ndecisions regarding young adults diagnosed with an autism spectrum disorder.\n5\nAbout Open SUNY Textbooks\nOpen SUNY Textbooks is an open access textbook publishing initiative established by State"
	},
	{
		"chunk_id" : "00004194",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "University of New York libraries and supported by SUNY Innovative Instruction Technology\nGrants. This initiative publishes high-quality, cost-effective course resources by engaging\nfaculty as authors and peer-reviewers, and libraries as publishing service and infrastructure.\nThe pilot launched in 2012, providing an editorial framework and service to authors, students\nand faculty, and establishing a community of practice among libraries."
	},
	{
		"chunk_id" : "00004195",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Participating libraries in the 2012-2013 pilot include SUNY Geneseo, College at Brockport,\nCollege of Environmental Science and Forestry, SUNY Fredonia, Upstate Medical University,\nand University at Buffalo, with support from other SUNY libraries and SUNY Press. The\n2013-2014 pilot will add more titles in 2015. More information can be found at\nhttp://textbooks.opensuny.org.\n7\nAcknowledgments\nFirst, Id like to acknowledge the students enrolled in my FYE100 course at SUNY Genesee"
	},
	{
		"chunk_id" : "00004196",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Community College during theduringthe 2014-2015 academic schoolyear. Your veryhonest\nand candid feedback truly helped to revolutionize academic success for generations of college\nstudents to come.\nMuch appreciation to both Kate Pitcher and Allison Brown at Geneseo for their patience and\nsupport as I worked through the Open SUNY Textbooks publication process.\nThank you, too, to Lindsey Dotson (SUNY Buffalo State, 2016) and Jeffrey Parfitt (SUNY"
	},
	{
		"chunk_id" : "00004197",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Genesee Community College, 2015) for their assistance in making FAS: WoW a reality.\nProps to Nicki Lerczak for giving the final draft the hairy librarian eyeball. Her words, not\nmine!\nFinally, many thanks goes to the following State University of New York (SUNY) students,\nemployees, and alumni for sharing their words of wisdom that frames the text:\nDr. Andrew Robert Baker\nDirector of Community Standards\nFinger Lakes Community College"
	},
	{
		"chunk_id" : "00004198",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Finger Lakes Community College\nGraduate of University at Buffalo, University at Albany, and SUNY Oneonta\nAmie Bernstein\nGraduate of Suffolk ounty Community College\nVicki L. Brown\nDirector of Student Activities\nHerkimer College\nGraduate of SUNY Potsdam\nDr. Kristine Duff\nPresident\nSUNY Adirondack\nGraduate of the College at Brockport\nJamie Edwards\nCareer Services Specialist\nSUNY Genesee Community College\nGraduate of University at Buffalo\nPaulo Fernandes\nStudent at Stony Brook University\nChristopher L. Hockey"
	},
	{
		"chunk_id" : "00004199",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Christopher L. Hockey\nAssistant Director of Student Mobility\nSUNY System Administration\nGraduate of SUNY Oswego\nFatima Rodriguez Johnson\nAssistant Dean of Students, Multicultural Programs & Services\n9\nSUNY Geneseo\nGraduate of SUNY Fredonia\nKristen Mruk\nAssistant Director of Student Activities\nSUNY Genesee Community College\nGraduate of University at Buffalo\nDr. Patricia Munsch\nCounselor\nSuffolk County Community College\nGraduate of SUNY Geneseo\nYuki Sasao\nStudent at SUNY Oswego"
	},
	{
		"chunk_id" : "00004200",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Yuki Sasao\nStudent at SUNY Oswego\nGraduate of SUNY Genesee Community College\nJacqueline Tiermini\nFaculty\nFinger Lakes Community College\nGraduate of SUNY Buffalo State\nSara Vacin\nHuman Services Adjunct Faculty\nSUNY Niagara County Community College\nJackie Vetrano\nGraduate of SUNY Geneseo\nNathan Wallace\nAssistant Project Coordinator, Assessment and Special Projects\nErie Community College\n10\nPreface\nSuccessdoesntcometoyouyougotoit.ThisquotebyDr.MarvaCollinssetsthestagefor"
	},
	{
		"chunk_id" : "00004201",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "thejourneyyouareabouttotake.Yoursuccess,howeveryouchoosetodefin it,iswaitingfor\nyou, and Foundations of Academic Success: Words of Wisdom (FAS: WoW) is your guide to your\nsuccess. Some may believe that success looks like a straight and narrow line that connects the\ndots between where you are and where you are going, but the truth is that success looks more\nlike a hot mess of twists and turns, curves and bumps, and hurdles and alternate pathways."
	},
	{
		"chunk_id" : "00004202",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Putting this textbook together was challenging because there is so much to tell you as you\nembark on your college journey. I have worked with college students on academic success at a\nnumber of college campuses, and have hunted for the most ef ective and most affordable college\nstudent academic success textbook but could never find everything I wanted to teach in one\nbook. So, I figured the answer was to write my own textbook!"
	},
	{
		"chunk_id" : "00004203",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Like any good research project, the outcome was not exactly what I expected. In addition\nto a host of true-to-life stories written by real people who have successfully navigated the\njourney through college, the first draft of the textbook included everything (and more) that\nthe other similarly themed textbooks about college student academic success do. The chapters\nwere framed by a slew of How To facts according to me (such as how to efficientl take"
	},
	{
		"chunk_id" : "00004204",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "notes during a lecture and how to effectively use your preferred learning style to help you\nlearn better) and research-based figuresaccording to researchers in the fieldof college student\nacademic success, such as rules like For every hour in class, successful college-level work\nrequires about two hours of out-of-class work: reading, writing, research, labs, discussion, fiel\nwork, etc. or A 15-credit course load is about equivalent to working a full-time job."
	},
	{
		"chunk_id" : "00004205",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Once the first draft was finished, I decided to test-drive my new textbook with the students\nin my First Year Experience class to see what they thought. I figured, who better to give me\nfeedbackonthetextbookthanactualstudentswhowouldusethetextbookinclass,right?Igave\nthefirs draftofthetextbook(factsandfigure andall)tomystudentstoread,review,andreflec\nupon. It turned out that the pieces that my students learned the most from were the true-to-"
	},
	{
		"chunk_id" : "00004206",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "life stories. They either didnt read or barely glanced over the facts and figures, but provided\nverypositivefeedback(andevenremembered)thewordsofwisdomfromrealpeoplewhohave\nsuccessfully navigated the journey through college.\nI guess it makes sense; students love when real-life stories are infused into the activities and\nlessons.Plus,asanumberofstudentstoldme,thefactsandfigure ontopicssuchasnote-taking\nand how many hours to study per week can be found by searching online and can vary per"
	},
	{
		"chunk_id" : "00004207",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "person. Whatreally mattered to students were the real-life wordsofwisdom that youcantfind\nonline.Thus, FoundationsofAcademicSuccess:WordsofWisdom(FAS:WoWasIlovinglycallit)\nemerged.\nI share this story because my intended outcome (to be the author of the worlds best open\naccess college student academic success textbook) was not exactly what I expected it to be.\nThe same is true of your journey through college, and youll read more about that in the"
	},
	{
		"chunk_id" : "00004208",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "stories right here in FAS: WoW. Youll find that this is not your typical college textbook full\nof concrete facts and figures, nor does it tell you how to succeed. No textbook can truly do\nthatforyousuccessisdefine differentl foreveryone.Thestoriesin FAS:WoWarerelevant,\nrelational, and reflective. The authors welcome you into their lives and offer ideas that ignite\nhelpful discussions that will help succeed.\n11\nFAS: WoW introduces you to the various aspects of student and academic life on campus and"
	},
	{
		"chunk_id" : "00004209",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "preparesyoutothriveasa successfulcollegestudent(sincethereisadifferenc betweenacollege\nstudent and a successful college student). Each section of FAS: WoW is framed by self-authored,\ntrue-to-lifeshortstoriesfromactualStateUniversityofNewYork(SUNY)students,employees,\nand alumni. You may even know some of the authors! The advice they share includes a variety\nof techniques to help you cope with the demands of college. The lessons learned are meant to"
	},
	{
		"chunk_id" : "00004210",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "enlarge your awareness of self with respect to your academic and personal goals and assist you\nto gain the necessary skills to succeed in college.\nIn the text,the authors tell stories about theirownacademic, personal, andlife-career successes.\nWhen reading FAS: WoW, consider the following guiding questions:\n How do you demonstrate college readiness through the use of effective study skills\nand campus resources?\n How do you apply basic technological and information management skills for"
	},
	{
		"chunk_id" : "00004211",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "academic and lifelong career development?\n How do you demonstrate the use of critical and creative thinking skills to solve\nproblems and draw conclusions?\n How do you demonstrate basic awareness of self in connection with academic and\npersonal goals?\n How do you identify and demonstrate knowledge of the implications of choices\nrelated to wellness?\n How do you demonstrate basic knowledge of cultural diversity?"
	},
	{
		"chunk_id" : "00004212",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "After you read each story, take the time to reflect on the lessons learned from your reading and\nanswertheguidingquestionsastheywillhelpyoutoconnectthedotsbetweenbeingacollege\nstudent and being a successful college student. Note your areas of strength and your areas of\nweakness, and develop a plan to turn your weaknesses into strengths.\nI could go on and on (and on) about college student academic success, but what fun is the\njourneyforyouifItellyoueverythingnow?Youneedtolearnsomestufonyourown, right?"
	},
	{
		"chunk_id" : "00004213",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "So, I will leave you to read and enjoy FAS: WoW with a list of tips that I share with college\nstudents as they embark on their journey to academic success:\n Early is on time, on time is late, and late is unacceptable!\n Get the book(s) and read the book(s).\n Take notes in class and when reading for class.\n Know your professors (email, office location, office hours, etc.) and be familiar w\nwhat is in the course syllabus.\n Put your phone away in class.\n Emails need a salutation, a body, and a close."
	},
	{
		"chunk_id" : "00004214",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : " Emails need a salutation, a body, and a close.\n Dont write the way you might textusing abbreviations and clipped sentences.\n Never academically advise yourself!\n Apply for scholarshipsall of them!\n Speak it into existence and keep your eyes on the prize.\nEnjoy the ride! Cheers,\n12\nTOM\nDr. Thomas C. Priester,tcpriester@genesee.edu\n13\nPart One: Your Solid Foundation\nThe Student Experience\nKristen Mruk\nWhen thinking about college, what comes to mind? Perhaps stereotypical images or"
	},
	{
		"chunk_id" : "00004215",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "misconceptions of college life, a friend or siblings story, or scenes from popular movies?\nThe Student Experience by Kristen Mruk\nInpopularculture,somemoviesdepictcollegelifeasapartyatmosphereinwhichstudentsbinge\ndrinkandwastetheirparentsmoneytohaveagoodtimewithoutconsequence.Filmsincluding\nNational Lampoons Animal House and Van Wilder as well as Accepted, to name a few, portray\nthe student experience as a blatant disregard for education coupled with excessive drunken"
	},
	{
		"chunk_id" : "00004216",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "buffoonery However,mypartyexperienceillustratesasidetocollegethatisnotgenerallyinthe\nlimelight.\nDuring my first weeks in college, I felt disconnected from the campus and feared that I would\nnot make friends or find my niche. I was commuting from my familys home and wanted to\ndo more on campus than just go to and from class. I was enrolled in a First-Year Experience\n(FYE) course that was intended to provide a framework for a successful undergraduate career"
	},
	{
		"chunk_id" : "00004217",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and beyond. In the class, we learned about student support services on campus (tutoring,\npersonalwellness,academicadvisement,etc.)aswellaspersonalsuccessskills(timeandfinancia\nmanagement, values exploration, etc.).\nBeing anewstudent, andacommuter,I was overwhelmed bythe amountofnew information,\nnew territory, campus culture, and unfamiliar processes. I asked my FYE instructor after class\none day if there was something I could do to feel more connected to campus. She opened my"
	},
	{
		"chunk_id" : "00004218",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "eyes to a side of college that I was missingthis was my invitation to the party.\n17\n18 Foundations of Academic Success: Words of Wisdom\nMy FYE instructor promptly led me to her office introduced me to the staff,and explained the\nvariety of involvement opportunities available through her office I was amazed that there was\nso much to do on campus! Because of that meeting, I decided to apply for a job in the Student\nUnion working at the information desk. This position was a catalyst for all of the additional"
	},
	{
		"chunk_id" : "00004219",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "parties I would be invited to throughout my time as an undergraduate student. With so many\npossibilities, I had to be diligent in prioritizing my time and energy.\nWhat My Friends Think I Do\nFriends knew me to be much like the girl in the meme above. I was juggling extracurricular\nactivitiesandtwojobsallwhilemaintainingafullcourseload.Ihadtobeproactiveanddiligent\nto coordinate activities and assignments and make sure I had the time to do it all. Finding a"
	},
	{
		"chunk_id" : "00004220",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "system was a trial and error process, but ultimately I found a method that worked for me. I was\nan undergraduate student when apps didnt exist and Facebook was just becoming popular, so\nmy organizational system included a planner, a pen, and a lot of highlighters. Whatever that\norganizational system looks like for you does not matter as long as you use it.\nThere are a variety of organizational methods and tools you can use to stay on track with all"
	},
	{
		"chunk_id" : "00004221",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "aspectsofyourlifeasastudent.SomeofthosearefeaturedintheStateUniversityofNewYork\n(SUNY) blog:http://blog.suny.edu/tag/apps\nWhat My Parents Think I Do\nIt may be difficul to discuss your studies and educational experience with a parent or someone\nthat has a significant interest in your academic achievement. This was the case for me; I was\nthe first kid in my house to enroll in college, and my parents were under the impression that\ngradeswouldbesenthomeliketheywereinhighschool.DuringtheNewStudentOrientation"
	},
	{
		"chunk_id" : "00004222",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "program, my Mom learned about FERPA (Federal Educational Rights and Privacy Act) and\nwhat that meant for my grades. FERPA gives parents certain rights with respect to their\nchildrenseducationrecords.Theserightstransfertothestudentwhenheorshereachestheage\nof 18orattends aschool beyondthehighschoollevel.In essence,parents cannot access grades\nor other restricted academic information unless you provide it to them (http://www2.ed.gov/\npolicy/gen/guid/fpco/ferpa/index.html?src=ft)."
	},
	{
		"chunk_id" : "00004223",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "policy/gen/guid/fpco/ferpa/index.html?src=ft).\nIwasfortunateenoughtohavemyparentsfinancia supporttowardtuition,sotheyfeltentitled\nto reviewing my grades at the end of each semester. I did not want to give them direct access\nto my grade report by filinga FERPA waiver, so after much deliberation, I agreed to share my\ngrades once released at the semesters end. If their standards were not met, there would have to\nbe a conversation about repercussions."
	},
	{
		"chunk_id" : "00004224",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "be a conversation about repercussions.\nIn the fall of my sophomore year I took my first online courseIntroduction to Computers\nand Statistics. All of the lectures and assignments were available online at anytime and exams\nwere administered in a computer lab on campus. I thought having the ability to view lectures\non my own time would be more conducive to my schedule as I was becoming more involved\non campus. For the first few weeks of classes I watched the lectures regularly and did the"
	},
	{
		"chunk_id" : "00004225",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "assignments on time. Slowly but surely I found myself prioritizing my time differently\nultimatelyputtingmyonlineclassonthebackburner,because(Itoldmyself)theworkcouldbe\ndoneanytime!BytheendofthesemesterIrealizedthatIwasgoingtofailtheclass.Noamount\nof extra credit, crying, or pleading could save my grade; I had earned an F.\nSeeing a failing grade on my transcript taught me two valuable lessons. First, I discovered that"
	},
	{
		"chunk_id" : "00004226",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "I needed the routine and accountability of an in-person class to ensure my participation in the\nmaterial.Second,IwasresponsibleforthegradesIreceived.Iprobablycouldhavecomeupwith\namillionexcusesforwhyIdidntwatchthelecturesordotheassignments,buttherealitywasI\nThe Student Experience 19\njustdidntdoit.Ididnotseekmyprofessorshelpduringtheiroffi hourswhenIstartedtofall\nbehind, I didnot go to thetutoring center on campus to get extra help,and I did not reach out\nto my classmates to form study groups."
	},
	{
		"chunk_id" : "00004227",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "to my classmates to form study groups.\nAlthough the F that I received will never disappear from my transcript, it is an important\nreminder of the gruesome conversation I had with my parents and the feeling of failure in the\npit of my stomach. Needlesstosay, thatwas the only online course Itook duringmycollegiate\ncareer, but it was absolutely worth the lessons learned.\nWhat My Professors Hope I Do\nProfessors do care about how you are doing in their class; they genuinely want you to succeed,"
	},
	{
		"chunk_id" : "00004228",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "buttheywillgiveyouthegradeyouearn.Therearepeopleandresourcesoncampusforyouto\nutilize so you can earn the grade you want.\nYour professors are one of those resources, and are perhaps the most important. Go see them\nduringoffi hours,askthemquestionsaboutthematerialandgetextrahelpifyouneedit.The\ncaveat here isthat you cannot wait until the last week of the semester to visit your professors to\nget help. Tears and pleading will not help you at the eleventh hour."
	},
	{
		"chunk_id" : "00004229",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Anotherresourcetoutilizecanbefoundinthecampuslearningcenter.Ifrequentedmycampus\nwriting center for assistance with papers and research projects. Initially, I was scared to be\ncritiqued, thinking my work would be perceived as inadequate. The first time I took a paper\nthere, I recall standing outside the door for about ten minutes thinking of an excuse not to go\nin. Thankfully I saw a classmate walk in and I followed suit. The experience was less dramatic"
	},
	{
		"chunk_id" : "00004230",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "than I imagined it to be; no one ripped my paper to shreds and told me that I would never\ngraduate. Instead I sat with an upper-class student who coached me through some pointers and\nsuggestions for improvement. Thanks to that first visit, I received an A- on the paper!\nWhat I Would Like to Do\nI thought I knew exactly what I wanted to do when I started college, but that changed three\ntimesbythetimeIgraduated.InitiallyIstartedasanInternationalBusinessmajorbutendedup"
	},
	{
		"chunk_id" : "00004231",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "receivingadegreeinCommunicationandcontinuedontograduateschool.Mygreatestadvice\nto you is to embrace feelings of uncertainty (if you have them) with regard to your academic,\ncareer, or life goals. Stop into the Career Services offic on your campus to identify what it is\nthatyoureallywanttodowhenyougraduateortoconfir youraffini toacareerpath.Make\nanappointmenttoseeacounselorifyouneedtoventorgetanewperspective.Doaninternship"
	},
	{
		"chunk_id" : "00004232",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "in your field;this can give you a first-handimpression of what your life might look like in that\nrole.\nWhen I chose International Business, I did not do so as an informed student. I enjoyed and\nexcelled in my business courses in high school and I had hopes of traveling the world, so\nInternational Business seemed to fit the bill. Little did I know, the major required a lot of\naccounting and economics which, as it turned out, were not my forte. Thinking this is what I"
	},
	{
		"chunk_id" : "00004233",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "wanted,IwastedtimepursuingamajorIdidntenjoyandacademiccoursesIstruggledthrough.\nSo I took a different approach. I began speaking to the professionals around me that had jobs\nthat appealed to me: Student Unions/Activities, Leadership, Orientation, Alumni, etc. I found\noutIcouldhaveasimilarcareer,andIwouldenjoytherequiredstudiesalongtheway.Making\nthat discovery provided direction and purpose in my major and extracurricular activities. I felt\nlike everything was falling into place."
	},
	{
		"chunk_id" : "00004234",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "like everything was falling into place.\n20 Foundations of Academic Success: Words of Wisdom\nWhat I Actually Do\nI would like to pause for a moment and ask you to consider why you are in college? Why did\nyou choose your institution? Have you declared a major yet? Why or why not? What are your\nplanspost-graduation? By frequently reflectin in thisway,you canassesswhetheror notyour\nbehaviors, affiliations, and activities align with your goals"
	},
	{
		"chunk_id" : "00004235",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "What you actually do with your student experience is completely up to you. You are the only\nperson who can dictate your collegiate fate. Remind yourself of the reasons why you are in\ncollege and make sure your time is spent on achieving your goals. There are resources and\npeople on your campus available to help you. You have the controluse it wisely.\nPractice, Practice, Practice\nDr. Kristine Duffy\nLife in college will be like no other time in your lifeI can guarantee you that! This is your"
	},
	{
		"chunk_id" : "00004236",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "timetoexplorewhoyouare,whoyouwanttobecome,andhowyouwishtoplayapartinthis\nworld. Dont squander this unique time in your life. I hope to share some thoughts that might\nhelp you avoid regrets when refecting on your college years.\nIwanttobecleartherearemanypathsthroughcollegeandweknowthatnoonepathisright\nfor all. You may be starting at a community college, taking courses part-time, starting college\nagainafteranunsuccessfulstart,orreturningtoeducationaftermanyyearsaway,butnomatter"
	},
	{
		"chunk_id" : "00004237",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "who you are or what path youve chosen, make the most of it.\nI took the fairly traditional path. I graduated from high school and went directly to college\n(which was three hours away from home). Because I wasnt really sure what else I should do,\nI chose to be a business major by default. My parents thought it was a good route to take and\nwould lead me to a good job (mainly to ensure I made some money and didnt live with them\nforever).\nThere are three things I learned quickly in college:"
	},
	{
		"chunk_id" : "00004238",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "1. I had lived a very nice life, but in a very homogeneous environment.\n2. There were people diferent from me.\n3. Although I was a decent student, I had a ways to go to be a good student!\nLearning to appreciate what you have is just as important as earning As on exams and papers. I\nsharethisbecausepartofcollegeispreparingforlife,notjustajob.Askyourselfsomequestions:\n Whats important to me and why?\n What do you know about other peoples lives, beliefs, and passions?"
	},
	{
		"chunk_id" : "00004239",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : " Are you confdent in your abilities to study, listen and learn, take notes, and be a\nlearner?\nWhats Important to Me and Why?\nIsitonlytomakemoneytobuythings?Ifso,doyoutrulybelievethatmoneymakeseverything\nbetter? Dont be fooled by that. Yes, money certainly makes life more comfortable, but it\nabsolutely doesnt buy happiness. I had friends in college that came from a signifcant amount\nof money and they would have traded it all to have a family they can depend upon and love in"
	},
	{
		"chunk_id" : "00004240",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "their homes. Consider this very carefully as you dream of the life ahead of you.\nWhat Do You Know about Other Peoples Lives, Beliefs, and\nPassions?\nYouarenotthecenteroftheworld.Youshouldbeconfdent andproudofwhoyouare,butbe\nhumbleandbeopentoothersexperiencesandworldviews.Takeclassesthatstretchyou,maybe\neven make you uncomfortable. In the end these types of classes will test your assumptions,\nbeliefs,andmakeyouamorewell-rounded andinterestingperson. Theroommateorclassmate"
	},
	{
		"chunk_id" : "00004241",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "that is diferent from you can teach you about yourself. Be open to this.\n21\n22 Foundations of Academic Success: Words of Wisdom\nAre You Confident in Your Abilities to Study, Listen and Learn, Take\nNotes, and Be a Learner?\nRemember, ifcollegewereeasy, everyone woulddoit!Youhavefullcontroland responsibility\nfor your learning. Yes,yourprofessors have theresponsibility ofteaching well and helping you\nlearn. But they cannot and should not do the work for you. Part of college is learning to learn:"
	},
	{
		"chunk_id" : "00004242",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "learningtostudy,listenbetter,takenotes,andmostimportantlyaskingforhelpwhenyouneed\nit.\nInmyownresearchIhavelearnedthatstudentsareconfrontedwithaparadoxicalsituation.On\ntheonehand,studentsinhighschoolarewarnedthatcollegewillbehardtheprofessorswont\ncareifyoudotheworkornot,andyouneedtodoitonyourown.Howeverinreality,college\nprofessors and support professionals do care and will tell you to come and see them if you need\nhelp."
	},
	{
		"chunk_id" : "00004243",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "help.\nSo what is a student to do? You may feel bad in class if you just arent getting it and are\nembarrassed to ask for help. Stop that thought in its tracks! Colleges offer many opportunities\nfor help and in almost all cases, for free! Professors offer offic hours specifically to address\nstudents questions and tutoring is available to help you do better, not to punish you for not\ngetting it. Remember you are paying a substantial amount of money for your tuition; find out"
	},
	{
		"chunk_id" : "00004244",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "what resources you have and take advantage of them. Be a mature learner, take advantage of\neverything your college offers, and hold your head high for doing so. There is no shame in\nasking for help. I always compare it to a job. When you start out on any job there is usually\nsometypeoftraining toteachyouhowtodothatjob. College isnodifferent Weareteaching\nyouhowtobeastudentyouvebeenpracticingsinceKindergarten,anddoesntendwhenyou\nget to college."
	},
	{
		"chunk_id" : "00004245",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "get to college.\nFinally,herearesomewordsofadvicebasedonsomeofmyregretswhenIreflec onmycollege\nexperience:\n1. I didnt study abroad during my four years of college.\n2. I didnt do any type of internship.\n3. I didnt get involved with many clubs or organizations.\n4. I didnt get involved with any type of research opportunities until graduate school.\nStudy Abroad\nWhether it is a short-term experience (some are as short as three weeks) or a semester to a"
	},
	{
		"chunk_id" : "00004246",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "yeardo it! This goes back to my point about understanding people different than you. The\nUnited States is a great nation, but we are not the only nation and our world is filled with\namazing stories to share. Oneof my favorite quotes by Neale Walsch is:Lifebegins at the end\nofyourcomfortzone(2010).Youwillnotmissmuchbeinggonefromyourcollegeforashort\nperiodoftime,andyouwillreturnfromyouradventureachangedperson.HowdoIknowthis\nifIdidntstudyabroadmyself?Iknowmanywhohaveandtheendresultisthesameforallno"
	},
	{
		"chunk_id" : "00004247",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "regrets, life changing moments, and better appreciation for the world we live in.\nInternships\nGoing to college in the 80s was different than today. The job market was relatively strong\nand the push for an internship or co-op was not as strong. But if I had gotten some hands\non experience and discovered my likes, dislikes, strengths, and weaknesses, I would had more\ndirection for my career when I graduated. In addition, there is nothing more frustrating for a"
	},
	{
		"chunk_id" : "00004248",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "collegegraduatethantogoonjobinterviewsonlytobetoldthatyoucantbehiredbecauseyou\nhave no real experience. So talk to your professors, academic advisors, counselors, and mentors\nPractice, Practice, Practice 23\nabout getting some internship experience while in school or during the summer. There are\nmany companies that welcome interns, and you may find the direction you are seeking.\nClubs and Organizations\nFor years employers have been surveyed by colleges to ask them what type of skills they are"
	},
	{
		"chunk_id" : "00004249",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "seeking in college graduates. Although having discipline specific skills are important (in other\nwords, the courses you take in your major), employers are very consistent in seeking out\nemployees with what they call soft skills, such as writing well, public speaking, getting along\nwith others, and having leadership abilities. Youll develop these skills in your courses, but you\ncan really hone and apply them by joining a club or organization on campus, where you will"
	},
	{
		"chunk_id" : "00004250",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "have opportunities to work with others, lead efforts, and have something to show for ita\ncampaign you ran, funds you raised, or an event you organized. Colleges offer many types of\nclubs to attract students in areas of interest. For example, if you are a business major, you could\njoin the business club. More than likely the activities the club offers will allow you to meet\nbusiness leaders, go on fieldtrips to learn more about the business world, and meet people who"
	},
	{
		"chunk_id" : "00004251",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "have similar interests as yourself. I was a college athlete so my time was limited, and while I\nsupport athletics in college as an opportunity to continue your passion and to grow and learn,\ntry to make time to join a special interest group. Take a leadership role in a group, and later,\nwhen you go on that job interview, talk about your leadership experience. The employer will\nbe impressed and it may determine whether or not you get the job.\nResearch"
	},
	{
		"chunk_id" : "00004252",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Research\nFinally, develop your research skills. You may think that research is most important in the\nsciences and medicine. But research occurs in all fields of study, and much of what you do\nin college is research in some form. If you are a music major you may need to research how\nother musicians developed their talent, the history of genres, or new ways music is applied in\nour world.Problem solvingthrough effectiv researchand knowing how totestyourideas and"
	},
	{
		"chunk_id" : "00004253",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "hypotheses will make you a very valuable employee and citizen of your community. If your\nprofessor offers a chance to work on a special research projectsign up.\nQuestion everything, and dont take the answers at face value. Question how people come to\ntheir conclusions, develop your own set of research questions, and be willing to dig to find\nthe answers. This is not only important as a student but as an employee as well. Strive to be"
	},
	{
		"chunk_id" : "00004254",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "an engaged citizen in our world and dont believe what everyone tells you. An adult needs to\nmake informed decisions to buy products, pay taxes, and vote for government leaders. Dont\nbe complacent and put your life in the hands of others without fully researching the pros and\nconsdraw your own conclusions.\nIn conclusion, come to the classroom with an open mind and a willingness to exercise your\nrighttotakefulladvantageofallacollegeoffers Donecorrectly,collegewillbechallengingand"
	},
	{
		"chunk_id" : "00004255",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "frustrating,andwilltesteverypartofyou.Lifewillbethesamewaysousethistimetopractice,\npractice, practice.\nReference\nWalsch, N. D. (2010). Neale Donald Walschs little book of life: A users manual. Charlottesville,\nVA: Hampton Roads.\nWhy So Many Questions?\nFatima Rodriguez Johnson\nI chose to attend a small liberal arts college. The campus was predominately white and was\nnestled in a wealthy suburb among beautiful trees and landscaped lawns. My stepfather and I"
	},
	{
		"chunk_id" : "00004256",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "pulled into the parking lot and followed the path to my residence hall. The looks we received\nfrom most of the families made me feel like everyone knew we didnt belong. But, he and I\ngreeted all we encountered, smiling and saying, Hello. Once I was unpacked and settled into\nmyresidence hall,hegave meahugandsaid,Goodluck.Iwasnt sureif he meant good luck\nwith classes or good luck with meeting new friends, but I heard a weight in his voice. He was"
	},
	{
		"chunk_id" : "00004257",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "worried. Had he and my mother prepared me for what was ahead?\nWith excitement, I greeted my roommate who I had already met through the summer Higher\nEducationalOpportunity Program(HEOP).SheandIwereveryhappytoseeeachother.After\ndecorating and organizing our room, we set out to meet new people. We went to every room\nintroducingourselves.Wewereprettysurenoonewouldforgetus;itwouldbehardtomissthe\nonly Black and Latina girls whose room was next to the pay phone (yes, in my day each floor\nshared one pay phone)."
	},
	{
		"chunk_id" : "00004258",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "shared one pay phone).\nEveryoneonourfloo wasniceandweoftenhungoutineachothersrooms.Andlikesomeof\nyou, we answered some of those annoying questions:\n Why does your perm make your hair straight when ours makes our hair curly?\n How did your hair grow so long (whenever we had weave braids)?\n Why dont you wash your hair everyday (the most intriguing question of all)?\nWe were also asked questions that made us angry:\n Did you grow up with your father?\n Arent you scared to take public transportation?"
	},
	{
		"chunk_id" : "00004259",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : " Have you ever seen anyone get shot (because we both lived in the inner city)?\nIt was those questions that, depending on the day and what kind of mood we were in, made a\nfellowstudenteitherwalkawaywithabetterunderstandingofwhowewereasBlackandLatina\nwomen or made afellow student walkaway red and confused. I guess thats why my stepfather\nsaid, Good luck. He knew that I was living in a community where I would stand outwhere"
	},
	{
		"chunk_id" : "00004260",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "I would have to explain who I was. Some days I was really good at answering those questions\nand some days I was not. I learned the questions were not the problem; it was not asking that\nwas troubling.\nMy roommate and I put forth a lot of effort to fit in with the communitywe spent time\nhanging out with our peers, we ate together almost every evening in the dining hall, and we\nparticipated in student organizations. We were invited to join the German Club, and were the"
	},
	{
		"chunk_id" : "00004261",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "onlystudentsofcolorthere.Indoingallthesethingswemadeourselvesapproachable.Ourpeers\nbecame comfortable around us and trusted us.\nAlthough my peers and I all had similar college stresses (tests, papers, projects, etc.) my\nroommate and I also had become a student resource for diversity. Not because we wanted to,\nbutbecausewehadtoo.Therewereveryfew studentsofcoloroncampus,andIthink students\nreally wanted to learn about people different from themselves. It was a responsibility that we\n24"
	},
	{
		"chunk_id" : "00004262",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "24\nWhy So Many Questions? 25\nhadaccepted.ThedirectorofHEOPwouldoftenremindusthatformanystudents,collegewas\nthefirs opportunity theyhad toaskthesetypesof questions.Hesaid wewouldlearnto discern\nwhenpeoplewerereallyinterestedinlearningaboutourdifference orinsultingus.Ifsomeone\nwas interested in insulting us, there was no need to respond at all.\nAlthough I transferred to another college at the end of my sophomore year, during those"
	},
	{
		"chunk_id" : "00004263",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "two years I learned a great deal about having honest conversations. Taking part in honest\nconversationschallengedmynotionsoftheworldandhowIviewedpeoplefromallwalksoflife\n(race, class, sexual orientation, ability, etc.). Those late nights studying or walks to the student\ncenter were when many of us listened to each others stories.\nMy advice is to take time to examine your attitudes and perceptions of people different from"
	},
	{
		"chunk_id" : "00004264",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "yourself, put yourself in situations that will challenge your assumptions, and lastly, when you\nmakeamistakedonotgetdiscouraged.Keeptrying.Itseasytostaywherewearecomfortable.\nCollege is such a wonderful experience. Take it all in, and I am sure you will enjoy it!\nThese Are the Best Years of Your Life\nSara Vacin\nThese are the best years of your life. I hope youve been told this a ridiculous amount of times\nand that you are finding this to be true! College provides an amazing opportunity to expand"
	},
	{
		"chunk_id" : "00004265",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "your mind, meet unique people you can deeply connect with, and discover new aspects of\nyourself.Beingawareofthisenergy and taking full advantage of these opportunitiescan belife\nchanging.\nYoulearnalotaboutyourselfwhenlivingonyourownforthefirs timeorstudyingtopicsthat\narecompletelytabooatyourhomeskitchentable.WhenItransferredtoafour-yearinstitution,\nI found the strength to come out. Realizing I was gay led me to question where I belonged in"
	},
	{
		"chunk_id" : "00004266",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the religion I was raised in and an enlightening journey ensued of exploring Buddhism, Native\nAmerican beliefs, and even New Age mysticism. This process of questioning what I believed\nhelped me to create a spiritual foundation that makes sense to me. I kept the best of what I was\nraised in and upgraded the rest!\nI also discovered that the college I attended had amazing tools to help me be as healthy as\npossible. I used the free gym and knew the counseling center was there if anything became"
	},
	{
		"chunk_id" : "00004267",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "tootough.Ialsochoseincredibleelectives(includingMountaineeringandModernDance)that\nstretched my physical capabilities. Additionally, I made deep connections with my professors,\nmany of whom remain friends. These smart, caring people validated my journey and were my\nsafety net as I grew out of my old, comfortable self.\nAnother incredible lesson learned was the importance of balance. I couldnt party every night\nand neglect my schoolwork without consequences. I figured out the hard way that I really did"
	},
	{
		"chunk_id" : "00004268",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "needsleepandIcouldntnourishmybodyoncoffe andpizzaalone.Inamomentofbrilliance,\nI also figured out that if I used time with my friends as a reward for finishing my work, I\nwouldstudyandcompleteassignmentsmoreefficientl Funcanbeagreatmotivatortry this;\nit works!\nIncollege,theemphasisisoftenonthemind.Doyourselfafavorandremembertoalsonourish\nyour spirit and take care of your body. Leave college brighter, healthier, and with a new"
	},
	{
		"chunk_id" : "00004269",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "understanding of yourself. Try that yoga or nutrition class. Join that new club. Trade in that\nsoda for water. Jump into that drum circle or improvisation group. Who knows what you will\ndiscoverit just may be greatness!\n26\nWith a Little Help from My Friends\nPaulo Fernandes\nWeoftenhearabouttheimportanceofrelationships:anecessaryaspectofintegrationinsociety.\nUnfortunately, we rarely follow that advice. Perhaps we live an excessively busy life or we"
	},
	{
		"chunk_id" : "00004270",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "alreadyhaveaclosegroupoffriendsanddonotfeelcompelledtomeetnewpeople.Ihavecome\ntolearnthroughmytimeincollegethatneglectingtocultivatenewrelationshipsisdetrimental\nto living a happy and successful life. I would like to offer this piece of advice: no matter how\ndifficul it seems at first, always try to make new friends. College is not always easy. However,\nhaving friends makes it much easier. Friends are a vital part of your life that can expose you"
	},
	{
		"chunk_id" : "00004271",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "to new subjects, cultures, and experiences while giving you the opportunity to do the same for\nthem.\nAtmycollege,therewasasmallspacethatthestudentscalledthebatcave.Itwasbynomeans\na first-class lounge, but it was a place where friends could help others better understand their\ncourse material. We gave it this peculiar nickname because it was our place to get together\nand conquer villains one after another. These were not your everyday super villains, however."
	},
	{
		"chunk_id" : "00004272",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Sometimes they were complicated homework assignments and other times they were difficul\nexams. No matter the challenge, someone was always willing to help. I went to the bat cave\nseveral times and every visit I learned something new. Professors and teaching assistants could\nnot relate to us like our friends could. That made a difference,because nothing was better than\nbeing taught by a friend.\nFriends are not only an essential support for your time in school, but also can be integral in"
	},
	{
		"chunk_id" : "00004273",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "helping realize post-college aspirations. During a visit to New York City, I visited the offic\nof the company Spotify. After touring their facilities I had the opportunity to talk to some of\nthe employees. One man I talked with was a senior employee who worked at Microsoft prior\nto joining Spotifys team. Our conversation stuck in my head because he gave a very striking\npieceofadvice:makefriends.Itnevertrulyoccurredtomethatthefriendsyoumakeincollege"
	},
	{
		"chunk_id" : "00004274",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "couldimpactyourfutureintheworkforce.Theycouldbepartnersinpotentialbusinessventures\nor help you land your dream job. In any case, having strong connections with friends can\nundoubtedly make a major difference in your career.\nThe best part of making new friends, however, is trading life experiences, skills, and interests\nwith them. For a year and a half before my final semester of college, I studied abroad in the\nUnited States. My family was concerned because typically, students search for firstjobs prior to"
	},
	{
		"chunk_id" : "00004275",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "graduation. I, on the other hand, had no trepidations about going because I knew that I would\nhavecountless,excitinglearningexperiences.Icansaytoday,withoutadoubt,thatmytripwas\na great decision. I met incredible people, and through knowing them, I grew and changed. I\nalso know that I was a positive feature in the lives of my new friends. The greatest thing that\nI learned was that meeting different people with different backgrounds, histories, perspectives,"
	},
	{
		"chunk_id" : "00004276",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "orevendifferen musicaltastes, inevitablychangesyouandletsyouseetheworldinanentirely\ndifferent way. You no longer see the world as simply a big, blue sphere with freezing winters\nor sizzling summers (although that certainly seems to be the case up North!), but as a place in\nwhich people like you live, learn, and love.\nGoing to college may seem hard, but it does not need to be. I have learned that the way I\nperceive my life as a student completely relies upon my relationships with my friends. They"
	},
	{
		"chunk_id" : "00004277",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "are not only the people that I like to spend time with, but also are essential in my growth and\ndevelopmentasahumanbeing.Thepagesinthisbookincludeinsightsfromothersjustlikeyou\n27\n28 Foundations of Academic Success: Words of Wisdom\nand me. They want to help you get through the common struggles of college with confidence\nand perseverance. Consider them your most recent new friends. I truly hope that this inspires\nyou in your quest for a great future.\nPart Two: You are the President\nand CEO of You"
	},
	{
		"chunk_id" : "00004278",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Part Two: You are the President\nand CEO of You\nCan You Listen to Yourself?\nYuki Sasao\nIt is almost impossible to find time away from information sources like TV, phone,\nadvertisements,orevenyourfriendsandfamilyinthismodernsociety.Canyouputyourselfin\na place that has no information at allyou alone, just yourself? If not, you should tryfindin\nthisquietmentalspacewillletyoutopracticelisteningtoyourself.Itisawonderfulwaytofin"
	},
	{
		"chunk_id" : "00004279",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "out who you truly are. Our society has become so loud that it is very difficul to listen to your\nown voice and extremely easy to lose it.\nIamaninternationalstudentfromaverysmalltowninJapan,andIamthefirs oneamongmy\nfamilymemberstostudyabroad.WhenItoldpeoplethatIdecidedtocometotheUnitedStates\nto study,every single oneofthem wasshocked and gave me their advice.Some said Americans\nareverybossyandtendtolookdownonpeople.SomesaidIwouldnotbeabletofin anyjobs\nthere."
	},
	{
		"chunk_id" : "00004280",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "there.\nIt does not matter who you are and what type of circumstances you are in. You will get some\ncomments and advice no matter what you do.\nWas the advice people gave me accurate? Im sorry, but mostly, no. People I met in the United\nStateswerenice,andtheadviceIreceivedreallydependedonwhetherornotthepersonlooked\ndownonothers.AmIstrugglingwithfindin ajob?No.Mymajorinaccountingprovidesme\nmoreopportunitiesthanIcantake.Lookingbackonthecommentsfrommyfriendsandfamily,"
	},
	{
		"chunk_id" : "00004281",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "I am very grateful that I was able to see what I truly wanted and stick with my decision.\nThe reason I could tune out those negative voices was not because I am lucky or intelligent. It\nisbecauseIlistenedtomyselfmyownvoice.However,thisdoesntmeanthatIdidntlistento\nothers.IconsideredwhatpeoplesaidtomeandIunderstoodthem.Ijustdidntagreewiththem,\nwhich was the most difficul part. In the process of building my own decisions, many pieces of"
	},
	{
		"chunk_id" : "00004282",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "advice actually helped me and I made some changes based on the advice from others combined\nwith my own thoughts.\nWhywasIabletosticktomydecisionsotightandlivethelifeIwanted?ItsbecauseItalkedto\nmyself and asked myself millions of questions.\nWhat do I want in my future?\nDo I really need it or just want it?\nAm I where I wanted to be? Yes? No? Why?\nWhere am I going?\nWhat am I doing?\nWhat would happen if I do this?\nWhy am I doing it?"
	},
	{
		"chunk_id" : "00004283",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Why am I doing it?\nIt is difficult frustrating, andtime-consuming to findyour raw voice in this very noisysociety,\nbut in doing so you will get through life with minimal regret and confidence in who you are\nand what you are doing. Pull yourself away from the massive amount of information, talk to\n31\n32 Foundations of Academic Success: Words of Wisdom\nsome people, understand them (never ignore them), and then talk with yourself. This is your"
	},
	{
		"chunk_id" : "00004284",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "life,andyoucannotrunawayfromyourselfforever.Youdbetterlearnhowtolistentoyourself\nand be able to stick with your own thoughts even after accepting what other say.\nFailure Is Not an Option\nNathan Wallace\nIn the movie Apollo 13, Ed Harris portrays NASA flight director Gene Kranz as he successfully\nguidesthecrewofadamagedspacecrafttosafety.InafamoussceneduringwhichKranzandhis\nstaffare attempting to overcome some extremely daunting challenges, Harris shouts, Failure is"
	},
	{
		"chunk_id" : "00004285",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "notanoption!ThissingularstatementperfectlyarticulatedthedeterminationofKranztobring\nthe Apollo astronauts back to Earth.\nThis failure is not an option credo was perfect for the life and death situation that NASA\nwas facing. Failure meant that the astronauts on Apollo 13 would never come home, and\nthat outcome was unacceptable. Attending college, on the other hand, shouldnt be a life or\ndeath experience, though it sometimes might feel like one. Failure, though never the intended"
	},
	{
		"chunk_id" : "00004286",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "outcome, can and sometimes does happen. Sometimes failure manifests itself in election results\nfor a student government post, in a test score, or even in a final grade.\nThroughout my life I have had many failures. In high school I drove my parents and teachers\ncrazy because of my lack of academic achievement. I even managed to get an F- in Spanish on\nmy report card. When I told my mom that it was a typo she responded, So you didnt get an"
	},
	{
		"chunk_id" : "00004287",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "F? No, I said, I definitelyearned the F, but theres no such thing as an F-. To this day Im\nnot so sure that my reply was accurate. I might have earned that minus after all.\nMyfailuresinhighschoolledtoonlyoneacceptancefromofallthecollegesIappliedtoattend.\nFurthermore, I was not accepted to the schools main campus, but to their branch campus.\nDuring my firstsemester there my effort wasnt much better than in high school, but since my"
	},
	{
		"chunk_id" : "00004288",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "parents were now paying for my education I did enough work to avoid academic probation. It\nwasnt until my second semester that I found my niche as a Religious Studies major and started\ngetting good grades, moved to the main campus, and eventually graduated with honors.\nSince graduating from college, my career path has taken me into higher education as a Student\nAffairs administrator. This career has exposed me to many great theories regarding student"
	},
	{
		"chunk_id" : "00004289",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "success,andmanyofthemgavemeinsightintomyowncollegeexperience.ButitwasStanford\npsychologist Carol Dweck who appeared to be thinking of me when she wrote the following\nabout fixed mindsets in the introduction to her book titled Mindset: The New Psychology of\nSuccess:\nBelievingthatyourqualitiesarecarvedinstonethefixe mindsetcreatesanurgencytoprove\nyourself over and over. If you have only a certain amount of intelligence, a certain personality,"
	},
	{
		"chunk_id" : "00004290",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and a certain moral characterwell, then youd better prove that you have a healthy dose of\nthem.Itsimplywouldntdotolookorfeeldeficien inthesemostbasiccharacteristics.(Dweck,\n2006)\nThisstatementwasarevelationtome.Ifinall understoodmyproblemthroughouthighschool\nand even in college. I earned good grades because I liked Religious Studies but never really\nchallengedmyselfinsideoroutsideoftheclassroom.MyproblemwasthatIhadafixe mindset"
	},
	{
		"chunk_id" : "00004291",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "aboutacademicsuccess.Ibelievedthatapersoniseithersmartortheyrenot,andnothingcould\nbe done to significantly change that. I also believed that I was one of the fortunate ones to be\ngifted with an abundance of intelligence.\nOnemightthinkthathavingconfidenc inyourintelligenceisawholelotbetterthanthinking\n33\n34 Foundations of Academic Success: Words of Wisdom\nthat youre stupid, but the result was the same. My fixedmindset was holding me back because"
	},
	{
		"chunk_id" : "00004292",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "it led to a paralyzing fear of failure. Since as far back as I could remember, my family, friends,\nandteacherswerealwaystellingmehowsmartIwas,andIbelievedthem.Butthatbeliefwasa\ndouble-edged sword. High school and college offeredmany occasions when self-confidencein\nmy inherent intelligence could be threatened. If I fail on this test or in this course it means that\nIm not the smart person I thought I was. If I fail, my family and friends will findout that they\nwere wrong about me."
	},
	{
		"chunk_id" : "00004293",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "were wrong about me.\nHowever, there was a way to avoid all of the risks of academic rigor. I could just not try.\nIf I dont try Ill get bad marks on my report card, but those wont be true indicators of my\nintelligence.Bynotputtingforthanyeffort myintelligencewouldneverbedisproven.Iwould\nalwaysbeabletosaytomyselfandothersthat,IcoulddotheworkandbeastraightAstudent,\nbutImjustnotinterested.Lookingbackonthistimeinmylife,itiscleartomethatthiswasnt"
	},
	{
		"chunk_id" : "00004294",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "a conscious decision to save face. It was fear, not logic, which was guiding my behavior.\nAfterreading Mindset Ihavemadeaconsciouseffor toidentifyandthwartanyremainingfixe\nmindset thoughts that I continue to hold. Dwecks book acts as a manual for rooting out fixe\nmindsetthoughts,becausesheexplainsthattheideaoffixe mindsetsisonlyhalfofhermindset\ntheory. There is another kind of mindset, and she calls it growth mindset. Dweck writes that,"
	},
	{
		"chunk_id" : "00004295",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Thisgrowthmindsetisbasedonthebeliefthatyourbasicqualitiesarethingsyoucancultivate\nthrough your efforts (Dweck, 2006). Dweck goes on to explain that we can choose to have a\ngrowthmindsetaboutanytypeofability,whetheritsmath,art,athletics,oranyotherskillthat\none wishes to cultivate.\nI put this theory to the test not long after reading the book. A few years ago I attended a\nmeetingonlytofin outthatitwasntanyordinarymeeting.Duringthismeetingwewouldbe"
	},
	{
		"chunk_id" : "00004296",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "brainstormingsolutionstoaspecifi problem.Thiswasgoingtobeatruebrainstormingsession,\nled by a facilitator trained in the science of soliciting uninhibited ideas from an audience. As\nsoonasIheardthewordbrainstormingIfroze.Ihavealwayshatedbrainstorming.Imthetypeof\nperson that likes to think things through two or three times before expressing an opinion. My\nfear of failing at this task in front of my coworkers paralyzed my mind. I couldnt think."
	},
	{
		"chunk_id" : "00004297",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Thats when it hit me. This was fixed mindset thinking. My belief in my brainstorming\ninadequacies was preventing me from even trying. So I flipped this thinking on its head and\ndecidedthebestwaytoimprovemybrainstormingabilitieswastoclearmymindandstartfirin\noutideas.Igaveitashot,andthoughtheideasdidntcomeoutattheprolifi rateofsomeofmy\ncolleagues,Ihadneverbeforehadsuchapositiveoutcomeandexperiencewhilebrainstorming."
	},
	{
		"chunk_id" : "00004298",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Through this experience I found that I really could choose to have a growth mindset, and that\nthischoiceproducesagreaterchanceofsuccess.Withagreaterchanceofsuccesscomesasmaller\nchance of failure.\nNevertheless,whenitcomestoacademicsuccessandsuccessinallphasesoflife,failureisalways\nan option. Though it can be painful, failure can lead to great learning and progress when a\nspecific failure is analyzed through the lens of a growth mindset. By focusing more on effort"
	},
	{
		"chunk_id" : "00004299",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "than onoutcomesanyone can learn andgrow,regardlessoftheirskilllevel. Therefore, to make\nthemostoftheirtimeincollege,studentsmustseekoutchallengesthatwillstretchtheirabilities.\nThese challenges can take many forms and they can occur in a variety of settings, both inside\nand outside of the classroom. When seeking out challenges there is always the possibility of\nagonizing defeat, but out of that defeat can be the seeds of great success in the future.\nReference"
	},
	{
		"chunk_id" : "00004300",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Reference\nDweck, C. (2006). Mindset: The new psychology of success. New York: Ballantine Books.\nThinking Critically and Creatively\nDr. Andrew Robert Baker\nCritical and creative thinking skills are perhaps the most fundamental skills involved in making\njudgments and solving problems. They are some of the most important skills I have ever\ndeveloped. I use them everyday and continue to work to improve them both.\nTheabilitytothinkcriticallyaboutamattertoanalyzeaquestion,situation,orproblemdown"
	},
	{
		"chunk_id" : "00004301",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "to its most basic partsis what helps us evaluate the accuracy and truthfulness of statements,\nclaims, and information we read and hear. It is the sharp knife that, when honed, separates fact\nfrom fiction, honesty from lies, and the accurate from the misleading. We all use this skill to\none degree or another almost every day. For example, we use critical thinking every day as we\nconsiderthelatestconsumerproductsandwhyoneparticularproductisthebestamongitspeers."
	},
	{
		"chunk_id" : "00004302",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Is it a quality product because a celebrity endorses it? Because a lot of other people may have\nused it? Because it is made by one company versus another? Or perhaps because it is made in\none country or another? These are questions representative of critical thinking.\nThe academic setting demands more of us in terms of critical thinking than everyday life. It\ndemands that we evaluate information and analyze a myriad of issues. It is the environment"
	},
	{
		"chunk_id" : "00004303",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "where our critical thinking skills can be the difference between success and failure. In this\nenvironment we must consider information in an analytical, critical manner. We must ask\nquestionsWhatisthesourceofthisinformation?Isthissourceanexpertoneandwhatmakesit\nso?Aretheremultipleperspectivestoconsideronanissue?Domultiplesourcesagreeordisagree\non an issue? Does quality research substantiate information or opinion? Do I have any personal"
	},
	{
		"chunk_id" : "00004304",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "biases that may affect my consideration of this information? It is only through purposeful,\nfrequent, intentional questioning such as this that we can sharpen our critical thinking skills\nand improve as students, learners, and researchers. Developing my critical thinking skills over\na twenty year period as a student in higher education enabled me to complete a quantitative\ndissertation, including analyzing research and completing statistical analysis, and earning my\nPh.D. in 2014."
	},
	{
		"chunk_id" : "00004305",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Ph.D. in 2014.\nWhile critical thinking analyzes information and roots out the true nature and facets of\nproblems, it is creative thinking that drives progress forward when it comes to solving these\nproblems. Exceptional creative thinkers are people that invent new solutions to existing\nproblems that do not rely on past or current solutions. They are the ones who invent solution\nC when everyone else is still arguing between A and B. Creative thinking skills involve using"
	},
	{
		"chunk_id" : "00004306",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "strategies to clear the mind so that our thoughts and ideas can transcend the current limitations\nof a problem and allow us to see beyond barriers that prevent new solutions from being found.\nBrainstorming is the simplest example of intentional creative thinking that most people have\ntriedatleastonce.Withthequickgenerationofmanyideasatoncewecanblock-outourbrains\nnatural tendency to limit our solution-generating abilities so we can access and combine many"
	},
	{
		"chunk_id" : "00004307",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "possiblesolutions/thoughtsandinventnewones.Itissortoflikesprintingthrougharacesfinis\nline only to find there is new track on the other side and we can keep going, if we choose.\nAs with critical thinking, higher education both demands creative thinking from us and is the\nperfect place to practice and develop the skill. Everything from word problems in a math class,\nto opinion or persuasive speeches and papers, call upon our creative thinking skills to generate"
	},
	{
		"chunk_id" : "00004308",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "new solutions and perspectives in response to our professors demands. Creative thinking skills\nask questions such asWhat if? Why not? What else is out there? Can I combine perspectives/\n35\n36 Foundations of Academic Success: Words of Wisdom\nsolutions? What is something no one else has brought-up? What is being forgotten/ignored?\nWhat about ______? It is the opening of doors and options that follows problem-identifcation."
	},
	{
		"chunk_id" : "00004309",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Consider an assignment that required you to compare two diferent authors on the topic of\neducation and select and defend one as better. Now add to this scenario that your professor\nclearlyprefersoneauthorovertheother.Whilecriticalthinkingcangetyouasfarasidentifying\nthe similarities and diferences between these authors and evaluating their merits, it is creative\nthinking that you must use if you wish to challenge your professors opinion and invent new"
	},
	{
		"chunk_id" : "00004310",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "perspectives on the authors that have not previously been considered.\nSo,whatcanwedotodevelopourcriticalandcreativethinkingskills?Althoughmanystudents\nmay dislike it, group work is an excellent way to develop our thinking skills. Many times I\nhaveheard fromstudents their disdain for working ingroups basedonscheduling, variedlevels\nof commitment to the group or project, and personality conficts too, of course. Trueits not"
	},
	{
		"chunk_id" : "00004311",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "always easy, but that is why it is so efective. When we work collaboratively on a project or\nproblemwebringmanybrainstobearonasubject.Thesediferent brainswillnaturallydevelop\nvaried ways of solving or explaining problems and examining information. To the observant\nindividualweseethatthisplacesusinaconstantstateofbackandforthcritical/creativethinking\nmodes.\nFor example, in group work we are simultaneously analyzing information and generating"
	},
	{
		"chunk_id" : "00004312",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "solutions on our own, while challenging others analyses/ideas and responding to challenges to\nour ownanalyses/ideas.Thisispartof whystudentstendtoavoidgroupworkitchallengesus\nas thinkers and forces us to analyze others while defending ourselves, which is not something\nwe are used to or comfortable with as most of our educational experiences involve solo work.\nYour professorsknowthisthats whyweassignittohelpyou growasstudents,learners, and\nthinkers!\nTime Is on Your Side\nChristopher L. Hockey"
	},
	{
		"chunk_id" : "00004313",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Time Is on Your Side\nChristopher L. Hockey\nThere I was, having just eaten dinner and realizing that I had less than twenty-four hours to\ngo before my capstone paper was due for my History of Africa class. This paper was the only\ngradefortheclassandallIhaddonewassomeresearch.Istillhadthirtypagesthatneededtobe\nwritten! How was I going to get this paper done?\nI came to the realization that I was going to have to skip some classes and work through the"
	},
	{
		"chunk_id" : "00004314",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "night. I kept my roommate up with the click clack of the keyboard and worked through the\nnight with breaks only to replenish the caffeine in my system. Morning came and I still had\nwork to do.\nIcontactedmyotherprofessorslettingthemknowthatsomethingcameupandIwouldntbein\nclass. Thankfully, I was in good standing in my other classes and could affordto miss one class.\nIsnuckinatwentyminutenapandkeptworking.Ifinall finishe aboutthirtyminutesbefore"
	},
	{
		"chunk_id" : "00004315",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the deadline. Exhausted and not terribly proud of myself, I trudged my way to class to drop of\nthepaperandcommittedtoneverworkinglikethisagain.Afterall,therewasasmalllikelihood\nthat I would get a decent grade; I was hoping for just a C to keep my GPA respectable. I went\nback to my room and slept for a long time. Imagine my amazement when I received my grade\nfor the paper (and ultimately the class) and there was an A- staring me back in the face! How\ncould this be possible?"
	},
	{
		"chunk_id" : "00004316",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "could this be possible?\nMy experience illustrates a very important lesson. Best practices do not always yield the best\nresults. Logic would tell us that to manage a thirty-page paper would require the student to\nspreadoutallthetasksoverthesemesteranddoalittlebitofworkoveralongperiodoftimeas\nopposedtoalot ofworkover ashortperiodoftime. Theproblemisthattimemanagementisa\npersonal thing. Everyone works differently and excels under different circumstances"
	},
	{
		"chunk_id" : "00004317",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "The important thing to remember about time management is that there is not one method.\nEveryone must find what works best for her or him. There are some strategies that have been\nused for yearsand others that are new. Whilethere are multiple perspectives onhow best to set\npersonalandprofessionalgoals,therearethreegeneralthemesthatinfluenc thedevelopmentof\npersonal time management plans: identifying priorities, managing time, and managing energy."
	},
	{
		"chunk_id" : "00004318",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "The concept of time management is actually personal management. Where you are going or\nwhat you are trying to accomplish is more important than how fast you get there. Personal\nmanagement demands organizing and executing around priorities. One thing to watch out for\nonyourcollegejourneyissomethingcalledtimefamine.Timefamineisthefeelingofhavingtoo\nmuch to do and not enough time to do it. This happens often to college students and without"
	},
	{
		"chunk_id" : "00004319",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "warning. This was certainly the case with my paper. I certainly felt overwhelmed with thirty\npages to write and not a lot of time available to write it in. However, theres one really helpful\naspect of timeyou always know how much you have in a day. You know that in any given\nday, you have twenty-four hours to accomplish everything you need to do for that day. With\nthat knowledge in handit becomes aneasy task to make smart choiceswhenplanning boththe"
	},
	{
		"chunk_id" : "00004320",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "schedule for the day, as well as the energy needed to complete the tasks.\nThe objective of successful time management is to increase and optimize controllable time.\nOnce you have a schedule made, dont change it unless something of some serious urgency\ncomes up. However, while managing time is challenging enough, theres another concept out\n37\n38 Foundations of Academic Success: Words of Wisdom\nthereaboutthemanagementofyourenergy.Thinkofenergyasmoneyandtimeaswhatyoud"
	},
	{
		"chunk_id" : "00004321",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "like to buy. If youre too tired (or energy broke) to be productive, its hard to accomplish (buy)\neverythingonyourschedule.Luckily,attheageoftwenty-two,Ihadlotsofenergyandstamina\nto pull an all-nighter and finish the paper. If I tried to do that today at thirty-five, I would be\nasleep on my keyboard after a few hours. In order to always have enough of time currency, its\nimportant that you are physically energized, emotionally connected, and mentally focused on\nyour purpose."
	},
	{
		"chunk_id" : "00004322",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "your purpose.\nWhile an understanding of these general principles is essential for the development of sound\ntime and energy management strategies, it is also important to focus on practical strategies that\ncan be implemented to improve the college experience. The first recommendation is to know\nwho you are and how you work. In this step, you need to examine all aspects of your current\ntime management skills. Take a look at personal practices such as where you work, how you"
	},
	{
		"chunk_id" : "00004323",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "organizeinformationandcoursematerials,howcurrentandfutureassignmentsandprojectsare\nprioritized, how commitments are balanced, and lastly, how you prevent burnout. Once you\nhave taken stock in your current practices, youll have a better idea of what you need to do to\nimprove.\nEven today, I try to space out large projects and assignments and find that I am not as focused\nor motivated. I struggle to complete the task and when I do, it never feels like I did it well."
	},
	{
		"chunk_id" : "00004324",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "However, when I revert back to that practice of waiting until the last minute, I am focused,\nenergized,andmotivatedandtheresultshavebeenverypositive.Inmyowndoctoralprogram,\nI have begun assignments a little too close to the deadlines but they ultimately get completed\nand I continue to be amazed at the high marks I get back. What does that tell me? It tells me I\nthrivein high-pressure situations where Ihavetofocusintenselyononething andstay focused"
	},
	{
		"chunk_id" : "00004325",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "foralongperiodoftime.Isthatmethodforeveryone?Certainlynot,butitworksforsomeand\nit may or may not work for you. You must examine your own work habits and practices and\nlook back at times that you have done well and times you have done poorly and identify habits\nthat led to those results.\nThe next strategy is to create a personal time management method to help prioritize projects\nandactivities.Trytoidentifyandeliminateactivitiesthatmaydetractfromeffectivel balancing"
	},
	{
		"chunk_id" : "00004326",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "your roles and responsibilities. In any given day, what are the most important things that need\nto be completed? What can be eliminated from your schedule that provides you the time you\nneedtobesuccessful?Iliketothinkofthisasthefive-year-ol plan.Myfive-year-ol lovesto\nplayinthemorningasherMomandIaregettingreadyforwork.Theproblemisthatweneed\nher to get ready for school, too. We put a plan in place that allows her to play in the morning,"
	},
	{
		"chunk_id" : "00004327",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "onlyaftersheiscompletelyreadyforschool.Youneedtimetoplay,havefun,andsocialize,but\nit should not come at the expense of higher priority tasks.\nThe next recommendation is to focus on the process of energy management. Create goals\nfocused on physical, mental, spiritual, and emotional renewal. These goals can include, but are\nnot limited to: getting seven to eight hours of sleep a night, taking small breaks during work"
	},
	{
		"chunk_id" : "00004328",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "sessions, eating healthy, exercising regularly, drinking lots of water, having a positive attitude,\nand practicing positive self-talk. Anytime I know I have a big work task or school task to\ncomplete, I am in the mindset of energy conversationmy energy. I make sure to get a good\nnightsleep,eatmyWheaties,andthinkgoodvibes.Thesehabitsallowmetocompleteprojects\nin a way that works for me.\nLastly, set up a reward system. One of the great things about creating prioritized lists of things"
	},
	{
		"chunk_id" : "00004329",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "that need to be done is the sense of accomplishment when you cross that item off the list.\nOnceyouveidentifie yourmajorgoalsandtasks,identifyarewardforeachofthesegoalsthat\nprovides an even greater sense of accomplishment. The reward should be personal and should\nTime Is on Your Side 39\nencourage you to continue your good habits. What are the things you love to do? Write them\ndown next to the major tasks and learn to practice delayed gratification by only doing those"
	},
	{
		"chunk_id" : "00004330",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "things once youve crossed the item off.\nInconclusion,practicalandtangiblestrategies fortimeandenergymanagementcanbethekey\nto success for any undertaking. While each concept related to time and energy management is\nunique and provides a starting point for you to begin to develop strong personal management\nskills, these methods and ideas are not one-size-fits-all, and you need to explore the strategies\nand discover which components of each best fityour lifestyle and circumstances. Through this"
	},
	{
		"chunk_id" : "00004331",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "exercise,youcandevelopapersonalmanagementplanthatisbestsuitedtoyourneedsandgoals.\nWhat Do You Enjoy Studying?\nDr. Patricia Munsch\nThere is a tremendous amount of stress placed on college students regarding their choice of\nmajor. Everyday, I meet with students regarding their concern about choosing right major; the\npaththatwillleadtoafantastic,high-payingpositioninagrowthindustry.Thereisahopethat\none decision, your college major, will have a huge impact on the rest of your life."
	},
	{
		"chunk_id" : "00004332",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Students shy away from subject areas they enjoy due to fear that such coursework will not lead\nto a job. I am disappointed in this approach. As a counselor I always askwhat do you enjoy\nstudying? Based on this answer it is generally easy to choose a major or a family of majors. I\nrecognize the incredible pressure to secure employment after graduation, but forcing yourself\nto choose a major that you may not have any actual interest in because a book or website"
	},
	{
		"chunk_id" : "00004333",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "mentioned the area of growth may not lead to the happiness you predict.\nWorking in a college setting I have the opportunity to work with students through all walks\nof life, and I do believe based on my experience, that choosing a major because it is listed\nas a growth area alone is not a good idea. Use your time in college to explore all areas of\ninterestandutilizeyourcampus resources to help you make connections betweenyourjoyina"
	},
	{
		"chunk_id" : "00004334",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "subject matter and the potential career paths. Realize that for most people, in most careers, the\nundergraduate major does not lead to a linear career path.\nAs an undergraduate student I majored in Political Science, an area that I had an interest in,\nbut I added minors in Sociology and Womens Studies as my educational pursuits broadened.\nToday, as a counselor, I look back on my coursework with happy memories of exploring new"
	},
	{
		"chunk_id" : "00004335",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "ideas, critically analyzing my own assumptions, and developing an appreciation of social and\nbehavioralsciences.Sotoimpartmywisdominregardstoastudentscollegemajor,Iwillalways\nask, what do you enjoy studying?\nOnce you have determined what you enjoy studying, the real work begins. Students need\nto seek out academic advisement. Academic advisement means many different things; it can\nincludecourseselection,coursecompletionforgraduation,mappingcourseworktograduation,"
	},
	{
		"chunk_id" : "00004336",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "developing opportunities within your major and mentorship.\nAs a student I utilized a faculty member in my department for semester course selection, and\nI also went to the department chairperson to organize two different internships to explore\ndifferent career paths. In addition, I sought mentorship from club advisors as I questioned my\ncareer path and future goals. In my mind I had a team of people providing me support and"
	},
	{
		"chunk_id" : "00004337",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "guidance, and as a result I had a great college experience and an easy transition from school to\nwork.\nI recommend to all students that I meet with to create their own team. As a counselor I can\ncertainly be a part of their team, but I should not be the only resource. Connect with faculty\nin your department or in your favorite subject. Seek out internships as you think about the\ntransition from college to workplace. Find mentors through faculty, club advisors, or college"
	},
	{
		"chunk_id" : "00004338",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "staff. We all want to see you succeed and are happy to be a part of your journey.\nAs a counselor I am always shocked when students do not understand what courses they need\nto take, what grade point average they need to maintain, and what requirements they must\nfulfill in order to reach their goalgraduation! Understand that as a college student it is your\n40\nWhat Do You Enjoy Studying? 41\nresponsibility to readyourcollege catalog and meetall of the requirements for graduation from"
	},
	{
		"chunk_id" : "00004339",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "your college. I always suggest that students, starting in their first semester, outline or map out\nall of the courses they need to take in order to graduate. Of course you may change your mind\nalongtheway,butbysettingoutyourplantograduationyouareforcingyourselftolearnwhat\nis required of you.\nIdothisexerciseinmyclassesanditisbyfarthemostfrustratingforstudents.Theywanttolive\ninthenowandtheydontwanttoworryaboutnextsemester ornextyear.However,formany"
	},
	{
		"chunk_id" : "00004340",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "students that I see, the consequence of this decision is a second semester senior year filled with\ncourses that the student avoided during all the previous semesters. If you purposefully outline\neach semester and the coursework for each you can balance your schedule, understand your\ncurriculum, and feel confident that you will reach your goal.\nPart Three: The Future You\nFighting for My Future Now\nAmie Bernstein\nInstead of completing high school, I elected to obtain my GED due to an anxiety disorder that"
	},
	{
		"chunk_id" : "00004341",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "kept me from being successful in the traditional school setting. At sixteen years old, I thought\nI had it figured out. I was going to go to college before my high school class graduated, and\nI would be ahead of the curve. But I learned quickly that college coursework is tremendously\ndifferen fromhighschool.Collegerequiresalevelofself-disciplinethatIhadnotyetdeveloped.\nMore importantly, college requires a substantial amount of courage and confidence that I was\nsorely lacking."
	},
	{
		"chunk_id" : "00004342",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "sorely lacking.\nIn the spring of 2005, I attempted to take my first two college classes. I withdrew from both\nwithinthefirs month.ImadeexcusesastowhyIcouldnotcompletethesemester.Itoldpeople\nthat it was theprofessors fault for makingthe information too difficul orteaching the material\ntooquickly.ItoldotherpeoplethatthereadingsweretooeasyandIwaswastingmytime.The\ntruth was that I was afraid to try. I was afraid that if I tried, the result would be failure."
	},
	{
		"chunk_id" : "00004343",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "After eight consecutive semesters, I had completed only five classes successfully, accrued\nseventeen withdrawals, and got three failing grades. In those four years, I was placed on\nacademic probation five times because I neglected to withdraw from classes and just received\na failing grade instead. I made the decision to find an entry-level job in an offic so I could\ngrow with a company to be successful instead of getting an education. Through a little bit of"
	},
	{
		"chunk_id" : "00004344",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "searching,IfoundwhatIwaslookingfor.Atthetime,itseemedliketheperfectplacetobe,and\nI was excited to start.\nI was working at an HVAC company in the offic part-time as a general offic assistant. My\ndutiesincludedansweringphonesandtakingmessages.Therewasntagreatamountofroomfor\ngrowth,howeverithadabettersalarythananyofthepreviousjobsIhadatfastfoodrestaurants\nandretailstores.IworkedatthecompanyfornearlytwoyearsbeforeIaskedforaraise.Ittook"
	},
	{
		"chunk_id" : "00004345",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "amonthofcontemplation,andtheownerfinall agreedtogivemearaise.Iwaitedformynext\npaycheck, excited to see the increase in pay even though I knew it wasnt going to be very big.\nTheraisemeantthatIhadaccomplishedsomething,butwhenIreceivedmynextpaycheck,the\npay rate wasnt changed and it felt like I had accomplished nothing at all. When I questioned\nthe owner about it, he said he forgot and he would change it for the next pay period. This"
	},
	{
		"chunk_id" : "00004346",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "same routine went on for two months until I made a big decision. The moment I received my\npaycheck, three months from when I originally asked for a raise, I walked out of the offic and\ndrovedirectly to the registrar onmylocalcommunity college.Iregistered for classes again, but\nthis time I promised myself it would be different; I was fighting for my future now\nI then started taking classes again in the fall of 2011 going part-time. I attended every class and"
	},
	{
		"chunk_id" : "00004347",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "studied as much as I could. I took every opportunity for extra credit assignments. I didnt stop\nto doubt myself. I just kept my thoughts focused on finishing the assignmentsone at a time.\nBefore I knew it, I had successfully completed the semester. I continued to take classes and try\nmy besttaking every challenge head-on.\nA year later, in the fall of 2012, I received a letter in the mail inviting me to join the honor"
	},
	{
		"chunk_id" : "00004348",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "society. Up until this point in my life I had let my anxiety disorder rule my life. This was proof\nthatIwasfinall ontherighttrack.Ireluctantly joinedanddecidedtocontinuetofurtherpush\nmyself outside my comfort zone to challenge my anxiety.\n43\n44 Foundations of Academic Success: Words of Wisdom\nNot only did I start going to meetings, but I participated in every event that the honor\nsociety had to offer. That included bake sales, volunteering for nonprofit organizations, and"
	},
	{
		"chunk_id" : "00004349",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "volunteering for the college itself. The opportunities came at me from every angle.\nI then started to be recognized by the college. In addition to being recognized for my good\ngrades, I was also recognized for my involvement with campus activities through the honor\nsociety. I received both a Distinguished Student Award from my college and a SUNY\nChancellors Award for Student Excellence. I joined everything I could after that, including"
	},
	{
		"chunk_id" : "00004350",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "two more honor societies, one for English and the other for Psychology. I enrolled in a non-\ncreditbearingleadershipclassoncampussoIcouldhaveevenmoreexperiencethatwouldhelp\nme with my future goals. I even went on to run for a regional office position in my honor\nsociety and won. I was able to travel to California, Florida, and Missouri, all because of campus\ninvolvement in the honor society.\nIstillhaveanxiety,butnowIamabletocopewithitwithoutlettingitdictatemyeverymove.I"
	},
	{
		"chunk_id" : "00004351",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "haveconfidence IalwaysthoughtthatbecauseIwasntthatcookiecutterall-Americanstudent,\nmy opportunities would be limited, but getting involved on campus opened so many doors for\nme. I learned so much about what it was to be a leader. I learned what it meant to be part of a\nteam and the value in building relationships. I learned what it meant to be engaged both inside\nand outside of the classroom.\nI developed a deeper sense of who I am through my campus involvement. Sometimes I think"
	},
	{
		"chunk_id" : "00004352",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "about what would havehappened if I never had taken that stepand joined the honor society or\nnever attended any meetings. Honestly, I probably would have been okay. I would have been\nsteadily gliding through my education. I would have shown up to class, taken notes, then gone\nhome and studied. I would have probably then gone on and found a decent job with a regular\namount of satisfaction. But who wants an okay, decent, or regular life? I dont; I need more. I"
	},
	{
		"chunk_id" : "00004353",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "want to love what I do and enjoy every moment I can.\nMaybe the honor society isnt something that you are interested in and thats okay. Do\nsomething different and learn about all of the opportunities that your campus offers and pick\none to try out. Make your life more than run of the milland start now.\nSomething Was Different\nJacqueline Tiermini\nIhaveearnedbothabachelorsandamastersdegreeandIhavenearlytwentyyearsofteaching"
	},
	{
		"chunk_id" : "00004354",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "experience. Would you ever guess that I contemplated not going to college at all? I originally\nthought about going to Beauty School and becoming a Cosmetologist. It was to me, honestly\nthe easy way out since I was sick of all the drama after high school. The thought of college\nseemed overwhelming. Why did I really need to have a college degree when all I ever wanted\nwas to get married and be a stay-at-home mom? My friends werent going to college either, so"
	},
	{
		"chunk_id" : "00004355",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "I often wondered if going would complicate our friendship.\nI decided to go anyway, and it did separate us a bit. While I was writing a ten-page paper for\nmy summer class in Genetics and Heredity, my friends were swimming in my pool. They also\nhad the chance to buy new cars and new clothes and to go on vacations. I just went to school,\ndriving my used Nissan Sentra, without much more than gas money and a few extra bucks."
	},
	{
		"chunk_id" : "00004356",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Again, why was I doing this? It would have been easier to just do what my friends were doing.\nLittle by little, semesters went by and I graduated with my bachelors degree in Education. I\nstarted substitute teaching immediatelyand withinsixmonthsIwasoffere afull-time job.Just\nlikethat,IhadmoremoneyandallkindsofnewopportunitiesandIcouldnowconsideranew\ncarorgoingonvacationjustlikemyfriends.Atthatpoint,Idecidedtocontinuemyeducation"
	},
	{
		"chunk_id" : "00004357",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and getmymasters degree. Yes, itwasalot of hard workagain, andyes, myfriendswondered\nwhy I wanted to go back again, but I knew then that this was the best choice for me. The\nchallenge wasnt knowing where I wanted my career to go, but rather overcoming the pull to\nsettle into a lifestyle or career because it was easy, not because it was what I wanted.\nBythetimeIgraduatedwithmymastersdegreeIrealizedthatsomethingwasdifferent Forall"
	},
	{
		"chunk_id" : "00004358",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "theyearsthatIfeltbehindorunable tokeepup withwhatmyfriendshad,Iwassuddenlyleaps\nand bounds ahead of them career-wise. I now had two degrees, a full-time teaching job, and a\nplantokeep mycareermovingforward.Iwas ableto doallofthethingsthat they haddoneall\nthose years and more. None of them had careers, just jobs. None of them had long-term plans.\nNoneofthemwereassatisfie withtheirchoicesanylongerandafewofthemevenmentioned\nthat they were jealous of my opportunity to attend college."
	},
	{
		"chunk_id" : "00004359",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Dont be fooled. Being a college student is a lot of work and, like me, most students have\nquestioned what they are doing and why they are doing it. However, the rewards certainly\noutweigh all of the obstacles. I used to hear, Attending college will make you a well-rounded\nperson or It sets you apart from those that do not attend, yet it never felt true at the time.\nEventually though, you will come to a point where you realize those quotes are true and you\nwill be on your way to earning that degree!\n45"
	},
	{
		"chunk_id" : "00004360",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "will be on your way to earning that degree!\n45\nTransferable\nVicki L. Brown\nI was supposed to be a teacher. Growing up, I had a classroom in the basement. I had a\nchalkboard,chalk,desks,textbooks,homeworkassignments,pens,pencils,paperyounameit,I\nhadit!MybrotherandsistercalledmeMissBrown.AllIeverwantedtobewasanelementary\nschool teacheruntil I went to college.\nAs an elementary education major in college, I participated in a variety of classesclasses on"
	},
	{
		"chunk_id" : "00004361",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "literacy, math and science, philosophies of teaching, child development theory, principles of\neducation, foundations of classroom behavior, and a whole list of others. We learned how to\nwrite a lesson plan, manage a classroom, how to set up a classroom, and much, much more.\nIn addition to my studies, I got involved in campus life. I joined the swimming and diving\nteam, participated in campus activities, and joined clubs. I served as a captain of the swimming"
	},
	{
		"chunk_id" : "00004362",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and diving team, became an Orientation Leader and a Resident Assistant, and completely\nimmersed myself in the college experience. It was through these co-curricular activities that I\nwasintroducedtotheworldofhighereducationandapotentiallynewcareerchoiceformyself.\nThrough my academic and co-curricular activities, I gained valuable knowledge from all those\nI came in contact withmy peers, professors, Residence Hall Directors, and many college"
	},
	{
		"chunk_id" : "00004363",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "administrators. They encouraged me to explore what it was that I really wanted to do with my\nlife. The more I got involved in my college experience, the more I learned about myself: what\nIm good at, what Im not good at, what I wanted to, and what I didnt want to do.\nAs I started to sort through my options, I continued my studies, receiving both a bachelors\ndegree and a masters degree in elementary education. While attending graduate school, I also"
	},
	{
		"chunk_id" : "00004364",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "worked asa Graduate Residence Hall Director. It was during that time when I finallymade the\ndecisiontopursueacareerinhighereducationadministration/studentaffair administrationand\nleave my plans of being an elementary school teacher behind.\nThe decision wasnt as difficul as one might think. When some listen to my story, I often hear\nyouvewastedallthattimeandmoneyBut,thetruthisIgainedvaluable,lifelongskillsfrom\nthepeopleImet,theclassesItook,thejobsIvehad,andtheactivitiesIinvolvedmyselfin.Each"
	},
	{
		"chunk_id" : "00004365",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and every skill you acquire is transferable. This is perhaps the best lesson Ive ever learned in\ncollege.\nThe countless lesson plans I had to write for my education classes and student teaching have\nhelped me prepare practice plans as the head coach for the mens and womens swimming and\ndiving team. The skills I learned while planning programs and activities for my residents as a\nResident Assistant, Hall Director, and Area Coordinator have helped me plan campus events as"
	},
	{
		"chunk_id" : "00004366",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the Director of Student Activities in the Center for Student Leadership & Involvement. The\nclassroom management techniques I learned in college have helped me to manage my office\nstaff, team, committees, etc. The communication and development theories Ive learned have\ntaughtmehowtohavemeaningfulconversationswithothersandhowbesttomeettheirneeds.\nEach and every skill you learn throughout your academic, personal, and professional career are"
	},
	{
		"chunk_id" : "00004367",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "valuable and transferable. Do not let your college degree definewho you are but rather, let the\nknowledge and skills youve acquired define who you are.\n46\nIts Like Online Dating\nJackie Vetrano\nSearching for a job, especially your first job, is a lot like online dating. It begins as a time\ncommitment,getsnerve-wrackingtowardsthemiddle,butendsinsuccessandhappinessifyou\nfollow the right process.\nLike manysingle people withaccess to current technology, Iventured intothe world of online"
	},
	{
		"chunk_id" : "00004368",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "dating. I went for coffee with potential mates who were instant no ways, some who left me\nscratching my head, and a few who I found a connection with.\nBut hang on. We are here to talk about professional development, not my love life.\nBeing on the job hunt is not easy. Many spend hours preparing rsums, looking at open\npositions, and thinking about what career path to travel. Occasionally, it is overwhelming and\nintimidating,butwhentakenonestepatatime,itcanbeamanageableandanexcitingprocess."
	},
	{
		"chunk_id" : "00004369",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Your Dating ProfileThe Rsum\nThe first step of online dating is the most important: create your dating profile. Your profile is\nwhereyouputyourbestfootforwardandshowofallofyourattractivequalitiesthroughvisuals\nand text. Online daters findtheir most flattering photos andthenseason the about me section\nof their profile with captivating and descriptive words to better display who they are and why\nother online daters should give them a shot."
	},
	{
		"chunk_id" : "00004370",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "other online daters should give them a shot.\nRsumsfollowthissamelogic.Yourrsumshouldbeclean,polished,andpresentyouinyour\nbest light for future employers. Like dating profiles,they are detailed andshould painta picture\nfor other prospective dates (or future employers) supporting why you deserve a chance at their\nlovean interview.\nThe unspoken rules of online dating profiles are very similar to the rules for writing a rsum."
	},
	{
		"chunk_id" : "00004371",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Whetheryoulikeitornot,youronlinedatingprofil andrsumbothserveasafirs impression.\nProfiles and rsums that are short, filled with spelling errors, or vague are usually passed over.\nUnless you are a supermodel and all you need is an enticing photo, your written description is\nvery important to display who you are.\nYour rsum should capture who you are, your skill set, education, past experiences, and\nanything else that is relevant to the job you hope to obtain. Knowing your audience is a key"
	},
	{
		"chunk_id" : "00004372",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "factor in crafting the perfect rsum. Logically, if my online dating profile presented studious\nand quiet personality traits, I would likely start receiving messages from potential mates who\narelookingforsomeonewhoisseekingthosetraits.Bytakingasimilarapproachwhilewriting\na rsum, you can easily determine the tone, language, and highlighted skills and experiences\nyou should feature. The tone of your rsum is dictated by the nature of the position you hope"
	},
	{
		"chunk_id" : "00004373",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "to obtain in the future. For example, hospitality jobs or positions that require you to interact\nwith many people on a daily basis should be warm and welcoming while analytical jobs, such\nas accounting or research positions, should reflect an astute attention to detail. Your choice in\nlanguage follows similar logicuse appropriate terms for the position you are seeking.\nUnlike online dating profiles,your rsum should include your important contact information,"
	},
	{
		"chunk_id" : "00004374",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "including email address, telephone number, and mailing address. Some advise refraining from\nlisting a mailing address, as this could create a bias due to some organizations that are looking\n47\n48 Foundations of Academic Success: Words of Wisdom\nfor a new employee who is already in the area. Unfortunately, this bias cannot be foreseen,\nwhich meansyoushould useyourbestjudgment when listing your contactinformation. Ifyou"
	},
	{
		"chunk_id" : "00004375",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "include this contact information on your dating profile, you may have some very interesting\ntext messages in the morning.\nFinding LoveThe Job Hunt\nSimply crafting an online dating profile doesnt necessarily mean you will find your one true\nlove,andthesameappliestoyourcareer.Onceyourrsumiscrafted,itisequallyasimportant\nto search the job market to find what you think would be a good fit based on your skills and\npreferences."
	},
	{
		"chunk_id" : "00004376",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "preferences.\nAn important part of online dating is setting the appropriate search filters. Sites allow users to\nsearchbygender,location,age,religiousbeliefs,orsocialpractices.Allthesearesmallpiecesthat\naffect the overall compatibility between two people, with some factors being more important\nthan others. By carefully choosing which filters are most important, youre sure to have better\nluck finding a perfect match that will make you happy and excited."
	},
	{
		"chunk_id" : "00004377",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "As you begin the job hunt, it is important to determine your filters when it comes to a career\nor first job. Some of these filters, like dating, may hold more weight to you than others. Many\njob search sites allow users to find job listings as defined by these filters, and they can include:\nlocation, type of organization, starting salary, potential for promotion, job responsibilities, etc.\nAlways establish filters. You may say, I dont care what I find, as long as I find something. All"
	},
	{
		"chunk_id" : "00004378",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "of us have a preference in our love lives as well as our careers, and being honest with yourself\nabout these filters will increase the likelihood for happiness in the end. These filters also allow\nyou to more quickly readthrough job postings, because you will be focusing on positions with\nthe qualities that you already determined are the most important to you.\nWhen you are searching for a posted position using an online service, enter your filters and"
	},
	{
		"chunk_id" : "00004379",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "try a variety of search phrases to find as many postings as possible. Even changing school\ncounselor to guidance counselor or only counseling may produce a different set of job\npostings, depending on the website.\nItisalsoimportant toremember,likeonlinedatingsites,noteveryjobpostingwillbeonevery\nemploymentsite.Experimentwithdifferen searchtechniquesandwebsites,andseektheadvice\nof others for the best resource for recent postings. Its easy to save these filtersand search results"
	},
	{
		"chunk_id" : "00004380",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "on most job search websites, allowing you to check back on a constant basis without resetting\nyour filters. Most sites also allow you to create a free account, providing you a way to receive\nemail alerts any time a new job is posted and fits in with your filters\nAfterthefilter areset,itistimetostarttheexcitingandnerve-wrackingpart:scrollingthrough\nprofiles.\nScrolling through ProfilesThe Job Postings\nYou will find attractive potentials with no description provided, others who exclusively take"
	},
	{
		"chunk_id" : "00004381",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "selfies, and a whole list of people who simply are not right for you based on their description.\nBut then, it happens. You find someone who may be a match, and your heart starts to flutter\nReading through a job description is equally as exciting. A good job posting provides a robust\ndescriptionofresponsibilities,minimumqualifications anddesiredqualification forcandidates.\nKnowing your own skill set, you can determine if youre a match or not. By having honest"
	},
	{
		"chunk_id" : "00004382",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "filters set before searching, its likely that you are.\nIts Like Online Dating 49\nSometimes,onlinedatingsimplydoesntwork.Manywillthenturntospeeddatingtomeetnew\npeopleinthearea.Thismethodallowsfordaterstoquicklydeterminewhetherornottheresany\nchemistry, without spending timesearching throughonline profiles.Similarly,job fairsprovide\nthis quick face-to-face advantage. If youre attending a job fair, be sure dress appropriately and"
	},
	{
		"chunk_id" : "00004383",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "havecopiesofyourrsumorbusinesscardsonhand.Throughjobfairs,youllbebuildingyour\nfirst impression right away, and may even be offered an interview on the spot\nSending a MessageThe Cover Letter\nAftersearching through dozensofprofiles onlinedaters generally findahandfulofpeoplethey\ncanpicturethemselveswith.Theresonlyonewaytofin outmoreabouttheperson,andthats\nby sending the first message.\nMy personal rule for online dating is to always send a thoughtful first message to those I want"
	},
	{
		"chunk_id" : "00004384",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "to meet. Its easy enough to send a short, impersonal hey, but its important to make a good\nimpression. Its obvious that the message I send, combined with my well-written profile, is\ngoing to continue to form a first impression of me. First impressions are very important in\ndating, job-hunting, and life overall.\nThechallengingpartofthefirs messageIsendthroughonlinedatingsitesisdeterminingwhat\ntosay.Ivenevermetthesepeoplebefore,butIdohaveaccesstotheirdatingprofile fille with"
	},
	{
		"chunk_id" : "00004385",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "their hobbies, hometowns, and more. This is a perfect starting point for my message, especially\nif we both root for the same football team or if the other person likes to run as much as I do.\nYourcoverletterservesasanintroductiontoyourfutureemployerandshouldcomplimentyour\nrsum to create a shining firstimpression. It is incredibly challenging to sit in front of a blank\nscreentryingtofin agoodstartingpoint,whichmeansyoushouldlookatthejobpostingand\norganizations website for ideas about what to include."
	},
	{
		"chunk_id" : "00004386",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Generally, these job postings provide a set of hard skills (such as proficiency with certain\ntechnology) and soft skills (such as public speaking, teamwork, or working in a flexible\nenvironment) required and desired for the posted position. This information provides you a list\nof what should be explained in your cover letter. Demonstrating your hard skills is a simple\nenough task by using examples or stating certifications, but describing your soft skills may"
	},
	{
		"chunk_id" : "00004387",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "requirealittlemorethought.Thesesoftskillscanbeexhibitedbydiscussingspecifi examplesof\npast experiences in previous jobs youve held, volunteer work, or work youve done in college\nclasses.\nAfter you have crafted your cover letter, you should send it to a few people you trust for their\nopinionandoverallproofreadingalongwiththejobpostingfortheirreference.Itsobviousthat\nyourcoverlettershouldbefreeofspellingandgrammarerrors,butthesetrustworthyindividuals"
	},
	{
		"chunk_id" : "00004388",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "will also be able to provide helpful insight about the examples youve used to display your soft\nskills.\nThe Hard PartWaiting\nYou just sent your first message to the love of your life, but now what? You wait. You will\nundoubtedly feel anxious, especially if you sit refreshing your inbox for hours at a time, but if\nyou made a good first impression and they like you as much as you like them, you will hear\nback.\nWhile you wait, take the time to do a little research. Search for the organization online and"
	},
	{
		"chunk_id" : "00004389",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "viewwhat informationtheyprovide. You willbestoringupsome goodfacts aboutyourfuture\npartner,whichissomethingyoucanbringupwhenyoureonyourfirs date.Thisresearchwill\n50 Foundations of Academic Success: Words of Wisdom\nalso allow you to understand the company better. The organization displays their values, work\nethic,andpersonalitythroughonlineandprintresources,whichallowsyoutoseeiftheirvalues\nmatch with yours.\nUnlikeonlinedating,itishelpfultofollowupwithanorganizationyouveappliedto.Generally,"
	},
	{
		"chunk_id" : "00004390",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the Human Resources department of an organization is the best place to start if you are unsure\nwhomtocall.Thisphonecallisanotherpieceofyourfirs impression,whichmeansyoushould\nbepreparedtotalk.Haveanymaterialsthatyouneedready,andbesureyouareinaquietplace.\nThe First DateThe Job Interview\nAfter what may feel like forever, you hear back from the love of your life. Congratulations! In\nthe online dating world, you may chat about common interests (because you wrote a stunning"
	},
	{
		"chunk_id" : "00004391",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "firs message),butintheworldofwork,youllbeaskedtovisittheorganizationforaninterview.\nI have been on many firstdates, and whether its in a coffee shop or over dinner, the firstface-\nto-facemeetingistremendouslyimportant.IfsomeoneIammeetingforthefirs timelookslike\nthey just came from the gym or rolled out of bed, my impression instantly changes. This same\ntheory can be directly applied to your first date with your future employer. You have worked"
	},
	{
		"chunk_id" : "00004392",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "hard on your cover letter and rsum, and you should not taint the sparkling first impression\nyou have created with the wrong choice in dress.\nWhat you wear to a job interview may change based on the position you have applied for,\nbut there are a set of basic rules that everyone should follow. Similar to meeting someone on a\nfirstdateforcoffee,youwanttobecomfortable. Some interviews may takeplacewithmultiple\npeopleinanorganization,meaningyouwillbewalkingtodifferen locations,sittingdown,and"
	},
	{
		"chunk_id" : "00004393",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "potentiallysweatingfromabrokenairconditioningunit.Considerthesefactorswhenchoosing\nyour outfitforyourinterview, and ifyoureconcernedaboutbeingunderdressed,rememberto\nalways dress a bit nicer than how youd dress for the job itself.\nThere is nothing worse than sitting alone at a coffee shop waiting for a mystery date to show\nup.Itsuncomfortableandaffect myoverallfirs impressionofwhomImabouttomeet.Avoid\nmaking your mystery employer annoyed and waiting for you by leaving at least ten minutes"
	},
	{
		"chunk_id" : "00004394",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "earlier than you need to, just in case you get stuck in traffic Arrive at least ten minutes early.\nThe interview will start out much better if you are early rather than nervous and running late.\nArriving early also gives you the time to have some coffee and review materials you may need\nfor the interview. Coming on time to an interview or a firstdate shows you respect the time of\nthe person you plan to meet.\nOn a firstdate, it is all about communication. Sometimes, there may be silences that cannot be"
	},
	{
		"chunk_id" : "00004395",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "filled or the person I have just met discloses their entire life story to me in less than an hour. If\nwecannotachieveaproperbalance,therewillnotbeaseconddate.Communicatingeffectivel\nin a job interview is equally as important, especially if you want a job offer!\nAll of the rules of dating apply to how you should behave in a job interview. The interviewer\nwill ask you questions, which means that you should look at them and focus on what is being"
	},
	{
		"chunk_id" : "00004396",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "asked. Your phone should be on silent (not even on vibrate), and hidden, to show that you are\nfullyattentiveandengagedintheconversationyouarehaving.Muchlikehavingaconversation\non a date, the answers to your questions should be clear and concise and stay on topic. The\nstoriesItellonmyfirs datesaremorepersonalthanwhatwouldbedisclosedinajobinterview,\nbut the mindset is the same. You are building the impression that the organization has of you,\nso put your best foot forward through the comments you make."
	},
	{
		"chunk_id" : "00004397",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Tomakethatgreatimpression,itisreallyimportanttoheavilyprepareandpractice,evenbefore\nIts Like Online Dating 51\nyou have an interview scheduled. By brainstorming answers to typical interview questions in a\ntyped document or out loud, later during the interview you will easily remember the examples\nof your past experiences that demonstrate why you are best for the job. You can continue to\nupdate this list as you move through different jobs, findingbetter examples to each question to"
	},
	{
		"chunk_id" : "00004398",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "accurately describe your hard and soft skills.\nThis interview is as much a date for your future employer as it is for you. Come prepared with\nquestions that you have about the company, the position, and anything else you are curious\nabout. This is an opportunity for you to show offthe research youve done on the organization\nand establish a better understanding of company culture, values, and work ethic. Without\nknowing these basics of the company or organization, what you thought was a match might"
	},
	{
		"chunk_id" : "00004399",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "only end in a tense breakup.\nAfter your interview is over, you continue to have an opportunity to build on the positive\nimpressionthatyouveworkedhardtoform.Sendingafollowupthankyounotetoeachperson\nyou interviewed with will show your respect for the time the organization spent with you.\nThese notes can be written and sent by mail or emailed, but either way should have a personal\ntouch, commenting on a topic that was discussed in the interview. While sending a thank you"
	},
	{
		"chunk_id" : "00004400",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "note after a firstdate may sound a little strange, you might not get asked to a second interview\nwithout one!\nIts OfficialThe Job Offer\nIn the online dating world, it takes a few dates to determine if two people are a match. In\nthe corporate world, you may have a one or two interviews to build a relationship. If your\nimpression was positive and the organization believes youre a match for the open position,\nyoull be offered a job."
	},
	{
		"chunk_id" : "00004401",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "youll be offered a job.\nWithajoboffe alsocomesthesalaryfortheposition.Itisimportanttoknowwhatareasonable\nsalary is for the position and location, which can be answered with a bit of research. One\ngood place to look is the Bureau of Labor Statistics website: http://stats.bls.gov/oes/current/\noessrcst.htm.Atthispoint,itisnotuncommontodiscussyoursalarywithyourfutureemployer,\nbut be sure to do so in a polite way.\nOnline dating sites provide the means for millions of people to meet future partners, and the"
	},
	{
		"chunk_id" : "00004402",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "number of people who use online dating is so large that there are sure to be disappointments\nalong the way. I have met people who I thought were compatible with me, but they did not\nfeel the same, and vice versa. This happens frequently while searching for a job, which can be\ndiscouraging, but should not hinder you from continuing to search! There are a great number\nof opportunities, and sometimes all it takes is adjusting your filtersor revising your rsum and"
	},
	{
		"chunk_id" : "00004403",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "cover letter. The clich theres plenty of fish in the sea may be true, but there is definitely a\nway for each person to start their career off right.\nLearn What You Dont Want\nJamie Edwards\nFor a long time, my plan had always been to be a kindergarten teacher. But when I began\nmy undergraduate degree I fell into that ever-growing pool of college students who changed\ntheir major three times before graduation. I was swayed by family members, my peers, and the"
	},
	{
		"chunk_id" : "00004404",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "economy,butIeventuallyrealizedthatIwasinvestingmyeducationinthewrongareasforthe\nwrongreasons.It shouldntjustbeabout salariesandjobsecurity. I needed tofin that personal\nattachment.\nAt eighteen, its hard to see your entire life spread out before you. College may feel like a free-\nfor-all at times, but the reality is that its one of the most defining times of our lives. It should\nnever be squandered. I started to imagine my life beyond collegewhat I found important and"
	},
	{
		"chunk_id" : "00004405",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the type of lifestyle I wanted in the end. I started thinking about the classes that I was actually\ninterested inthe ones that I looked forward to each week and arrived early to just so I could\nget a seat up front.\nAturningpointformewaswhenItooktheadviceofacampusmentorandenrolledinacareer\nexploration course. I learned more about myself in that class than I had in my entire three years\nat college prior to taking it.Itshowedme that my passion was something I had always thought"
	},
	{
		"chunk_id" : "00004406",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "about but never thought about as a career. In high school, I could sit in the Guidance Offi\nforhoursonend.Ienjoyedlisteningtoothershearingandhelpingpeopleworkthroughtheir\nstruggles.\nI had seen firsthand how detrimental the absence of career classes can be to someones future.\nThrough this realization and my participation in my career exploration class, I saw a viable\nfuture in the Higher Education Administration field. As I dove deeper, I was opened to"
	},
	{
		"chunk_id" : "00004407",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "an incredible amount of unique and diverse opportunities to work with students. My main\napproach was to get a taste anything to do with student services: I shadowed a career counselor\nin a career services office attended graduate school fairs and informational sessions, discussed\nthe Higher Education Administration Program with several staff at my college, and most\nimportantly, I talked with my internship coordinator (my mentor). From there, I completed an"
	},
	{
		"chunk_id" : "00004408",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "internship in my prospective field, which gave me a wealth of insight and skills that directly\nrelated to my future career goals.\nFrom where I sit nowmy former personal and professional struggles in towI offer up some\npiecesofadvicethatwerecrucialtogettingmewhereIamtoday.Whetheryoureanundecided\nmajor who is looking for guidance or a student with a clearly definedcareer path, I suggest the\nfollowing:\n1. Find a mentorFor me, everything began there. Without my mentor, I wouldnt have done any of"
	},
	{
		"chunk_id" : "00004409",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "the other items Im about to suggest. Finding the right mentor is crucial. Look for someone who\ncan complement your personality (typically someone whos the opposite of you). My advice\nwould be to look beyond your direct supervisor for mentorship. Its important to create an open\nforum with your mentor, because there may be a conflict of interest as you discuss work issues and\nother job opportunities. Potential mentors to consider are an instructor on campus, your academic"
	},
	{
		"chunk_id" : "00004410",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "advisor, a professional currently working in your prospective field, someone you admire in your\ncommunity, or anyone in your network of friends or family that you feel comfortable discussing\nyour future goals with.\n2. Enroll in a Career Exploration/Planning course, or something similarEven if you do not see the\n52\nLearn What You Dont Want 53\neffects of this course immediately (such as dramatically changing your major), you will notice the"
	},
	{
		"chunk_id" : "00004411",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "impact down the road. Making educated career choices and learning job readiness skills will\nalways pay off in the end. Through my career exploration class, I learned how to relate my\npersonality and values to potential career fields. These self-assessments changed my entire thought\nprocess, and I see that influence daily. Beyond changing the way you think, the knowledge you\ngain about effective job search strategies is invaluable. Learning how to write purposeful rsums"
	},
	{
		"chunk_id" : "00004412",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and cover letters, finding the right approach to the interview process, and recognizing your\nstrengths and weaknesses are just a few of the benefits you can gain from these type of courses.\n3. Complete a Job Shadow and/or Informational InterviewNo amount of online research is going to\ngive you the same experience as seeing a job at the front line. In a job shadow or an informational\ninterview, youre able to explore options with no commitment and see how your in-class"
	},
	{
		"chunk_id" : "00004413",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "experience can carry over to a real world setting. Additionally, youre expanding your professional\nnetwork by having that personal involvement. You never know how the connections you make\nmight benefit you in the future. My only regret about job shadowing in college is that I didnt do\nit sooner.\n4. Do an InternshipA main source of frustration for recent grads is the inability to secure an entry-\nlevel position without experience. How do I get a job to gain experience when I cant get a job"
	},
	{
		"chunk_id" : "00004414",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "without experience? This is how: do an internship or two! Most colleges even have a course\nwhere you can obtain credit for doing it! Not only will you earn credits towards graduation, but\nyoull gain the necessary experience to put on your rsum and discuss in future interviews.\nHaving completed four internships throughout my college career, I cant say they were all great.\nHowever, I dont regret a single one. The first one showed me the type of field I didnt want t"
	},
	{
		"chunk_id" : "00004415",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "work in. The second confirmed that I was heading in the right direction with my career. My third\nand fourth internships introduced me to completely different areas of higher education which\nbroadened my knowledge and narrowed my search simultaneously.\nMytakeawayisthatsometimesyouhavetolearnwhatyoudontwantinordertofin outwhat\nyou do want. The more informed you are about career options through real life conversations\nandexperiences,thebetterpreparedyouwillbeforyourfutureandthemoreconfiden youwill"
	},
	{
		"chunk_id" : "00004416",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "be in your career decisions. Always explore your options because even if you learn you hate it,\nat least youre one step close to finding what you love.\nConclusion\nIn the text, the authors told true-to-life stories about their own academic, personal, and life-\ncareer successes. When reading FAS: WoW, you explored the following guiding questions:\n How do you demonstrate college readiness through the use of effective study skills\nand campus resources?"
	},
	{
		"chunk_id" : "00004417",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and campus resources?\n How do you apply basic technological and information management skills for\nacademic and lifelong career development?\n How do you demonstrate the use of critical and creative thinking skills to solve\nproblems and draw conclusions?\n How do you demonstrate basic awareness of self in connection with academic and\npersonal goals?\n How do you identify and demonstrate knowledge of the implications of choices\nrelated to wellness?"
	},
	{
		"chunk_id" : "00004418",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "related to wellness?\n How do you demonstrate basic knowledge of cultural diversity?\nNowthatyouveread FAS:WoW,itstimetopayitforwardbycomposing yourown Words of\nWisdom story to share with college students of the future. Reflect on the lessons learned during\nyour own college experience this term and use the guiding questions to develop a true-to-life\nstory that can help other college students connect the dots between being a college student"
	},
	{
		"chunk_id" : "00004419",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "and being a successful college student. Submit your story to be considered in the next edition\nof Foundations of Academic Success: Words of Wisdom by emailing your name, institution, and\na draft of your short story to opensunyfas@gmail.com. Submissions will be reviewed as they\nare received, and you will be contacted directly if your submission is reviewed and selected for\npublication.\nThe options for textbooks focusing on college student success in college are overwhelming;"
	},
	{
		"chunk_id" : "00004420",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "manytextbooksexistatvaryinglevelsofrigorandcost(somewellover$100).The FAS:WoW\nseriesoftextbooksprovidescollegestudentsopenaccesstextbooksthatarestudent-centeredand\nreadable (dare I even say enjoyable). FAS: WoW supports the open access textbook philosophy\nto help students reduce the cost of attending colleges and universities.\n54"
	},
	{
		"chunk_id" : "00004421",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nGraphs of Compositions\nTags: GeoGebra\nAlignments to Content Standards: F-BF.A.1.c\nStudent View\nTask\nFor each function in this task, assume the domain is the largest set of\nreal numbers for which the function value is a real number.\nf f(x) = x2 g\nLet be the function defined by . Let be the function\ng(x) = x\ndefined by .\ny = f(g(x))\na. Sketch the graph of and explain your reasoning.\ny = g(f(x))"
	},
	{
		"chunk_id" : "00004422",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "y = g(f(x))\nb. Sketch the graph of and explain your reasoning.\nIM Commentary\nThis task addresses an important issue about inverse functions. In this\nf g g\ncase the function is the inverse of the function but is not the\nf f\ninverse of unless the domain of is restricted.\nThis task includes an experimental GeoGebra worksheet, with the\nintent that instructors might use it to more interactively demonstrate\nthe relevant content material. The file should be considered a draft"
	},
	{
		"chunk_id" : "00004423",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "version, and feedback on it in the comment section is highly\nencouraged, both in terms of suggestions for improvement and for\nideas on using it effectively. The file can be run via the free online\napplication GeoGebra, or run locally if GeoGebra has been installed on\na computer.\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nSolution\na.\nWe have\nf(g(x)) = (x)2 = x.\nx\nThe domain of is all non-negative real numbers and so the graph\n(0,0) x"
	},
	{
		"chunk_id" : "00004424",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "(0,0) x\nstarts at the point and then includes all positive values of in its\ndomain:\nb.\nWe have\n\ng(f(x)) = x2.\nThe square root  takes non-negative real numbers as input and gives\n\nx2 = x x\nnon-negative numbers as output. So if is non-negative and\n\nx2 = x x g(f(x)) = |x|,\nif is negative. In other words the absolute\nvalue function.\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a"
	},
	{
		"chunk_id" : "00004425",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nInscribing a circle in a\ntriangle I\nNo Tags\nAlignments to Content Standards: G-C.A.3\nStudent View\nTask\nThe goal of this task is to show how to draw a circle which is tangent to\nall three sides of a given triangle: that is, the circle touches each side of\nABC\nthe triangle in a single point. Suppose is a triangle as pictured\n \nAO A BO\nbelow with ray the bisector of angle and ray the bisector of\nB\nangle :"
	},
	{
		"chunk_id" : "00004426",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "B\nangle :\n  \nM AC OM AC\nAlso pictured are the point on so that meets in a right\n  \nN AB ON AB\nangle and similarly point on is chosen so that meets in a\nright angle.\nAOM AON\na. Show that triangle is congruent to triangle .\n \nOM ON\nb. Show that is congruent to segment .\n\nP BC\nc. Arguing as in parts (a) and (b) show that if is the point on so\n  \nOP BC OP OM\nthat meets in a right angle then is also congruent to ."
	},
	{
		"chunk_id" : "00004427",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "O |OM|\nd. Show that the circle with center and radius is inscribed\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nABC\ninside triangle .\nIM Commentary\nThis task shows how to inscribe a circle in a triangle using angle\nbisectors. A companion task, ''Inscribing a circle in a triangle II'' stresses\nthe auxiliary remarkable fact that comes out of this task, namely that\nABC O\nthe three angle bisectors of triangle all meet in the point . In"
	},
	{
		"chunk_id" : "00004428",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "order to complete part (d) of this task, students need to know that the\np\ntangent line to a circle at a point is characterized by being\np\nperpendicular to the radius of the circle at .\nThis task is primarily intended for instruction purposes but parts (a) and\n(b) could be used for assessment. The teacher is encouraged to draw\nmany different triangles both to provide a broader context for the\nproblem and to see where the inscribed circle lies depending on the"
	},
	{
		"chunk_id" : "00004429",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "shape of the triangle. If students no how to bisect an angle and draw\nthe segment from a point to a line, meeting the line in a right angle,\nthen they could perform the entire construction with straightedge and\ncompass. This particular construction has a great reward at the end,\nnamely being able to produce the inscribed circle inside the triangle.\nSolution\na.\n\nAO A MAO\nSince ray bisects angle this means that angle is congruent\nNAO AMO ANO\nto angle . We also know that angles and are both"
	},
	{
		"chunk_id" : "00004430",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "to angle . We also know that angles and are both\nright angles and so they are congruent. Since\nm(MOA) = 180m(MAO)m(AMO)\nm(NOA) = 180m(NAO)m(ANO)\nm(MAO) = m(NAO) m(AMO) = m(ANO)\nSince and we\nm((MOA) = m(NOA) MAO NAO\nconclude that . Triangles and\n\nAO\nshare side and so by ASA they are congruent.\nMAO NAO\nAlternatively, triangles and are right triangles. Their\nEngage your students with effective distance learning resources. ACCMESAS ORESOURCES>>"
	},
	{
		"chunk_id" : "00004431",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "hypotenuses are congruent and they share the congruent angles\nNAO\nand . If students are familiar with the Hypothenuse Angle\ncongruent theorem, this is sufficient information to conclude that\nMAO NAO\ntriangles and are congruent.\n \nOM ON\nb. Segments and are corresponding sides of congruent\nMAO NAO\ntriangles and , by part (a), and so they must be congruent.\nc.\n \nBO CO\nWe will use a new picture with ray replaced by ray\nThe argument from part (a) can be produced word for word with the\nP N"
	},
	{
		"chunk_id" : "00004432",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "P N\npoint replacing . Alternatively, looking at the argument for parts (a)\n\nABC AO\nand (b) they used the facts that is a triangle, that bisects angle\n\nA BO B ABC\nand that bisects angle . Since the vertices of triangle can\nB\nbe labelled in any way we choose, we could switch the labels for and\nC AOM AOP\nand then part (a) would show that triangles and are\n \nOP OM\ncongruent and part (b) would conclude that is congruent to ."
	},
	{
		"chunk_id" : "00004433",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "As in part (a), students can also use the Hypotenuse Angle congruence\n\nCOM COP CO\ntheorem to show that triangles and are congruent ( is\n\nCO MCO PCO\ncongruent to and angle is congruent to angle ). It then\n \nOM OP\nfollows that segments and are congruent as they are\ncorresponding parts of congruent triangles.\nd.\n \nO |OM| AB N AC M\nThe circle with center and radius meets at , at ,\n\nBC P\nand at . Moreover, the radii of the circle at these three points are"
	},
	{
		"chunk_id" : "00004434",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " \nAC AB\nperpendicular to the segments. This means that the lines , , and\n\nBC |OM|\nare all tangent lines for the circle. So the circle radius and\nO ABC\ncenter is inscribed in triangle as pictured below:\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a\nCreative Commons Attribution-NonCommercial-\nShareAlike 4.0 International License."
	},
	{
		"chunk_id" : "00004435",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "National Admission Tests\nMany professional schools require an entrance exam as part of the application process. Be sure to learn everything\nyou can about your national exam and follow all the test day rules. Timing of the test: For undergraduates hoping to\nbegin professional school in the fall following graduation, the test should be taken in the spring/summer prior to\nentering their final year of classes, approximately 15-months prior to starting a professional program. Score"
	},
	{
		"chunk_id" : "00004436",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "Release: When provided the opportunity, please release your score to Pre-Professional Advising. This allows us to\nbetter help Purdue students. Pre-Professional Advising maintains the confidentiality of your score and will only use\nscores in aggregate form with no names attached.\n*All costs accurate as of March 2025\nField Specific Admission Tests\nMedical College Admission Test (MCAT)--Sponsored by the Association of American Medical Colleges (AAMC), the MCAT"
	},
	{
		"chunk_id" : "00004437",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "is required at all U.S. allopathic (MD) and osteopathic (DO) medical school, and most Canadian medical programs. Podiatric\nmedical schools and some anesthesiologist assistant programs also require the MCAT.\n The Test: 4 multiple-choice sections: National Sciences (Biological and Biochemical Foundations of Living Systems and\nChemical and Physical Foundations of Living Systems); Psychological, Social, and Biological Foundations of Behavior;\nCritical Analysis and Reading Skills"
	},
	{
		"chunk_id" : "00004438",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "Critical Analysis and Reading Skills\n When: Offered on specific dates January to early September (register early, 60+ days in advance)\n Test Modality: Taken on computer at a test center\n Where: Pearson VUE test centers https://home.pearsonvue.com/\n Cost: $345* if register at least 29 days in advance\n Fee Assistance through the AAMC: https://students-residents.aamc.org/applying-medical-school/applying-medical-school-\nprocess/fee-assistance-program/"
	},
	{
		"chunk_id" : "00004439",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "process/fee-assistance-program/\n Accommodations: https://students-residents.aamc.org/applying-medical-school/taking-mcat-exam/mcat-exam-\naccommodations/\n Scoring: The sections are scored from 118-132 with a midpoint of 125. The total score range is 472-528. Scores are\navailable approximately one month following the test and are generally good for 3 years.\n Competitive Score: Above 125 in each section with an overall score above 500. Most MD programs require 509+ and DO\nprograms a 505+"
	},
	{
		"chunk_id" : "00004440",
		"source" : "Foundations of Academic Success_ Words of Wisdom_ College_Life_Prep.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "academic_skills",
		"content" : "programs a 505+\n MCAT Information: https://students-residents.aamc.org/applying-medical-school/taking-mcat-exam/\n MCAT Essentials: https://students-residents.aamc.org/media/11711/download\n Test Repeat Rules: Can take it 3 times in 1 year. No more than 4 times over 2 years and 7 times total. Schools receive\nscores on all test attempts.\nLaw School Admission Test (LSAT)--required at most law schools in the United States and Canada and is sponsored by\nthe Law School Admissions Council (LSAC)."
	},
	{
		"chunk_id" : "00004441",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : "the Law School Admissions Council (LSAC).\n The Test: Consists of 2 parts. Four 35-minute multiple choice sections plus a writing sample administered separately. The\nmultiple-choice sections include Reading Comprehension, Logical Reasoning and an unscored section that could be in\nany one of these areas.\n When: Typically offered in a test cycle of April, June, August, September, October, November, January, February, and April."
	},
	{
		"chunk_id" : "00004442",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Test Modality: Taken as a remote test on a computer of your choice with strict monitoring and room requirements or in a\ntesting center.\n Where: Taken as a remote test in pre-approved locations. Also available at Prometric test centers.\nhttps://www.prometric.com/\n Cost: $238.00* Can also sign up for Score Preview for $45 which allows 6 days to cancel score.\n Fee Waivers: https://www.lsac.org/jd/lsat/fee-waivers"
	},
	{
		"chunk_id" : "00004443",
		"source" : "\"Graphs of Compositions.pdf\"",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "mathematics",
		"content" : " Accommodations: https://www.lsac.org/lsat/lsac-policy-accommodations-test-takers-disabilities\n Scoring: LSAT scores range from 120 to 180 and are available in 3-4 weeks after the test. The writing sample is not scored\nbut is sent to schools. Scores are valid for 5 years.\n Competitive Scores: Range from 155-170+ depending on the school\n More Information on LSAT: https://www.lsac.org/lsat\n Registration deadlines for each test are one month prior to the exam"
	},
	{
		"chunk_id" : "00004444",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Test Repeat Rules: The LSAT may be taken 3 times in a single testing year (June-May), 5 times in 5 years and a lifetime\ntotal of 7 times. The LSAT Flex exam does not count toward lifetime testing limits. Schools will receive scores from all\nattempted tests.\nGraduate Record Exam (GRE)--The GRE general exam is intended to measure readiness for graduate and professional\nprograms. It is offered by the Educational Testing Service (ETS). Required for many PT, OT, PA and other grad programs"
	},
	{
		"chunk_id" : "00004445",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " The Test: verbal reasoning, quantitative reasoning, and analytical writing\n When: Almost any day of the year (schedule well in advance)\n Where: An approved ETS test center https://www.ets.org/gre/test-takers/general-test/schedule.html or at a pre-\napproved site of your choice (at-home version)\n Test Modality: Taken on computer at testing center or as an at home. The at-home version is also available\nhttps://www.ets.org/s/cv/gre/at-home/"
	},
	{
		"chunk_id" : "00004446",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "https://www.ets.org/s/cv/gre/at-home/\n Cost: $220* for first 4 schools (plus $27 fee for each additional score report)\n Fee Reduction Program: https://www.ets.org/gre/test-takers/subject-tests/register/fees.html\n Accommodations: https://www.ets.org/gre/revised_general/register/disabilities\n Scoring: Verbal and quantitative reasoning scores range from 130 to 170. Analytical writing score ranges from 0-6.\nScores are available at your account within 10-15 days after testing and are valid for 5 years."
	},
	{
		"chunk_id" : "00004447",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Competitive scores: Varies widely, 80th percentile would be 150 to 160 in verbal/quantitative, & 4.2 in analytical writing\n generally you want to be above the national average\n More Information on GRE: https://www.ets.org/gre/revised_general/about\n Test Repeat Rules: Must allow 21 days between tests. Can retake the test up to 5 times in a continuous year. With\nScoreSelect, you decide which scores are sent."
	},
	{
		"chunk_id" : "00004448",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "ScoreSelect, you decide which scores are sent.\nDental Admission Test (DAT)--The DAT is sponsored by the American Dental Association (ADA) and is required for\nprograms in the U.S. and Canada. To register, obtain a DENTPIN at https://www.ada.org/education/manage-your-dentpin\n The Test: 4 sections: survey of natural sciences (biology, general and organic chemistry), perceptual ability, reading\ncomprehension, and quantitative reasoning.\n When: Schedule 60-90 days before desired test date"
	},
	{
		"chunk_id" : "00004449",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Where: Prometric Test Centers https://www.prometric.com/\n Test modality: Computerized exam at test center\n Cost: $560* (includes all schools selected at application)\n Partial Fee Waiver: Partial fee waivers are available and can be requested by logging into the DENTPIN account and\nselecting submit request\n Accommodations: see DAT Candidate Guide below\n Scoring: As of March 2025, new scoring for the DAT will range from 200-600. Results prior to March 2025 range from 1 to"
	},
	{
		"chunk_id" : "00004450",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "30. Scores are generally valid for 3 years depending on the school.\n Competitive Score: Under previous scoring 18-20+. This is now about 400-430+\n More Information on DAT: https://www.ada.org/education/testing/exams/dental-admission-test-dat\n DAT Candidate Guide: https://www.ada.org/-/media/project/ada-organization/ada/ada-\norg/files/education/dat_examinee_guide.pdf?rev=bc38c656b5a8486dbae6a51bc5b5d43b&hash=58B13D825DD060B1A012016DC0876B4B"
	},
	{
		"chunk_id" : "00004451",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Test Repeat Rules: May take 4 times in 12-month period. Must wait 60 days between testing attempts.\nOptometry Admission Test (OAT)--The OAT is required by most optometry programs in the United States and Canada and\nis sponsored by the Association of Schools and Colleges of Optometry (ASCO) To register, obtain an OAT Pin at\nhttps://oat.ada.org/apply-to-take-the-oat.\n The Test: 4 sections: survey of natural science (biology, general and organic chemistry); reading comprehension, physics,"
	},
	{
		"chunk_id" : "00004452",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "and quantitative reasoning\n When: Schedule 60 to 90 days in advance\n Where: Prometric Test Center https://www.prometric.com/\n Test Modality: Computerized exam at test center\n Cost: $520 through 6/20/25 (includes all schools selected at application)\n Partial Fee Waiver: Partial fee waivers are available. Check OAT Candidate Guide below.\n Accommodations: See OAT Candidate Guide below"
	},
	{
		"chunk_id" : "00004453",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Accommodations: See OAT Candidate Guide below\n Scoring: Scores range from 200 to 400, 300 is the national average. Unofficial scores are available immediately following.\nScores are generally valid for 2-3 years depending on the school.\n Competitive Score: 300 is the average, competitive scores range from 330 to 350\n More Information on OAT: https://oat.ada.org/\n OAT Candidate Guide: https://oat.ada.org/-/media/project/ada-"
	},
	{
		"chunk_id" : "00004454",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "organization/ada/oat/files/oat_examinee_guide.pdf?rev=1484a4f99b8c465fa6cef58bfa0b2349&hash=BB60D91651FC92B128FBEA898678ABAC\n Test Repeat Rules: Can take 4 times in 12 months. A minimum of 60 days is required between tests.\nPreparing for your Giant Leap\npurdue.edu/preprofessional\nNational Admission Tests\nPharmacy College Admission Test (PCAT)--The PCAT is sponsored by the American Association of Colleges of\nPharmacy (AACP). Most schools no longer require the PCAT and the test was retired in 2024."
	},
	{
		"chunk_id" : "00004455",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "Physician Assistant College Admission Test (PA-CAT)--The PA-CAT is accepted at about 35 PA programs.\n The Test: 240 questions covering 9 areas: anatomy, physiology, general biology, biochemistry, general chemistry, organic\nchemistry, microbiology, behavioral science, genetics and statistics\n Who: Applicants to some PA programs. Participating programs. https://exammaster.aweb.page/pacat-participating-\nprograms\n When: Flexible scheduling, but can take 6 weeks to receive scores"
	},
	{
		"chunk_id" : "00004456",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Where: Prometric Test Center or can take as a remote test https://www.prometric.com/\n Test Modality: computer test at test center or remote\n Cost: $257\n Fee Assistance: https://exammaster.aweb.page/feeassistance\n Accommodations: https://www.pa-cat.com/wp-content/uploads/2020/01/Special-Accomodations_Jan2020.pdf\n Scoring: You receive two scores: a composite score for the entire exam and 3 sub-scores for anatomy and physiology,\nbiology, and chemistry.\n Competitive scores: 501-550"
	},
	{
		"chunk_id" : "00004457",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Competitive scores: 501-550\n More Information on PA-CAT: https://www.pa-cat.com/about-the-pa-cat/\nNational Admission Test Tips\n Start looking into your test earlyeven your first yearto become familiar with the topics that will be on the exam. (You dont\nhave to start studying that early just become familiar with the test).\n Since most people take their test at the end of the junior year, you need to start your junior year with a solid plan for how"
	},
	{
		"chunk_id" : "00004458",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "you will study for the test as you begin your junior year and for the additional expenses of the study materials and the test\nitself. Remember that there are fee assistance programs for most of the tests if you qualify for them (generally you need to\nsubmit a FAFSA for this).\n Sign up well in advance of your test as dates do fill quickly.\n Do not blow off studying any of the sections of the test. Traditionally the reading comprehension sections have the worst\nscores nationally"
	},
	{
		"chunk_id" : "00004459",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "scores nationally\nOther Testing Required by Some Healthcare Programs\nPREview Professional Readiness Exam--This test is relatively new and is a professional readiness exam from the AAMC.\n Who: Applicants to programs requiring OR recommending this test including MD and a few DO schools as well as some\npodiatric and dental programs. https://students-residents.aamc.org/aamc-preview/participating-medical-schools\n When: The test is offered on certain dates in April-September\n Cost: $100"
	},
	{
		"chunk_id" : "00004460",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Cost: $100\n Additional information can be found at the general PREview website https://students-residents.aamc.org/aamc-\npreview/aamc-preview-professional-readiness-exam. PREview Essentials https://students-\nresidents.aamc.org/media/13296/download\nCasper Situational Judgement Test--Casper is required by a growing number of healthcare programs. Typically, it is\nrequired before you go to an interview. You should prepare for it by becoming familiar with the format of the test and taking a"
	},
	{
		"chunk_id" : "00004461",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "practice test instructions on the website. To take this test you will need a quiet place and a computer with a webcam and\nmicrophone.\nCasper has 2 component tests and the field you are applying for will determine which of these you will be required to take. Even\nif both sections are not required, you may need to take them as you cant go back and take the other section later if another\nschool should require both later on. It may be to your benefit to just do both."
	},
	{
		"chunk_id" : "00004462",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "Casper is a situational judgement test which requires you to watch videos then respond with written descriptions and with video\nresponses. in writing describe how you would respond to those scenarios. The test has 2 sections and 11 scenarios. Four\nscenarios have a video response and 7 have a written response. https://acuityinsights.app/casper/\nDuet assesses what you value in a program and compares that with what programs offer. Duet should be completed within 14"
	},
	{
		"chunk_id" : "00004463",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "days of taking Casper. https://acuityinsights.app/duet/\nIf you need to take more than just Casper, the Altus system will alert you to this and you will need to take Snapshot and/or Duet\nwithin the next 10 days after completing Casper.\nWe do not recommend paying for any preparation materials for Casper, but these videos are useful YouTube video 1 and\nYouTube video 2. https://www.youtube.com/watch?v=Pmv-HrQ7Q6E and https://www.youtube.com/watch?v=62uzltImdXs\n Who: Applicants to some healthcare programs"
	},
	{
		"chunk_id" : "00004464",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : " Who: Applicants to some healthcare programs\n When: Summer/Fall of application year\n Cost: $85 which includes distribution to 7 schools and $18 per additional school. This includes Duet.\n Score: You will not receive your exact scorethis is sent to schools. You will receive a quartile rank: first, second, third or\nfourth. Being in fourth quartile means your responses were ranked as better than 75% of your peers. If you are in the first"
	},
	{
		"chunk_id" : "00004465",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "quartile, inversely, it means 75% of your peers were ranked higher in terms of empathy and comprehensive answers.\n Additional information on Casper: https://acuityinsights.app/casper/\nKira Talent AssessmentKira is not an admission test. It is, however, part of the holistic review process at requiring\nprograms.\nApplicant Help Center: Check your invitation from the school at which you are applying to learn the type of Kira Assessment they\nare using. https://support.kiratalent.com/"
	},
	{
		"chunk_id" : "00004466",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "are using. https://support.kiratalent.com/\n Asynchronous: Some schools will use Kira for asynchronous assessments. This may include video questions where you are\nprovided a pre-recorded video and you will answer with a timed, recorded video answer. You may also have written\nquestions. You will see a question displayed through text on the screen and you will respond to that question.\n Live Interview: In live interviews, you are connected directly with an interviewer and move through different questions or"
	},
	{
		"chunk_id" : "00004467",
		"source" : "National_Admission_Tests_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "stations automatically. This is a two-way interview rather than there being an avatar providing interview questions, you are\nspeaking with a real individual.\n Multiple Mini Interview (MMI): Kira also accommodates the MMI format. An MMI involves moving through a series of real-\nworld scenarios and discussing them with an interviewer. You will move through rooms and chat with several different live\ninterviewers about your topics.\nPreparing for your Giant Leap\npurdue.edu/preprofessional"
	},
	{
		"chunk_id" : "00004468",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 1: Prehistory\nApproach + License\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. A few considerations:"
	},
	{
		"chunk_id" : "00004469",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "I'm designing this course for World History students. If that's you: good. This should line up with your\ncoursework, at least to some degree. If you're taking my course, it should line up considerably. I'm going to\nuse a conversational tone. I think it's wise to acknowledge my audience and make the content easy to\napproach, rather than trying to impress you with haughty lexicon. For the same reason, I'm going to format"
	},
	{
		"chunk_id" : "00004470",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "this text in the style of a magazine, which should be easy to digest (it's also the only template I have).\nI think this is a necessary endeavor. I'm going to take this opportunity to make something useful and\nworthwhile that everyone can access.\nPatterns of Behavior\nHistory class has a reputation inventions in the Industrial\nfor being boring. It doesn't Revolution. We're just looking\nhave to be boring. I think part for patterns in the way\nof the disconnect has to do people behave. Change is"
	},
	{
		"chunk_id" : "00004471",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "with our (teachers, that is) interesting to us; especially\nmethods for communicating when it shapes the world we\nand benchmarking the know. And we do like to see\ncontent. You (students, that how people, events, and\nis) need to know everything ideas foster that change;\nthat happened in every especially when it's relevant\nhistorical period, figure out to our experiences.\nwho shot who (and with what\nRight now, I want you to\ngun), memorize the dates,\nsimply look for patterns in\nand then somehow put all"
	},
	{
		"chunk_id" : "00004472",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "and then somehow put all\nthe way people behave. We're\nthe specifics into context.\ngoing to look at critical\nNobody should expect that of\nperiods in human history,\nyou, because it's not what\nand take note when the way\nhistorians do.\npeople behave leads to\nHistory is a social science change. I might ask you to\nthat looks at human behavior remember a few things, but\nthrough a temporal (as in never names, dates, or shoe\ntime-based) perspective.[ 1 ] sizes. Instead, I'm going to"
	},
	{
		"chunk_id" : "00004473",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "We're not actually all that ask you to look at the\ninterested in listing the patterns and reflect critically\nbattles in the Sino-Japanese about what you see.\nWar or matching inventors to\nRift Val l ey\nOur story starts in a part of Africa called the Great Rift Valley, near Kenya and\nEthiopia. We start existing around four million years ago. Then things get weird.\nChange Over Time\nOkay, so it's not like \"snap"
	},
	{
		"chunk_id" : "00004474",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "our change over time (and every other living thing's) is called\n\"Lor em sit"
	},
	{
		"chunk_id" : "00004476",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "there's an even greater chance of passing it on. Because you're\nnot dead.\nOver the last four(ish) million years, we've gained a few\nadaptations that have helped us, well, adapt. Here are a few\nimportant ones:\n-Bipedalism (standing on two legs) may have helped us see\npredators in savannas with tall grasses\n-Changes in our hands helped us grip objects with both precision\nand power\n-Our brains got much bigger; that helps\nThese adaptations make it possible to"
	},
	{
		"chunk_id" : "00004477",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "These adaptations make it possible to\n-Changes in our legs helped us become able to run long distances do many things that my cat can't.\n-The development of language helped us work together [3]\nMy cat can't do this.\nPaleolithic\npeople spent a\nlot of time and\nenergy\nfollowing\nlarge game\nanimals from\nplace to place,\nwithout\nGoogle Maps\nNow would be a good time to period of human behavior, so archaeologists have to do a lot"
	},
	{
		"chunk_id" : "00004478",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "have you memorize our entire we can let the term live. This of extra science and detective\nevolutionary trajectory from period takes us from 4 mya work to reverse extrapolate the\nAustralopithecus afarensis to (million years ago) to about 12 past. What's interesting,\nmodern Homo sapiens, but it's kya (thousand years ago). however, is that we can learn a\nnot important to this discussion. Prehistory is broken up into tonne about the lives of"
	},
	{
		"chunk_id" : "00004479",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "I would like you to remember a three smaller periods, called the prehistoric people by observing\nfew examples of human Paleolithic, the Mesolithic, and patterns of changes in the stone\nadaptations, however, and to be the Neolithic. These fabulously tools they made over time. Let's\nable to reflect critically on their clever terms mean \"old stone"
	},
	{
		"chunk_id" : "00004480",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "on to behavior; as historians, guessed already, these periods We have evidence of stone tools\nthat's what we really care about, are defined by the different from as early as 3.3 mya, and\nanyway. things people did with stones at these tools continue to indicate\nthose times. [4] similar patterns of behavior until\nPrehistory\nabout 20 kya. That is to say,\nIt seems weird, right? But here's\nFor the record, the usefulness of humans had very little change in\nthe thing: while I'm sure people"
	},
	{
		"chunk_id" : "00004481",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the thing: while I'm sure people\nthe term prehistory for our lifestyle for the vast majority of\ndid all kinds of stuff that had\npurposes is hanging by a thread. human history.[ 4 ] So, what do\nnothing to do with stones, it\nIt means, loosely, \"stuff that the stones tell us?"
	},
	{
		"chunk_id" : "00004482",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Luckily, the development of\nbone decay rapidly, while stones the chopper, or hand-axe. It's a\nwriting coincides with a swath of\nstay in tact. This is called limited diamond-shaped, hand-held\nother important changes that\nevidence, and it means that tool used for cutting and\nbrought in a whole new critical\nscraping (not to mention\nchopping).[ 5 ] There are a few\nother tools in the archaeological\nrecord, but by and large, it was\nchoppers for days (millions of\nyears, actually). We have found"
	},
	{
		"chunk_id" : "00004483",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "years, actually). We have found\nsuch an abundance of these\nthings that we can use them to\ntrack migration patterns, figure\nout what people were eating,\nand even decipher clues about\ntheir behavior by comparing\nthem with nearby artifacts\n(things that people left behind).\nWhat we've learned is that\nbehavior during this time was\nlimited to a few activities. People\nlived in bands (families of\naround 10) and were nomadic,\nwhich means they moved\naround from place to place, with\nno permanent home. They"
	},
	{
		"chunk_id" : "00004484",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "no permanent home. They\nfollowed around herds of large\ngame, like bison, and hunted\nthese animals for food, while\nalso gathering fruit and nuts to\nsupplement their diets. We have\nanother really creative name for\nthis way of living: we call it the\nhunter-gatherer lifestyle. [6 ]\nThat's how humans lived for,\nwell, most of the human story.\nNot very complex, but we did\nthrive; and spread throughout\nEurope and Asia at this time. It\ngets better at this next one.\nM esolithic"
	},
	{
		"chunk_id" : "00004485",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "gets better at this next one.\nM esolithic\nAt about 20 kya, humans start The Great Zab, a river\ndiversifying the types of stone\nflowing through Iraq, is home\ntools we made; which means we\nto many paleolithic sites,\nmust have started doing more\nthings. The biggest pattern we including Shanidar Cave in\nsee in the tools is that they got\nthe Zagros Mountains\nsmaller. It's at this time that we\nsee the development of blades,\nwhich are thin cuts of stone\nused for knives, spear tips, and"
	},
	{
		"chunk_id" : "00004486",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "used for knives, spear tips, and\narrow heads. There is a larger variation in the tools\nwe have, especially when it comes to hunting. At this\ntime, we find implements that would be appropriate\nfor different styles of hunting. This is also when we\nstart to see fish hooks.[ 7]\nSo, the big pattern here is that we're still using stone\ntools, but they're getting smaller and more diverse. If\nwe follow the evidence, we should be able to draw\nsome pretty keen guesses as to what that means to"
	},
	{
		"chunk_id" : "00004487",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "some pretty keen guesses as to what that means to\nhuman behavior in this era. In this case, it looks to\nus like smaller hunting tools means we're hunting\nsmaller game. And more tool diversity means we're\nengaging in a broader range of activity.\nIt makes sense as far as I'm concerned. If the name\nof the game is survival, and you figure out a way to\naccess a larger pool of food options, then you just\nleveled up. In the Mesolithic, humans invented a\nnumber of tools that gave them access to food that"
	},
	{
		"chunk_id" : "00004488",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "they couldn't access before, like small game and fish.\nSo, what are people doing in the Mesolithic? More.\nWe're also moving around less, because our\nnewfound access to a more diverse pool of food\nChoppers were used in the Paleolithic, probably for\nshearing meat from bone. means we now rely less on following herds of big\ngame. No permanent settlements, though.\nI want to use this as an example of the type of\ncritical thinking I'm looking for in this course. As"
	},
	{
		"chunk_id" : "00004489",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "scientists, one of our main goals is to follow patterns\nlike this in the data when we hypothesize about\nwhat's going on in human behavior. You already\nhave excellent tools for recognizing patterns. I want\nyou to trust that your thoughts and ideas are\nimportant to us (all of us, that is) in figuring out the\nbig picture. Good pep talk? Great, because this next\none's a doozy.\nNeolithic\nI know what you're thinking, and you're right. In the\nNeolithic (which starts around 12 kya), the tools get"
	},
	{
		"chunk_id" : "00004490",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "even smaller and even more diverse, which means\nwe're accessing an even larger pool of food\npossibilities, and engaging in increasingly complex\nbehaviors. In addition to the blades of the\nMesolithic, the Neolithic adds an even smaller\nsub-set of tools called microliths. These include tiny\n(around 1 cm), intricate blades, needles for sewing,\nand even saw teeth. Another radical innovation in\nA core from the Neolithic. We even found the same hand\nthis era is called a core. This is a hand-held stone"
	},
	{
		"chunk_id" : "00004491",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "model to share it with you.\ncylinder, often made of flint, that Remember that talk we had a human experience were born\nis used as a portable quarry for few pages ago about natural from the seeds of agriculture. It\nmaking other stone tools. A selection? How, if the was so revolutionary to our way\nsecond \"hammer stone\"\" is used circumstances around a living of living that archaeologists refer"
	},
	{
		"chunk_id" : "00004492",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "that can be shaped into blades. [7] mutation work in it's favor, then agriculture (and the influx of\nBecause people could take cores it gains considerable change that followed) as the\nwith them and then create new survivability? Well, it's coming Neolithic Revolution.[ 8 ] We'll\ntools as-needed, they are known back around in the Neolithic, but talk more about the impact of\nto archaeologists as the this time for the plants! agriculture on human behavior"
	},
	{
		"chunk_id" : "00004493",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "\"Neolithic Swiss-Army Knife.\"\" in the next chapter; but for now"
	},
	{
		"chunk_id" : "00004494",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "via a monumental accident that because we would go on to\nwould ultimately reshape the spread, replant (and um, \"Lor em sit"
	},
	{
		"chunk_id" : "00004495",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "seed.\nrealized that we could plant\nOr, maybe you'd call it a kernel. them in the ground directly and\nAt any rate, there were a bunch therefore grow our own food, -Quote Author\nof seeds and grains and kernels which is called agriculture. [8]\nhanging out in the Levant\nSo, if more access to food meant\n(modern-day Iraq) at this time\nmore complex behavior in the\nbetween the Tigris and the\nMesolithic, what do you think\nGrandma's grits might be as old a\nEuphrates rivers. Bother rivers"
	},
	{
		"chunk_id" : "00004496",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Euphrates rivers. Bother rivers\nhappens when we have complete\nrecipe as she claims\nflooded often, so the land\ncontrol of our food?\nformed a sort of fertile crescent\nPande-freakin'-monium, that's them, and think critically about\n(I totally came up with that term\nwhat. Controlling our food what they might reflect. What\non my own). Anyway, people\nmeant we could stay in the did it mean for Paleolithic\nwere in the area too, and they\nsame place and build people to be controlled by their"
	},
	{
		"chunk_id" : "00004497",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "found that this abundance of\npermanent cities. We did. Food food, and for Neolithic people to\nfood all over the place went\nsurpluses meant we could gain control of it? How might the\nnicely with their plans to not\ndiversify the labor force and tools we leave behind reflect on\nwork very hard.\navoid shortages in the winter. our behavior and way of life?\nAnd that was all well and good We did. We even invented And what might you be able to"
	},
	{
		"chunk_id" : "00004498",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "for a while. People had access to ceramics to store the excess. guess about someone if you had\nfood surpluses and could stay in Extra food means we can have a chance to look through their\none place (this is called being merchants and trade. We did. garbage?\nsedentary). But then, the most We invented writing to keep\ncurious thing happens. track of transactions. In fact,\nhundreds of changes to the\nPaleolithic petroglyphs in the Atlay Mountains, Russia\nReferences"
	},
	{
		"chunk_id" : "00004499",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "References\n1. Professor Richard J. Evans (2001). \"The Two Faces of 5. \"\"Oldowan and Acheulean Stone Tools.\"\" Museum of"
	},
	{
		"chunk_id" : "00004500",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Favoured Races in the Struggle for Life (6th ed.). Cambridge University Press.\nLondon.\n7. Bahn, Paul, ed. (2002). The Penguin archaeology\n3. Cameron, David W. (2004). Hominid Adaptations and guide. London: Penguin Books.\nExtinctions. Sydney, NSW: UNSW Press.\n8. Pollard, Rosenberg, and Tigor (2015). Worlds\n4. Fagan, Brian. 2007. World Prehistory: A brief together, worlds apart concise edition vol.1. New York:\nintroduction New York: Prentice-Hall. W.W. Norton & Company."
	},
	{
		"chunk_id" : "00004501",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 2: Neolithic Revolution\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nFrom Stone Tools to\nStarbucks\nIn the last chapter, we left off at"
	},
	{
		"chunk_id" : "00004502",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Starbucks\nIn the last chapter, we left off at\nthe start of the Neolithic Period,\nwhich lasted from about 12 kya\nto about 7 kya. I should mention\nthat this particular period is a\nlittle difficult to date, as it\nhappens at different times for\ndifferent groups of people. [1]\nRegardless, this is when humans\nhad a species-wide mid-life\ncareer change: from nomadic\nhunter-gatherers to sedentary\nfarmers.\nAs difficult as it is to picture\ndropping everything to become\na farmer, it was a sweet deal for"
	},
	{
		"chunk_id" : "00004503",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "a farmer, it was a sweet deal for\nhunter-gatherers, who were\nburning as many calories to\nacquire food as there were in\nthe food itself. Agriculture, on\nthe other hand, the process of\nplanting and harvesting\ndomesticated crops, gave\npeople the ability to be in\ncomplete control of their food.[ 2]\nThe implications of this power\nare so monumental, I'm writing\na whole chapter on it.\nOn top of that, let's not forget\nthat the whole invention of\nagriculture was an accident that\nstarted with people in the"
	},
	{
		"chunk_id" : "00004504",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "started with people in the\nLevant choosing to eat bigger\ngrains instead of smaller ones;\nincidentally jump-starting\nevolutionary change and\nsecuring a permanent food\nsource for the entire region. [2 ]\nThis was a gift in a way: one that\nwould foster more change in our\nspecies than any other\ndevelopment. We took the gift,\nand all that came with it. And we\ndidn't look back.\nAgr icul tur e\nEarly Levantine agriculture gave people the ability to control their food. This is no"
	},
	{
		"chunk_id" : "00004505",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "small feat: it's the hinge-pin to survival and longevity. In nature, if you control the\nfood, you control everything.\nCouch Potato Revolution\nFor the sake of context, let's million years, and our best\ntake a second to remember ideas were the result of a\nthe Paleolithic. It lasted from hankerin'. More options\n4 mya to about 20 kya. In that meant less work, and less\ntime, most of human movement for that matter.[3]\nbehavior consisted of\nAgriculture in the Neolithic is\nfollowing big game animals"
	},
	{
		"chunk_id" : "00004506",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "following big game animals\nno different, but it's way\nfrom place to place, hunting,\nmore intense. The cereal\nand foraging. In terms of\ngrains domesticated in the\nsheer percentage, that's a\nFertile Crescent were enough\nvast majority of human\nto provide a permanent and\nhistory. And when things\nsustainable supply of food;\nfinally change, what solicits\none that could produce\nthe change? Food, glorious\nenough for us to eat, replant,\nfood!\nand even save. This gave us a"
	},
	{
		"chunk_id" : "00004507",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "food!\nand even save. This gave us a\nIn the Mesolithic, when couple of luxuries we had\npeople start developing new never experienced before:\ntools, they do so to gain food surpluses (more food\naccess to a wider pool of food than we could eat), and the\noptions. That would be like ability to be sedentary (to\ninventing a car so that you stay in one place). Ultimately,\ncan go to the drive-through. this newfound control would\nBlades, arrows, and fish facilitate irrevocable change,"
	},
	{
		"chunk_id" : "00004508",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "hooks would yield access to serving as a catalyst that\nthe availability of deer, prompted each major human\nrabbits, and fish sticks. Four development to follow. [3]\nThe M other of Invention\nLike last chapter, let's start by illustrating the behaviors, and then\nlook for the patterns. When agriculture became an option for late\nMesolithic Levantine folks, it was followed by an explosion of new\nbehaviors. Among those were: living in permanent settlements"
	},
	{
		"chunk_id" : "00004509",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "(eventually buildings), the development of ceramics, and the\ninvention of writing. Not long after (at least by 5,000 BCE), we\nstart to see large-scale building projects, organized government,\norganized religion, and evidence of social stratification (also\n[4]\nknown as social classes or division of labor). In terms of\nchange-over-time, this is the most rapid introduction of new\nbehaviors and technology in human history (I'm allowed to call it\nhistory now that people are writing)."
	},
	{
		"chunk_id" : "00004510",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "history now that people are writing).\nI mean, I get that control over food supply is a big deal, but did it\nreally usher in every behavior that defines civilization? As the\nKool-Aid Man would say: oh, yes. Let's dig into that thought.\nNotwithstanding all the modern convenience, agriculture also\ncame with problems. First, the abundance of food and new\nsedentary lifestyle led to a major surge in population. To suit the\nneeds of these increased numbers, people established building"
	},
	{
		"chunk_id" : "00004511",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "complexes. Since food crops could only grow in certain seasons,\npeople needed to intentionally create surpluses to save for the\nwinter. Ceramics were developed to store the excess. And finally,\nas settlements started dividing labor, trade became a part of\neveryday life. Writing was invented as a way of keeping track of\ntrade and resources. [4]\nSo here's the pattern I'm seeing: new complexities of agricultural\nsociety would come to pose new problems. And, somewhat"
	},
	{
		"chunk_id" : "00004512",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "ironically, the solutions we developed to address these problems\nwould consistently integrate more complex behaviors into\nsociety. Rinse and repeat this process for a few thousand years Housing complex at Ur, Sumer\nand suddenly society starts to look a lot like civilization.\nEarthenware style pottery\nRammed-Earth architecture at Gansu Province, China\nBuilding\nDuring the Mesolithic, we have evidence of These complexes, like the well-preserved"
	},
	{
		"chunk_id" : "00004513",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "semi-permanent dwellings (some of which are atalhyk site in Turkey, were usually constructed\nmade of interesting materials like mammoth of alluvial clay and would sun-bake into a\nbones). Archaeologists have hypothesized that long-lasting ceramic compound.[ 6 ] Today, we often\npeople may have stayed in these homes use the Spanish word adobe to describe this style\nseasonally, based on the availability of food from of construction. Oh, speaking of which, I'm about"
	},
	{
		"chunk_id" : "00004514",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "region to region. But nobody had any need for a to stick the landing on this transition!\npermanent home until farming gave us the ability\nC eramics\nto lead sedentary lives.[5]\nCeramics are durable, long-lasting material crafts\nEven the first permanent settlements in the Levant\nthat are formed by chemically altering clay (and\nwere cleverly constructed housing complexes able\nother organic components) by applying heat. The\nto support the fast-growing populations that"
	},
	{
		"chunk_id" : "00004515",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to support the fast-growing populations that\nprocess creates compounds with molecular bonds,\nagricultural society could support. Or, maybe they\nmeaning that they provide a versatile building and\nweren't that clever: I'll let you decide. All rooms\ncrafting material that can survive for thousands of\nwere adjacent to one-another in a quasi-grid. This\nyears. Though an ancient invention, this would\nmeant that when new rooms were added, only two\nbecome one of the most widely-useful"
	},
	{
		"chunk_id" : "00004516",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "become one of the most widely-useful\nor three walls needed to be built. Nice. Only, how\ntechnologies in human history. [7]\nwould you get to you get to your room if you lived\nin the middle? The head-scratching continues While there are a few known ceramic figures that\nwhen I mention that there weren't any doors. predate the Neolithic period, the technology was\nPeople had to enter their rooms through a uncommon and certainly in its infancy before it"
	},
	{
		"chunk_id" : "00004517",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "trap-door in the ceiling and would then climb had a role in society. When permanent agricultural\ndown on ladders.[ 6 ] Dorm life! settlements came around, the technology\nexploded into our lives.\nFarming is a multi-season process that requires\ncareful planning and timing. Crops like the cereal\ngrains of the Levant are usually planted in the spring\nand harvested in the fall. Few food crops survive the\nwinter, and so early agriculturalists would plant huge"
	},
	{
		"chunk_id" : "00004518",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "surpluses with the intention of storing the extra food\nin ceramic vessels during the winter months. As\nsociety became more complex, we start to see more\ninteresting applications of ceramic technology.[ 7]\nThe civilization that was born from the first\nLevantine farmers is called Sumer. Sumerians used\nceramic technology for transporting resources, as\nbuilding materials, and to create works of art.\nCeramic technology applied to an adobe building.\nSumerians would even invent the potter's wheel to"
	},
	{
		"chunk_id" : "00004519",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Sumerians would even invent the potter's wheel to\nmake the process faster and more uniform.\nCeramics had both practical and intrinsic value to\nSumerian culture. Their popularity during the\nevolution of an increasingly complex society would\nultimately lead to the emergence of art as a cultural\nvalue and even an artisan social class.[ 7]\nW riting\nHere is a good place to mention that it wasn't all\noatmeal all the time for Levantine agriculturalists. The\npattern of diversifying food sources that began in"
	},
	{
		"chunk_id" : "00004520",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the Mesolithic continued in the Neolithic. And for\nthat matter, we weren't just domesticating plants:\nwe were also breeding farm animals for meat, milk,\nand wool. We had even domesticated dogs as\nSumerian housing complex. hunting companions well before the Neolithic. Add\nthis to the builders and artisans and we've got plenty\nof diverse resources to trade.\nAnd trade is going to make society exceedingly\ncomplex. It will build us up, divide us, spur the\ndevelopment of government and religion, and,"
	},
	{
		"chunk_id" : "00004521",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "development of government and religion, and,\nrelevant to this passage, facilitate the development\nof writing. We were all hoping that the first Sumerian\nwriting would be some sort of detailed account of\nthe history of civilization. But the earliest known\nwriting is actually just a bunch of receipts. Sumerians\nkept excellent records of their trades, and that\naccounted for most of their need for writing for a\ngood portion of the development of their society."
	},
	{
		"chunk_id" : "00004522",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "good portion of the development of their society.\nArchaeologists aren't complaining either: it made it\neasy to figure out what early pictographs stood for\nThis style of pottery is known as attic blackware. when most of the talk was about goats and chickens.\nWe use that word, pictograph,\nto describe writing in which the\nsymbols are like pictures that\n[8]\nresemble the word-in-question.\nSame goes with the Greek term\nhieroglyphic, both of which\nroughly translate to \"picture"
	},
	{
		"chunk_id" : "00004523",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "writing.\" Early Sumerian writing"
	},
	{
		"chunk_id" : "00004524",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Linguists are super interested in\nwriting systems with arbitrary\nsymbols because they signify\ncomplex thinking. In our\nlanguage, you can convey pretty\nmuch any thought by learning\n26 corresponding symbols and\nsounds. Future archaeologists\nare going to see emojis and be\nvery confused as to why we\nCuneiform found on Sumerian brickwork.\ndecided to go back to\nhieroglyphics.\nOver time, as society became\nmore complex, Sumerians\nstarted writing more than trade\nrecords. The demands of"
	},
	{
		"chunk_id" : "00004525",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "records. The demands of\ncivilization facilitated the need\nfor new applications of writing.\nScribes began to make records\nof laws and government,\nreligious texts, and even\nliterature. Writing found its place\nin society as an artistic\nendeavor, and scribes were\nrevered as artisans.[8]\nCuneiform's arbitrary symbols could convey complex ideas.\nBy the end of\nthe Neolithic,\nSumer had a\nhighly complex\nsociety, as\nevidenced by\narchaeological\nremains like\nthe famous\nZiggurat of Ur.\nM ore Oatmeal, M ore Problems"
	},
	{
		"chunk_id" : "00004526",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Ziggurat of Ur.\nM ore Oatmeal, M ore Problems\nBased on the timing of these couple more examples before represented natural\ndevelopments, a clear pattern we wrap. phenomena, and worship (in the\nemerges. It's a cycle of gifts and eyes of the Sumerians) was tied\nMajor increases in population\nproblems. After four million to agricultural success or lack\nwould necessitate some form of\nyears of not-much-change, thereof.\ncontrol of the distribution of\nagriculture comes along and"
	},
	{
		"chunk_id" : "00004527",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "agriculture comes along and\nresources. From this, a system Finally, as cities grew, people\nsuddenly we can control our\nof laws and a power structure began to fill different roles to\nfood and dramatically increase\n(we call this government) [9] meet the needs of society\nour survivability. But that\nemerged. Somehow we landed at-large. This led to division of\nlifestyle creates new problems\non a monarchy, allowing one labor, where many people had\nfor society, and we need"
	},
	{
		"chunk_id" : "00004528",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "for society, and we need\nperson to have singular control different jobs, each of which\ncomplex solutions like buildings,\nover all the oatmeal. When he served a specific need. [9 ] As\nceramics, and writing to address\ndied, his son would inherit his society became more complex,\nthem. As the cycle continues,\noatmeal kingdom (that should be builders, bakers, and farmers\nsociety is becoming rapidly more\na Wes Anderson movie). Well, gave way to increasingly more\ncomplex: which means even"
	},
	{
		"chunk_id" : "00004529",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "complex: which means even\ngood for that guy, I guess. specialized jobs. Incidentally,\nmore gifts, and even more\npeople began to value certain\nproblems. As people sought to make sense\njobs more than others. This is\nof the world around them,\nAnd that's why we call it the called social stratification (or\nexplanations and myths would\nNeolithic Revolution: never in social classes). Social scientists\nevolve into organized religion.[ 9]\nhuman history has so much consider this a marker of"
	},
	{
		"chunk_id" : "00004530",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Early Sumerian religion was\nchange happened in so little extremely complex human\npolytheistic (many gods), and\ntime. It's the Oatmeal Big Bang interaction. As such, we begin\nfunctioned to both explain\n(not a real term but do enjoy it). here to move from the term\nnature and (if all went well)\nSo, that's the main idea I'm society in describing human\nafford people a sense of\ntrying to share, but let's hit a cultures to the term civilization.\ninfluence over it. Gods\nOne Last Thing"
	},
	{
		"chunk_id" : "00004531",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "influence over it. Gods\nOne Last Thing\nBefore we break, I want to make a note on the nature of my goals\nPS: There were in these chapters and, ultimately, on the limitations of these\ngoals. My primary objective is to build a general understanding of\ndomesticated\nhuman behavior over time by looking at patterns of change that\nsnails found at\ncharacterize and define each of the critical periods in human\nnumerous development. I hope that this will provide a framework for"
	},
	{
		"chunk_id" : "00004532",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "sites in the understanding human history: providing compartments with\nwhich to compartmentalize the many specifics you will learn in\nNeolithic\nthe future.\nMediterranean.\nHowever, that means we're in for some limitations as well. A\nAre you feeling\nbroad approach means that we can't learn everything about\nhungry now?\neverything. I invite you to engage independently to start\nYou're gathering more examples of these patterns of behavior. For"
	},
	{
		"chunk_id" : "00004533",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "welcome. instance, if you research other early civilizations like Egypt and\nthe Indus Valley, you'll find more evidence of the patterns we\nillustrated in Sumer; and be able to note how other cultures\napproached similar problems.\nThe other limitation is that we'll miss out on some of the\nexceptions to these patterns. For example, there are cultures that\ndeveloped Neolithic technology without agriculture. The Jomon in\nJapan were able to fish so effectively that they could support"
	},
	{
		"chunk_id" : "00004534",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "large populations without the aid of farming. And the Erteblle of\nDenmark did it with eel trapping. But, notwithstanding, most\npeople needed farms to get sedentary.\nThe patterns I describe in these chapters won't be able to\nencompass all of human activity. But I also don't think that trying\nto do so would be a very useful goal. For our purposes, I want\nyou think about the big picture when it comes to these critical\nperiods and the patterns of behavior that define them."
	},
	{
		"chunk_id" : "00004535",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Sumerians ideas like the arch spread quickly in the Neolithic.\nBarley, emmer, and einkorn are thought to be the first\ndomesticated grains. References\n1. Jean-Pierre Bocquet-Appel (July 29, 2011). \"When the 5. Francis Ching"
	},
	{
		"chunk_id" : "00004536",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "2. Graeme Barker (2009). The Agricultural Revolution in 1999, Anatolian Archaeology, vol. 5, pp. 4-7, 1999\nPrehistory: Why did Foragers become Farmers?. Oxford\n7. Frankfort, Henri, The Art and Architecture of the\nUniversity Press.\nAncient Orient, Pelican History of Art, 1970, Penguin\n3. Scarre, Chris (2005). \"The World Transformed: From"
	},
	{
		"chunk_id" : "00004537",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Human Past: World Prehistory and the Development of\nLibrary of Virginia.\nHuman Societies. London: Thames and Hudson.\n9. Ascalone, Enrico. 2007. Mesopotamia: Assyrians,\n4. Gordon Childe (1936). Man Makes Himself. Oxford\nSumerians, Babylonians (Dictionaries of Civilizations).\nuniversity press.\nBerkeley: University of California Press."
	},
	{
		"chunk_id" : "00004538",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 3: Golden Age\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nAt the Height of\nCivilization\nNeolithic technology left us with\nincreasingly complex societies;"
	},
	{
		"chunk_id" : "00004539",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "increasingly complex societies;\nowing much thanks to a cycle of\nintricate problems and solutions\nspurred by the development of\nagriculture (and the sedentary\nlifestyle that came along for the\nride). Sumer was the first society\nto grow into a civilization,\ncomplete with ceramics, writing,\nhousing complexes, organized\ngovernment and religion, large-\nscale building projects, and\nsocial stratification.\nSomewhere around 7 kya (or\n5,000 BCE as the kids are calling\nit) we stop calling it the Neolithic"
	},
	{
		"chunk_id" : "00004540",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "it) we stop calling it the Neolithic\nPeriod and change over to\nsomething like The Age of Ancient\nCivilizations. I'm not really into\nthis term, because, even though\nit's informative, it implies that\nwe need to give equal time to\neach of the civilizations that\npopped up after Sumer got up\nand running. We don't (and\nwhen we try, it usually just turns\ninto a list of their stuff).\nSo I'm opting for a term that the\nGreeks used when describing a\ntime of prosperity: The Golden\nAge.[ 1 ] This was an era in which"
	},
	{
		"chunk_id" : "00004541",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Age.[ 1 ] This was an era in which\nnew cultures (like Egypt, Greece,\nand Rome) grew and thrived,\ncommon people had access to\nhigh standards of living, and\ngreat ideas would spread across\ncontinents. In this chapter, we'll\nthink about how ideas disperse,\nwhat ideas can increase the\nquality of life of a population,\nand what types of ideas had the\ngusto to last into the world we\nknow today.\nTr ue Gr it\nCulture was incredibly diffuse during this era as people traded, traveled, and"
	},
	{
		"chunk_id" : "00004542",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "shared ideas. The best ideas had the moxie to last through the modern age.\nSharing is Caring\nNot long after agriculture facilitated the development of\ncivilization in Sumer, we start to see the same patterns sprout up\n(see what I did there?) throughout Africa, Europe, and Asia. First,\nsustainable food, then civilization. The curious thing about these\nrising cultures is that they all seem to have the same technologies\nand behaviors associated with agriculture that Sumer did. This"
	},
	{
		"chunk_id" : "00004543",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "begs the question: are these cultures borrowing ideas from\nSumer, or independently inventing them?\nThe answer is that, while we have evidence of both forces at play,\nborrowing is going to play a much more pronounced role in\nshaping the patterns of behavior that define the Golden Age for\nthese ancient civilizations. Still, it's worth noting that behaviors\nand technology like agriculture were independently invented in\nseveral places (like America); and when this happened, these"
	},
	{
		"chunk_id" : "00004544",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "cultures ended up with the entire Neolithic Party Pack of\nbehaviors and technologies (necessity being the mother of\ninvention and all).[ 2]\nWhen ideas and culture spread from place to place (via\nborrowing), we call it cultural diffusion. Tracking the diffusion of\ntechnology and culture is a major tool for historians, and it can\nhelp us identify all kinds of interesting patterns of behavior. In\nthe Golden Age, good ideas become considerably diffuse as"
	},
	{
		"chunk_id" : "00004545",
		"source" : "Open_History_Ch_1_Prehistory.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "cultures passed information throughout Europe, Africa, and Asia.[ 3]\nThe better the idea, the more it spread, benefiting more cultures\nand people. The very best ideas during the Golden Age not only\nbenefited people of the ancient world, but ultimately had the\nstaying power to last into the modern era. So what types of ideas\nGreek architecture brought art and\nhad the gusto to make it into the world we know today? The\nbeauty into the lives of regular people."
	},
	{
		"chunk_id" : "00004546",
		"source" : "Open_History_Ch_2_Neolithic_L5MxV2D.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "beauty into the lives of regular people.\npattern we see is that the longest-lasting, most impactful ideas\nwere those that improved the quality of life for ordinary people.\nRuins at Pompeii., Italy.\nRome carried on the Greek theatre tradition.\nPower to the People\nThere's a good reason that regular people hold the change. But chances are a little slim that such a\nkeys to lasting culture: we are the culture. We're person would just so happen to be the guy whose"
	},
	{
		"chunk_id" : "00004547",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the ones having the ideas and we're the ones great accomplishment was being born. Yeah, kings\nspreading them. That's how the good stuff gets so don't usually like change anyway. Wouldn't want to\ndiffuse in the first place. Ideas and technologies risk upsetting the balance of inequality that put\nthat stand to benefit us find themselves in high them in power in the first place.\ndemand, and all parties (merchants, scholars,\nOkay, got a little off track there. Let's get back on"
	},
	{
		"chunk_id" : "00004548",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "artisans, etc) collect the spoils. Really long-lasting\nthe rails. The next item on the agenda is to dig in:\nideas, contributions to the arts, math, science,\nhaving a look at the lasting contributions of\ncivics, and engineering, are the result of high\nancient civilizations, and taking note of how these\ndemand in the ancient world. And nothing could\nideas improved the quality of life of regular folks.\nsolicit greater demand than items bearing\nThe Arts\nsignificant benefit to the people demanding them."
	},
	{
		"chunk_id" : "00004549",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "significant benefit to the people demanding them.\nWhich reminds me- here's another place where I think this is a pretty good place to start, as art is\nthis document is going to part from traditional inherently beneficial to society; a sort of cultural\ntextbooks: I have little interest in the kings, armies, food that brings happiness and meaning to our\nand court politics that seem to garner so much lives while delineating our values and ideals. Art"
	},
	{
		"chunk_id" : "00004550",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "attention in these books. History is a social science has been a part of our lives since at least the late\nthat looks at patterns of behavior amongst people. Paleolithic. If you go to the Lascaux and Chauvot\nThat's people, not just the rich and powerful, who archaeological sites, you'll see that French people\naccount for so little of the population and shape so have been the world's premiere artists since well\nlittle of our culture. Certainly, there were before there even was a France. That being said,"
	},
	{
		"chunk_id" : "00004551",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "individuals who were responsible for significant the achievements of Golden Age artists had a\nsignificant impact on the quality of people's lives, so\nmuch so that many of the traditions established in\nthis era continue to improve the standard of living in\ntoday's world.\nWhich brings us to Greece. While every civilization in\nthe ancient world lays claim to glorious and timeless\nartistic achievements, it was the Greeks who\ndeveloped traditions in the arts that would become"
	},
	{
		"chunk_id" : "00004552",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "staples of public life. The Greeks invented theatre,\nan art form that brought literary works to life\nthrough actors who played the roles of characters in\npopular stories and mythology. Greek theatre was\nlarger-than-life, featuring big casts wearing\nover-sized, exaggerated masks to represent the\nAcropolis, Athens emotions of their characters.[ 4 ] Playwrights like\nAeschylus, Euripides, and Sophocles drew on history\nand mythology to produce comedies or tragedies"
	},
	{
		"chunk_id" : "00004553",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "and mythology to produce comedies or tragedies\nwhose themes resonated with audiences (and still\ndo). Large public production centers throughout\nGreece, called amphitheatres, reveal how crucial\nthis art form was to all members of society. The\ntradition of theatre lives on today, in its original form\nas well as by extension in film and television.\nStorytelling was an important element in the lives of\nancient Greeks. Mythology, which to them was more\nof a combination of religion and history, was largely"
	},
	{
		"chunk_id" : "00004554",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "gleaned through narratives told by a special class of\nprofessional storytellers called rhapsodes.[ 5 ] While\nwriting was commonplace in ancient Greece, many\nstories weren't written down, but were passed on\nAthena and Heracles\nthrough public performance. Some important Greek\nliterature has sadly been lost because it was never\nwritten down. Fortunately, several of the works of\nHomer (the ancient rock star of this tradition) were\nwritten down by Persian scribes (when he was on"
	},
	{
		"chunk_id" : "00004555",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "written down by Persian scribes (when he was on\ntour), including the Iliad and the Odyssey.[6]\nSomewhere between the epic poetry, collected\nmyths, the singer-songwriter tradition, and theatre,\nthe Greeks mastered the art of telling a story. The\ninfluence of Greek storytelling is widespread\nthroughout literature and the arts: from the\nstructure of a play to stock characters, literary\ndevices, the story arc, and many timeless themes.\nThis is one of their most lasting impacts on our"
	},
	{
		"chunk_id" : "00004556",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "This is one of their most lasting impacts on our\nworld, and consequently a facet that dramatically\nimproves people's quality of life.\nTheatre at Ephesus, Turkey\nEngineering + Technology\nOkay, sure, you're right. Egyptians had large-scale building on\nlock. And, there's stuff they did- we don't don't know how (with\nancient tools and all). But that's actually going in another\ncategory. I mean, unless the pyramids themselves are a staple of\nanyone's quality of life, we can come back to them later. Anyone?"
	},
	{
		"chunk_id" : "00004557",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Great. Let's talk about toilets.\nThe Romans were a culture that, for much of their history, had a\nmarked focus on improving the standard of living of regular folks.\nAnd I see no better example of this in the technology of the\nancient world than toilets. Now, in a society like Sumer or Egypt\nwith more emphasis on stratification (no offense, my dudes), we\nmight expect expensive technology like this in the hands of kings\nand aristocrats. But not the Romans: their bathrooms were all"
	},
	{
		"chunk_id" : "00004558",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "public facilities (please bring your own wiping stick). That's\nalready saying something for a populous minded vision, but it\ngets even better.\nRoman plumbing (networks of water distribution) required\nrunning water throughout their cities, many of which were built\nuphill of water sources.[ 7 ] To address this, they built aqueducts: a\nsimple but expensive technology that brought water pipes\nhundreds of feet in the air so as to make better use of gravity in"
	},
	{
		"chunk_id" : "00004559",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "redirecting the water. Aqueducts left the Roman government with\nhefty infrastructure bills, all so that every Roman citizen had\naccess to clean, sanitary public facilities.[ 7 ] And it doesn't even\nstop there. Public bath houses, complete with large swimming\npools, stores for shopping, and hot tubs (I'm not kidding), showed\nan emphasis on public quality of life bordering on obsession.\nMan, this section is super Rome-y, and I'm not even done with"
	},
	{
		"chunk_id" : "00004560",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "them. Roman leaders cared a lot about beauty in everyday life, so\nLegit Roman hot tubs.\nit's not surprising that there was an emphasis on architecture.\nRoman baths at Caracalla.\nAqueducts let Romans hydrate anywhere.\nTo make it possible to have to as many neighboring cultures (not literally, though: Greek\nbeautiful, long-lasting cities, as possible, and so built an money was mostly coins).\nRomans developed new elaborate system of roads.[ 7] Greece was the first society to"
	},
	{
		"chunk_id" : "00004561",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "techniques, while perfecting a Roman roads were so well develop a government run by\nfew that they borrowed from planned and maintained, we still the people themselves. This is\nneighboring civilizations. retain the old aphorism all roads called democracy (which\nSumerian arches and Greek lead to Rome. literally means people power),[ 8]\ncolumns were staples in Roman Civics and it's probably the most\narchitecture, but the Romans important development of the"
	},
	{
		"chunk_id" : "00004562",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "were able to take these concepts Okay, Rome dominated that last ancient world; which makes\nfurther with the development of section, and I'm not sad about it. sense, because it bears the most\ncement and concrete, This was a culture that cared direct impact on people's\ntempered stone building deeply about the lives of regular standard of living.\nmaterial that could be poured citizens. It's worth noting that\nDemocracy was first developed\ninto shape.[ 7 ] This technology many Roman leaders were"
	},
	{
		"chunk_id" : "00004563",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "by the people of Athens, when\nstacks nicely with ceramic enamored with Greek culture,\nGreeks were organized into\nmolds, which could help and did what they could to\ncity-states. Athenians had a\nbuilders produce intricate replicate the success of the\ndirect democracy, in which all\nfacades, which often covered Greeks (Roman Emperor Nero\ncitizens would eventually serve\ngood old-fashioned brick work. was almost as obsessed with the\non the Senate, a committee that\nGreeks as my dad- and almost"
	},
	{
		"chunk_id" : "00004564",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Greeks as my dad- and almost\nThe Romans were also would make and enforce the\nas crazy). So why were the\nconsidered a cosmopolitan laws. Good, but not perfect,\nGreeks so good at improving the\nsociety, one which welcomed considering that not everyone\nlives of citizens?\nand connected to other cultures was considered a citizen.[ 8 ] When\nthrough trade and travel. Because the Greeks put their Rome sought to adopt\nRomans wanted to have access money where their mouths were democratic ideals, they found"
	},
	{
		"chunk_id" : "00004565",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "direct democracy difficult to\nmanage in a society with such a\nhigh population. Their solution\nwas to divide society into\ndistricts, and to elect a\nrepresentative from each one.\nThis is called a republic (or\nrepresentative democracy) and\nit's the form we use today. [9]\nOkay, I promised Egyptian\npyramids, and an explanation of\nwhy they're in my section on\ncivics. The Greek historian\nHerodotus, upon seeing the\nmarvel of Egyptian architecture,\nconcluded that pyramids could\nonly have been accomplished by"
	},
	{
		"chunk_id" : "00004566",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "only have been accomplished by\nslave labor. And he was wrong.\nArchaeologists like Mark Lehner\nhave discovered sites that\nserved as workers' villages\nalongside the great pyramids,\nand the evidence suggests that\nnot only were workers paid, but\nthey were skilled state\nemployees taking part in a\nnational (maybe international)\ncivic project.[ 1 0 ] This implies that\nlarge-scale building in the\nancient world was mutually\nbeneficial to the people and the\npeople in charge. Projects like"
	},
	{
		"chunk_id" : "00004567",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "people in charge. Projects like\nthe pyramids, the Acropolis in\nAthens, and the Roman\nColosseum served as landmarks\nof cultural identity, civic pride,\nand, importantly, as a source of\nreadily-available employment. Archaeological sites of\nOne Last Thing builders' villages reveal a\npaid, skilled workforce.\nFrom art to engineering to civic\nlife, we can see that societies in Projects like this were the\nthe ancient world had a clear\nopposite of slave labor: they\nfocus on the quality of life of"
	},
	{
		"chunk_id" : "00004568",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "focus on the quality of life of\nwere sources of ready\neveryday people. Culture was\nemployment and civic pride.\nextremely diffuse in this era, and\nas we've discussed, the chances\nof an idea becoming so popular\nRuins at Delphi, Greece.\nas to bear saturation amongst ancient civilizations drive cars and use computers, while pagers and\nand a lasting impact on our world today, were rollerblades fall into decline? Then and now, it's"
	},
	{
		"chunk_id" : "00004569",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "greatly increased for those ideas in demand. And the stuff that improves the standard of living, that\nwe're in charge of demand. It stands to reason that makes life better for regular people.\nthe stuff we want is the stuff that makes our lives\nThat being said, the next chapter is going to be a\nbetter; and that's still true. We see this force at\nlittle annoying, because it marks an era of greed,\nplay constantly in the modern world: a world\nwar, and inequality that puts the progress of the"
	},
	{
		"chunk_id" : "00004570",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "war, and inequality that puts the progress of the\noverrun with new ideas and technologies. What\nGolden Age in utter decline. In the mean time, try\nsticks? Why do some artifacts stay relevant in our\nto enjoy your toilet and your democracy.\nculture, while others are forgotten? Why do we still\nSumerian arches made it possible to build large, open\nstone structures. References\n1. Robin Hard - The Routledge Handbook of Greek 6. Bahn, E. & Bahn, M.L. (1970). A History of Oral"
	},
	{
		"chunk_id" : "00004571",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Mythology, Psychology Press, 2004 Interpretation. Minneapolis, MN: Burgess.\n2. Lamb, David; Easton, S. M. (1984). Multiple Discovery: 7. Casson, Lionel. Everyday Life in Ancient Rome,\nThe Pattern of Scientific Progress. Amersham: Avebury revised and expanded edition. Baltimore: The Johns\nPublishing. Hopkins University Press, 1998.\n3. Rogers, Everett (1962) Diffusion of innovations. New 8. Thorley, J., Athenian Democracy, Routledge, 2005\nYork: Free Press of Glencoe, Macmillan Company."
	},
	{
		"chunk_id" : "00004572",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "York: Free Press of Glencoe, Macmillan Company.\n9. Flower, Harriet I. (2004). The Cambridge Companion\n4. Ley, Graham. A Short Introduction to the Ancient to the Roman Republic.\nGreek Theatre. University of Chicago, Chicago: 2006.\n10. Lehner, Mark 1997. The Complete Pyramids.\n5. Davidson, J.A., Literature and Literacy in Ancient Thames and Hudson. New York.\nGreece, Part 1, Phoenix, 16, 1962."
	},
	{
		"chunk_id" : "00004573",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 4: Dark Ages\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nAshes, Ashes, W e All\nFall Down\nIn the last chapter, we discussed\nan era known to many historians"
	},
	{
		"chunk_id" : "00004574",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "an era known to many historians\nas Classical Antiquity. Ranging\nfrom about 5,000 BCE to the\n300's CE, this is a time marked\nby a high standard of living for\naverage citizens in ancient\ncivilizations that were growing,\nthriving, and sharing ideas\nacross continents. In these\ncultures, there was a clear focus\non maintaining a high quality of\nlife for the people. This makes\nsense. We're the ones actively\nmaking the culture, and so we\nset the tone for its ideals.\nWhat happens next doesn't"
	},
	{
		"chunk_id" : "00004575",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "What happens next doesn't\nmake sense. For the next\nthousand years, societies go into\ndecline. The advantages that\nbenefited citizens in the Golden\nAge were all but lost. People\nstopped reading and writing.\nThe middle class dried out,\nleaving most people stuck in\nunskilled lines of work, mostly in\nthe agricultural sector. [1]\nA Renaissance scholar named\nPetrarch looked back and called\nthis time The Dark Ages.[ 2 ] Modern\nhistorians don't really use this\nterm anymore, opting for more"
	},
	{
		"chunk_id" : "00004576",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "term anymore, opting for more\npolitically correct monikers like\nthe Middle Ages or the Medieval\nPeriod. Technically, the\nachievements of classical\nantiquity weren't lost, they just\nweren't available to most\npeople; who were too poor to\nafford them and too busy to\ntake them back for themselves.\nTo me, that sounds pretty dark,\nso I'm opting for Petrarch's\nphrase to describe it.\nGoing Sol o\nThe Dark Ages are marked by a shift in cultural priorities from the needs of the"
	},
	{
		"chunk_id" : "00004577",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "community to the principle of self-interest. The plague didn't help, either.\nThe One and Only\nThe Golden Age of Classical Antiquity exhibited a distinct cultural\nfocus on the quality of life of the populous at-large. Greece and\nRome especially displayed a sense of public solidarity: that the\nmembers of a community could be stronger if they worked\ntogether. By the 300's CE, we start to see the decline of Rome,\nand the rise of several small but wealthy kingdoms (that for some"
	},
	{
		"chunk_id" : "00004578",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "reason we call the Germanic Tribes).[ 3 ] As these kingdoms (France,\nEngland, Germany, Spain) grow in wealth, power, and influence,\nthey bring along a shift in cultural values; and that sense of\ncommunity is soon replaced with a new focus on the individual.\nThis is part of a larger pattern of consolidation in the Middle\nAges, in which a focus on multiplicity became a focus on\nsingularity. A big world of many things became a small world of\noneness. Much of the change in this time followed this motif."
	},
	{
		"chunk_id" : "00004579",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Religious beliefs shifted from a world of many gods to a world of\none. Democracy, a government of the people, was replaced with\nmonarchies, which were ruled by a single family line. The middle\nclasses, with their countless artisans, poets, and scholars, went\ninto decline as people became bound to a system of unskilled\nlabor.[ 4 ] The world was getting smaller for most of us.\nBut don't get me wrong: the Dark Ages were more than a shift in\npriorities. Most of the achievements that made society so golden"
	},
	{
		"chunk_id" : "00004580",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "in antiquity were lost to a majority of the European population.\nAverage people were no longer reading or writing, sanitation and\nclean water weren't available, a new system of land distribution\nwiped out the middle class and sent poverty skyrocketing, and, to\nadd insult to injury, the plague led to our first decline in world\nIconic castle arrow loops leave behind\npopulation.[ 4 ] Last chapter, I told you that the people propel the\na picture of dangerous times."
	},
	{
		"chunk_id" : "00004581",
		"source" : "Open_History_Ch_3_Golden_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "a picture of dangerous times.\nculture, that demand was a driving force in the height of\ncivilization. What would convince us to give up the keys?\nDonan Castle, Scotland\nMoats kept out any invaders who didn't like to get wet.\nSafe as Houses\nThe answer takes us back to the notion of self It's in conditions like these that people have to\ninterest, a growing cultural priority in the Middle make one of the most difficult choices society has"
	},
	{
		"chunk_id" : "00004582",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Ages. The pattern of consolidation we see in this to offer: whether or not we are willing to give up\nera isn't an aberration, it's a reflection of an some of our freedom for the sake of safety. In this\nemerging sense of self in Medieval society. As time, where reading and writing were in decline,\nRoman values of public life start to decline, we see and king and clergy seemed an insurmountable\nthe rise of the Germanic Tribes: European force, people tended toward the relative safety of"
	},
	{
		"chunk_id" : "00004583",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "kingdoms seeking to acquire wealth, power, and submission over the bleak alternatives.\ncultural influence. The Franks (who will eventually\nSo there it is: a perfect palate for people to act out\nbe France, as well as predecessors for the German\nof self interest. Kings built empires in their names,\nand Dutch cultures) are the first to rise to power,\nchurch officials sought influence, soldiers fought\nand set the tone for empire-building as a cultural\nfor notoriety, and everyone else was just staying\nvalue.[5]"
	},
	{
		"chunk_id" : "00004584",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "value.[5]\nalive. And when you spend so much time and\nThe Germanic Kingdoms made Europe a rather energy looking out for yourself, the world\ndangerous place, and violence crept into every becomes a small, lonely place indeed.\ncorner of life. Wars were waged over territory and Religion\nreligious beliefs. Laborers were strong-armed into\npoor work conditions and managed by an armed The first major societal element to be affected by"
	},
	{
		"chunk_id" : "00004585",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "cavalry. People that sought to maintain elements the pattern of coalescence of the Middle Ages was\nof Roman culture, notably pagan religious religion. Before late antiquity, religious beliefs\npractices, were persecuted and, in some cases, were almost universally polytheistic, meaning\ntortured or killed. [6] that people believed in a system of many gods.\nIt's worth noting that religion in the Golden Age was\nfocused on explaining nature, describing what\npeople saw as history, and sharing songs and"
	},
	{
		"chunk_id" : "00004586",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "people saw as history, and sharing songs and\nstories. Greek and Roman Paganism (the primary\nreligious practice of the time) did not include\nimplications or prescriptions of morality, nor\ndenotations of right and wrong. In fact, it was\nprobably a bad idea to follow the examples of the\ngods, who were seen as amoral (without morals) and\ndid all kinds of awful, selfish things; including, but\nnot limited to, lying, patricide, delivering plagues out\nof spite, and throwing baby Hephaestus off Mt.\nOlympus.[ 7]"
	},
	{
		"chunk_id" : "00004587",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Olympus.[ 7]\nThings started to change, however, during the late\nDome of the Rock, Israel\nRoman Republic, as people began to introduce\nmonotheistic religious systems, which honored a\nsingle god. These new religions, including Judaism,\nChristianity, and Islam, asked followers to adhere to\nmoral guidelines designed mostly to help members\nof society to coexist peacefully. That being said,\nreligion was the focus of a lot of violence in the\nMiddle Ages, including a series of crusades (holy"
	},
	{
		"chunk_id" : "00004588",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Middle Ages, including a series of crusades (holy\nwars) and inquisitions (religious trials) aimed at\nsuppressing paganism and securing sacred lands.[6]\nSo, that seems a little off, right? How could a series\nof moral guidelines aimed at promoting peaceful\nsociety become a source of animosity and violence?\nAnd, if this was a pattern in human behavior, why\nhadn't it happened already? Well, the new element\nin this era is called fundamentalism, a belief that a\nAngelic iconography of the Middle Ages"
	},
	{
		"chunk_id" : "00004589",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Angelic iconography of the Middle Ages\nspecific religion was the one true faith, which comes\nwith an implication that other religions are, by\ndefault, false.[ 6 ] This led to problems that we didn't\nhave in the ancient world, because there weren't any\npolytheistic fundamentalists. Nobody had impetus\nto say \"my gods are the only gods.\"\" When pagans"
	},
	{
		"chunk_id" : "00004590",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "your moral code.\nThe religions that came to be during the Middle Ages\nare still predominant faiths in the world. And, issues\nsurrounding faith and fundamentalism that arose\nduring this time are still sources of tension between\nvarious world cultures. So, I guess we're still working\non that one.\nHeadstones at a Christian cemetery\nGovernment\nIf you have a strong sense of justice, I'd recommend some deep\nbreathing about now: this one's a doozy. Following our pattern of"
	},
	{
		"chunk_id" : "00004591",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "consolidation and self interest, the monarchy comes into fashion\nagain in the Dark Ages. This system grants power to a single\nleader, typically a king or queen; and the power is inherited\nthrough the family line. Most monarchies in this time gave\nabsolute power to the first born male in the royal family;\nhowever, other members of these families could become very\npowerful as well.[8]\nAs populations got bigger, kingdoms became harder to control,\nand a new system of delegating power through land distribution"
	},
	{
		"chunk_id" : "00004592",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "became very popular. Feudalism, as it was called, divided the\nking's land into sections, and appointed nobles (in the family, of\ncourse) to oversee them. These vassals had power over their\ndominions. This is also where we got all of the lovely terms for\nnobility and their territories. Counts watched over their counties,\nBarons looked after their baronies, and (my favorite) Dukes took\ncare of their duchies. Now, even this was a lot to manage, so\nthese nobles hired knights, who also needed to be of noble"
	},
	{
		"chunk_id" : "00004593",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "birth, to serve as ersatz property managers to keep people in\norder and keep the land productive.[9]\nSo, I'm not sure if you had an image of chivalrous courage and\nselflessness in your mind associated with knights, but their job\nwas actually a little boring. They mostly collected taxes and\ntribute from the peasants, who owed a (large) percentage of their\ncrops and goods to the state in exchange for the right to live and\nwork on the king's land. Cruel as it was, this system had built-in"
	},
	{
		"chunk_id" : "00004594",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "incentives for everyone. Nobles had plenty of opportunities to\nattain wealth and notoriety (often through corrupt practices), and\nIt's only missing a ladder of hair. peasants had the incentive to, well, not die.\nA peasant's-eye-view of Edinburgh\nCastle ruins in the Scottish Highlands\nThis seems like a good place to culture. Even the artisan class with incentive to make these\nnote that these changes didn't would soon become nearly privileges available to society"
	},
	{
		"chunk_id" : "00004595",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "take place overnight; they impossible to access, as a didn't have the means to make it\nhappened gradually over the master-apprentice system happen.\ncourse of several hundred years. would allow businesses to favor\nIn a dangerous world, it's not\nIt was the Romans themselves family lines; which made training\neasy to obtain such means.\nwho gave up democracy in favor inaccessible to everyone else.\nConstant war, a corrupt police\nof Emperors. And while those The only widely available work"
	},
	{
		"chunk_id" : "00004596",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "force (not all knights wore white\nEmperors at first sought to was in construction and\nsatin), and infectious disease\nuphold the Roman culture of agriculture, and feudalism made\nmade it even harder to just live\ncommunity and solidarity, it was it very difficult to live\nour lives. The bubonic plague (a\nthey who would eventually cave comfortably in those lines.\npandemic known colloquially as\nto self interest; which would be\nHere's where we can really see the Black Death) wiped out at"
	},
	{
		"chunk_id" : "00004597",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "a primary contributing factor\nthe decline of Golden Age least a third of Europe's\nthat led to Rome's decline. That,\nprosperity in full focus. Those in population.[ 1 0 ] Since the plague\nand malaria. Oh, and maybe\ncontrol had strong incentives to was spread by fleas on rats, it hit\nlead poisoning from the pipes\nstay that way. Reading and the hardest in populations with\nand cosmetics.\nwriting are the tools of poor sanity conditions. While\nThe Fall revolution. Dark Age scholars disease wasn't directly caused"
	},
	{
		"chunk_id" : "00004598",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "wanted these tools as far away by feudalism and inequality, it\nFeudalism created a great divide\nfrom the people as possible; and did make a harsh world harsher,\nin society between those of\ntheir positions made it possible deepening divides and making\nnoble birth and, well, everyone\nto keep it that way. Plumbing, upward mobility that much less\nelse. Upward mobility in society\nclean water, and sanitation were accessible.\nbecame increasingly scarce as\namenities reserved for those"
	},
	{
		"chunk_id" : "00004599",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "amenities reserved for those\nthe concept of royal blood None of the achievements of\nwho could afford them. Those\nbecame more prevalent in the antiquity were lost, exactly. They\njust weren't available to the\naverage citizen. Scholars wrote\nhistories and religious texts.\nArchitects designed magnificent\ncathedrals. Engineers built\nmechanical marvels (siege\nweapons mostly, which makes\nsense). And knights took part in\nthe glory of campaigns and\ntournaments. The rest of us\neither weren't invited to the"
	},
	{
		"chunk_id" : "00004600",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "either weren't invited to the\nparty or we couldn't afford a\nticket.\nOne Last Thing\nThe cultural significance of\nnobility made these boundaries\ncrystal clear: you either had a\ncoat of arms or you didn't. This\nlevel of control over the cultural\nnarrative made it possible for\nthe noble classes to maintain\npower structures that, under\ndifferent circumstances, the\npeople simply wouldn't tolerate.\nWhile cultural divides kept the\nworld small, feudalism provided\nincentives for everyone, noble or"
	},
	{
		"chunk_id" : "00004601",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "incentives for everyone, noble or\notherwise, to keep it that way.\nBut it didn't last forever. Over\ntime, growing populations would\nstart to become concentrated in\nurban centers. Cities were\nharder to control than rural\nareas, especially through\nfeudalism, which capitalized on\nthe value of the land itself. In\ncities, people were able to find\nmore diverse ways of making a Art from the Middle Ages\nliving. By the 1300's, people presented a curious\nstarted to look back to antiquity\nintersection of faith and"
	},
	{
		"chunk_id" : "00004602",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "intersection of faith and\nas a model for society, and took\nviolence. Artists have a knack\naction to bring about a second\nGolden Age. This wasn't going to for capturing the people's\nbe possible in a small, selfish voice during troubled times.\nworld. We needed a change in\nperspective if we wanted to\npaint a better picture.\nCathedral architects captured a sense of otherworldly\npower in a time when faith took center stage. References"
	},
	{
		"chunk_id" : "00004603",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "1. Backman, Clifford R. (2003). The Worlds of Medieval 6. Asbridge, Thomas (2012). The Crusades: The War for\nEurope. Oxford, UK: Oxford University Press. the Holy Land. Simon & Schuster.\n2. Mommsen, Theodore (1942). \"Petrarch's Conception 7. Owen Davies (2011). Paganism: A Very Short"
	},
	{
		"chunk_id" : "00004604",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "3. Collins, Roger (1999). Early Medieval Europe, Royalty: Power and Ceremonial in Traditional Societies\n300?1000. New York: Palgrave Macmillan. (1992).\n4. Heather, Peter. The fall of the Roman Empire. A new 9. Brown, Elizabeth A. R. (October 1974). \"The Tyranny"
	},
	{
		"chunk_id" : "00004605",
		"source" : "Open_History_Ch_4_Dark_Ages.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Creation and Transformation of the Merovingian World. 10. D. Herlihy, The Black Death and the Transformation\nNew York: Oxford University Press, 1988. of the West (Harvard University Press: Cambridge."
	},
	{
		"chunk_id" : "00004606",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 5: Renaissance\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nInto the Light\nThe Middle Ages lasted from\naround 300 to 1300 CE, so that's"
	},
	{
		"chunk_id" : "00004607",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "around 300 to 1300 CE, so that's\nabout a thousand years where\nlife was rather splendid for\nnobility, high church officials,\nand skilled tradespeople. And\nnot-so-much for everyone else,\nwho accounted for a vast\nmajority of the population. I've\ntaken the liberty of restoring\nPetrarch's Dark Ages moniker for\nthe purposes of this publication\nas it best describes the average\nexperiences of the average\nperson at this time; which is to\nsay dark.\nA lot of historians are beginning\nto abandon characterizations of"
	},
	{
		"chunk_id" : "00004608",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to abandon characterizations of\ndarkness and light in these\nperiods, but I rather like them,\nbecause (first) they tend to take\non the perspective of the\npopulous, and (second) the light\nperiods almost always seem to\ncorrespond with times when\npeople were reading. That's\ninteresting, right? When people\nare reading more, we tend to\nhave more control over our\ncircumstances.\nAnd that is to say that there is a\nchoice involved in these affairs.\nWe can choose to learn new\nskills and ideas, to make"
	},
	{
		"chunk_id" : "00004609",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "skills and ideas, to make\nourselves better. We can study\nand practice and grow, taking\nfate into our own hands. The\nperiod that follows is a time\ncharacterized by that very ideal.\nThings were never going to get\nbetter if we sat around waiting\nfor change. In this time, regular\npeople chose to make change\nwithin in themselves. And that, it\nturns out, was a very good place\nto start.\nEur eka!\nCried Archimedes, realizing that the sudden, unexpected solution to his problem"
	},
	{
		"chunk_id" : "00004610",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "was, in fact, himself. In the Renaissance, we followed suit. We also bathed more.\nBack to the Future\nIn the Dark Ages, power structures like feudalism ensured that\nthe rich and powerful stayed that way. The middle classes were\nall but squeezed out of the equation; and between the church,\nthe law, and the culture of skilled trade, it seemed very difficult to\ntake them back. I'll note here that a robust middle class is, to\nhistorians, a marker of a strong, lasting society. People longed to"
	},
	{
		"chunk_id" : "00004611",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "improve their station in society, but few avenues were available.\nSome people, however, began to read and study the past. And it\nwasn't long until we started looking back to antiquity as a better\nmodel for society.\nBefore long, people stopped accepting the notion that they were\nbound to the conditions they were born into. Instead of gaining\napprenticeship to learn a trade, people sought knowledge from\nbooks, and began to figure out how to attain a better life through"
	},
	{
		"chunk_id" : "00004612",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "hard work and reasoning. We started to see the Golden Age as\nthe height of civilization, and took it upon ourselves to bring it\nback. We call this period (1300-1700 CE) the Renaissance, a\nrebirth of Greek and Roman culture which sought to restore the\nmiddle classes, bring beauty into the lives of citizens, and raise\nsociety to its highest standard of living. [1]\nBut this wasn't going to be handed to anyone. To accomplish a\ntrue revival, people needed to take it upon themselves. Scholars"
	},
	{
		"chunk_id" : "00004613",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "looked to Greek philosophy, and developed a new way of\nthinking, which they called humanism.[ 2 ] This school of thought\nhad several core ideas: to solve problems with logic and reason,\nto challenge authority, and to think for ourselves. These simple\ntenets would resonate throughout the Renaissance. When\napplied to any art or trade, they put people in the driver's seat,\nDante, author of the Divine Comedy.\nenabling them to improve their lives on their own terms."
	},
	{
		"chunk_id" : "00004614",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Florence, Italy. Birthplace of the Renaissance.\nWhere we're going, we don't need roads.\nW herefore Art Thou?\nHere's the pattern we're looking for in the and architects were starting to develop scientific\nRenaissance: people improving their own lives principles to improve the quality of their works.[ 3]\n(and the lives of others by extension) through the They looked not only to restore the masterpieces\nuse of Renaissance Thinking: when we solve our of ancient Greece and Rome, but to accomplish"
	},
	{
		"chunk_id" : "00004615",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "problems with logic and reason, when we new feats as well. When applied, Renaissance\nchallenge authority, and when we think for Thinking enabled them to raise their crafts to a\nourselves.[ 2 ] In a time still ruled by the authority of higher level.\nthe king and the scrutiny of the church, this was a\nOthers would follow. Scientists began to apply new\npotentially dangerous game. In certain\nmethods of discovery to uncover the true nature\ncircumstances, a challenge to the old ways could"
	},
	{
		"chunk_id" : "00004616",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "circumstances, a challenge to the old ways could\nof the universe. Engineers would design\nlead to dire straits. So, how were people able to\nmonuments and plan urban centers. And\nrestore a high quality of life with the Eye of Sauron\ncomposers, writers, and philosophers would make\n(just a metaphor, not a real thing) always looking\nthese new ideals accessible to everyone. The keys\ninto the library window?\nto this revolution were anyone's to take. And every"
	},
	{
		"chunk_id" : "00004617",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Very sneakily, it turns out. The first Renaissance time someone did, things got better for all of us.\nthinkers were artists, and most of their Art + Architecture\ncommissions were from nobility and the church.\nThey applied the ideals of humanism to their work Okay, let's get something out of the way. Yes, each\nand its quality improved rapidly. But the subject of the Ninja Turtles are named after Renaissance\nmatter didn't change. As far as the nobles and artists. Curiously enough, Eastman and Laird (who"
	},
	{
		"chunk_id" : "00004618",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "church officials were concerned, they were just created the turtles) did a decent job of curating a\ngetting better art for their money. What they didn't few important figures of the era and keeping them\nrealize was why the art was getting better. Artists culturally relevant (in their own odd little way). I do\nwish we had space to talk about each and every\nRenaissance turtle- or, I mean, artist. Alas, we're\nhunting big patterns on this safari. Let's get back on"
	},
	{
		"chunk_id" : "00004619",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "track. Painting, especially oil on canvas, was starting\nto become popular during the Middle Ages as nobles\nand church officials were commissioning artists to\ncreate portraits and religious images. For the most\npart, it wasn't very good. The proportions were all\noff. Compositions lacked balance. Foregrounds and\nbackgrounds were confusing. Sadly, apprentices\nwere learning the trade from masters, and so there\nwas little room for innovation.\nUntil, that is, artists like Filippo Brunelleschi started"
	},
	{
		"chunk_id" : "00004620",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to apply Renaissance Thinking to the craft. Filippo\nRaphael's School of Athens\nBrunelleschi started to apply principles of math and\nscience to the page, beginning with the development\nof perspective, a series of principles and techniques\nthat enable the representation of three-dimensional\nimages on two-dimensional (flat) surfaces. Applying\nthese ideas made it possible for painters to create\nrealistic images that could accurately portray the\nworld as well as give vivid life to images we could"
	},
	{
		"chunk_id" : "00004621",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "only imagine.[ 4 ] Masters like Michaelangelo and\nRaphael would apply Brunelleschi's techniques to\nproduce some of the most beloved works of all time.\nBrunelleschi's influence became paramount as his\nprolific work brought new life to the heart of the\nRenaissance: Florence, Italy. Perhaps his most well\nknown contribution was his architectural design for\nthe Duomo, the Cathedral of St. Maria del Fiore.[ 5]\nThe Duomo in Florence\nAside from its beauty, Brunelleschi's dome was of"
	},
	{
		"chunk_id" : "00004622",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Aside from its beauty, Brunelleschi's dome was of\ngreat significance to the philosophy of the\nRenaissance. This was the first Roman-style dome\nbuilt in a thousand years. It was a literal harbinger of\nrebirth of the classical world. A reminder to all of us\nthat we could bring back the Golden Age. If I could\nname a fifth turtle, it would be Brunelleschi (and his\nshell would be done in a red brick fresco).\nSculpture in the Renaissance not only brought back\nthe Greek ideal, it made improvements. Have you"
	},
	{
		"chunk_id" : "00004623",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the Greek ideal, it made improvements. Have you\nnoticed a missing-limb problem in some of the great\nfigures of Greece like the Venus de Milo? The Greeks\nhad some trouble creating evocative posture while\nmaintaining enough structural integrity to keep all\nthe pieces attached. Renaissance masters like\nMichaelangelo and Donatello sought to master the\nAn angel in perfect contropposto\nGreek idea of contrapposto, the off-balance pose.[ 6]\nThis could provide a sense of\nreality, of fluid movement; a"
	},
	{
		"chunk_id" : "00004624",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "reality, of fluid movement; a\nmoment between the steps.\nRenaissance sculptors\ndeveloped new techniques and\nused new materials to uphold\nthe Greek ideal without\ncompromising structural\nstrength.\nI would be remiss to do a\nchapter on the Renaissance\nwithout mentioning Artemisia\nGentileschi, the first female\nartist admitted to the Florence\nSchool of Art. Artemisia (along\nwith Caravaggio and later\nRembrandt) was a master of\nChiaroscuro, a technique that\ncould create vivid, realistic fields"
	},
	{
		"chunk_id" : "00004625",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "could create vivid, realistic fields\nof light by contrasting them\nagainst stark, black, darkness. [7]\nHer work not only reached new\ntechnical heights, but\nphilosophical ones as well. She\ndepicted women in positions of\npower as well as the other-\nworldly violence of Greek\nmythology. Artemisia is a perfect\nexample of a Renaissance\nthinker: she challenged the old\nways while carving a new place\nin the world for herself.\nScience\nSoon, other fields would borrow\nthe ideals of the Renaissance,"
	},
	{
		"chunk_id" : "00004626",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the ideals of the Renaissance,\ncreating new spaces in society\nfor critical thinking along the\nway. In the Middle Ages, science Artemisia's masterwork,\nwasn't its own distinct pursuit, Judith and her Maidservant\nbut was rolled into other fields\nwith the Head of Holofernes,\nlike engineering and design. As\ncontains a little of everything\nRenaissance thinkers started to\nbreak from traditional methods, that typified the Renaissance\nscience found its place in all ideal.\nwalks of life. Oddly enough, it"
	},
	{
		"chunk_id" : "00004627",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "walks of life. Oddly enough, it\nwas philosophers who invited\nscience to the party.\nAstronomical Clock. Prague, 1410.\nNew interest in Greek and now with the weight of careful you said that, you'd get in\nRoman culture led to the observation and testing. This trouble with the church.\ndiscovery of philosophical texts process came to be known as Ironically, the church got the\nfrom antiquity. These works, the scientific method, and is idea of geocentrism (an"
	},
	{
		"chunk_id" : "00004628",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "especially those from Aristotle now the cornerstone for Earth-centered universe) from\nand Ptolomy, brought about a establishing our working Aristotle. Copernicus ended up\n[8]\nresurgence of logic, a process understanding of the universe. waiting to publish his ideas until\nfor discerning truth through he was on his death bed.[9]\nIt was a gradual process,\nquestions, argument, and\nhowever, with a lot of wild turns Later scientists, like Kepler and\nreasoning. The idea that a"
	},
	{
		"chunk_id" : "00004629",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "reasoning. The idea that a\nand conflicts along the way. For Galileo would use scientific\nsystematic process (like\nstarters, one of the first scientific observation and testing to prove\nAristotle's syllogism) could be\nobservations that we couldn't it. Galileo used a telescope to\nused to draw conclusions that\nmake heads or tails of was the make more detailed\nwe could accept as truth was\nretrograde motion of Mars (it observations of the stars and\nvery interesting to Renaissance"
	},
	{
		"chunk_id" : "00004630",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "very interesting to Renaissance\nappears to go backward for a planets. These observations\nthinkers; who did what they did\nwhile). An astronomer named could be used to test his\nbest and one-upped it.[8]\nCopernicus thought about it hypotheses about the true\nAristotle's Syllogism used a and came to the conclusion that nature of the solar system.[ 9]\nseries of declarations to draw Mars looks like it's going Galileo's telescope helped him"
	},
	{
		"chunk_id" : "00004631",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "new conclusions based on backwards because the Earth is discover craters on the moon,\naccepted facts. Scientists in the passing it. [9] the phases of Venus, sun spots,\nRenaissance built on this model, and Jupiter's moons. While\nAnd he was right. But\nusing facts and observations to these discoveries helped put the\nunfortunately, he couldn't really\nsystematically test hypotheses. final nails in the geocentric\nsay so, because that would\nLike syllogisms, this could coffin, they also got him thrown"
	},
	{
		"chunk_id" : "00004632",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "mean that the sun was at the\nestablish new knowledge, but in jail. Still, though. Savage.\ncenter of the solar system, and if\nLiterature\nThe Renaissance breathed such a powerful resurgence of new\nlife into society because its ideals applied to everyone.\nRegardless of line of work or socio-economic conditions,\nchallenges to the old ways and the use of human reason led to\npopulations of free-thinking people. And when free-thinking\npeople start reading and writing, we accelerate the"
	},
	{
		"chunk_id" : "00004633",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "transmission of ideas throughout society; equipping the best\nones to once again improve our standard of living.\nThat being said, I have a sticker for Johannes Gutenberg (it says\n\"Good Job!\"\" and has an exorbitant amount of glitter)"
	},
	{
		"chunk_id" : "00004634",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "became experts at needlepoint (unconfirmed, but my point is,\nnow it's possible).\nWriting gave the Renaissance to everybody. Political thinkers\nlike Niccolo Machiavelli reintroduced us to Greek and Roman\ndemocracy while philosophers like Pico della Mirandola wrote\ntreatises on humanism and critical thinking. And speaking of\naccess, nobody was more prolific in the spread of the\nRenaissance than The Bard. William Shakespeare wrote\npoetry, verse, and about 38 plays. His work was highly diffuse"
	},
	{
		"chunk_id" : "00004635",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "throughout Europe, and even those who couldn't read could\nstill engage with his works through the resurgence of live\ntheatre. Shakespeare wove universal themes into complex and\nprovocative narratives that made literature exciting and\navailable to all. He even extended Europe's cultural fascination\nwith classical antiquity, dramatizing the histories of Pericles,\nShakespeare's Globe Theatre Antony and Cleopatra, and Julius Caesar. [10]\nA Gutenberg-style block press.\nOne Last Thing\nThe reason the Renaissance"
	},
	{
		"chunk_id" : "00004636",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "One Last Thing\nThe reason the Renaissance\nspread so rapidly and became\nso diffuse throughout European\nsociety is that it was a cultural\nmovement: one brought upon\nby the people themselves and\nwhose principles and products\nhad a direct benefit to society.\nWhen Brunelleschi applied logic\nand reason to his art practice,\nthe benefits applied to himself\nand the rest of us. When Galileo\nchallenged authority, his risks\nwould improve our lives. Like\nthe Golden Age, the Renaissance\nbrought a high quality of life to"
	},
	{
		"chunk_id" : "00004637",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "brought a high quality of life to\naverage people. But unlike the\nfirst run, this one put the agency\nfor change in the hands of the\npeople themselves.\nThe benefits of Renaissance\nthinking were equally effective in\nall applications. Whether applied\nto art, engineering, music, or\nliterature, human reason,\nchallenges to authority, and free\nthinking made the work better.\nDuring the late Renaissance, this\nschool of thought would seep\neven further into our culture\nand society. By the 1500's,"
	},
	{
		"chunk_id" : "00004638",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "and society. By the 1500's,\nRenaissance ideals would even\nbe applied to the church, leading\nMartin Luther to quell\ncorruption and eventually\ndevelop new sects of religious\npractice in a movement known\nas the Protestant Reformation.\nAnything the Renaissance\ntouched improved the lives of\ncitizens and brought society to a\nhigher level. By the 1700's, there\nwere only a few corners of\nsociety that the Renaissance had\nyet to influence. But don't fret.\nNext chapter, we eat cake."
	},
	{
		"chunk_id" : "00004639",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 6: Enlightenment\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nA Castle M ade of\nStraw\nIn the last chapter, we left off in\na pretty hopeful place, and for"
	},
	{
		"chunk_id" : "00004640",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "a pretty hopeful place, and for\ngood reason. The Renaissance\n(1300-1700 CE), a cultural\nmovement of revival and free\nthinking brought upon a new\nGolden Age for many European\npeople. It's a heartening chapter\nin history because people\nbrought light to a darkened\nworld on their own accord, by\nchanging their way of thinking\nand striving to bring their work\nto a higher standard.\nThe tenets of Renaissance\nthinking were simple but\nrevolutionary. As people began\nto solve problems with logic and"
	},
	{
		"chunk_id" : "00004641",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to solve problems with logic and\nreason, challenge authority, and\nthink for themselves, they\nopened the door for countless\nbreakthroughs; pushing art and\nscience and philosophy to new\nheights while carving a path for\na revitalized middle class. What\nstarted with artists soon bled\ninto other fields, revitalizing and\nadvancing everything it touched.\nSoon, Renaissance thinking\nwould bring light to nearly every\naspect of society.\nNearly. But there was one place\nwhere the Renaissance wasn't"
	},
	{
		"chunk_id" : "00004642",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "where the Renaissance wasn't\nwelcome; one structure that had\nreason to fear a free-thinking\npopulation that had the power\nto improve the quality of life of\nregular citizens. The last holdout\nwas government. Like crumbling\npillars, kings and nobles did\neverything they could to hold\nonto the power structures that\nsustained their livelihoods.[ 1 ] But\nthe castle was already falling.\nHit the Books\nThe Enlightenment is a period characterized by the power of the written word."
	},
	{
		"chunk_id" : "00004643",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Ideas, bound in iron gall ink, had the capacity to topple kingdoms.\nPainting the Roses Red\nIf I've been properly foreshadowing, you might guess that this\nnext period is called the Age of Revolution, and yes, the story is\nheading that direction. But there's more going on here than an\nangry populous rushing towards the aristocracy with gardening\ntools (I've said too much). This period (1700-1850) is often termed\nthe Age of Enlightenment as it is characterized by a wave of new"
	},
	{
		"chunk_id" : "00004644",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "philosophical ideals that saw reason as the primary mechanism\nof truth and promoted the values of human liberty, progress, and\nconstitutional government in society. [1]\nIn many ways, this sounds like the last pieces of the Renaissance\nfalling into place, and a lot of historians see no need for a\ndistinction. The ideals of the Enlightenment aren't really any\ndifferent than those of the Renaissance, but should rather be\nseen as a continuation of this way of thinking. The Renaissance"
	},
	{
		"chunk_id" : "00004645",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "was about bettering your community by bettering yourself, and\nso it applied to anything and everything the people controlled;\nwhich was a lot. As art and architecture brought beauty to our\ncities, scientific thought and religious revolution brought freedom\nto our minds. Anything within the people's grasp was fair game\nfor revitalization and rebirth.\nBut the government wasn't in the people's grasp. In fact, it was\nspecifically designed to ensure that existing power structures"
	},
	{
		"chunk_id" : "00004646",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "stayed strong by maintaining wealth inequality through tithe and\ntribute. Yes, feudalism was still happening, even through the\nRenaissance.[ 2 ] So, that's why we're switching to a new term for\nthis era: because when people apply Renaissance thinking to the\ngovernment, something happens. Suddenly, democratic ideals\nIvy grows on the Palladium window of a\nare in vogue. People start to demand a government that honors a\nuniversity library.\nsocial contract, protects their human rights, and ensures liberty"
	},
	{
		"chunk_id" : "00004647",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "and equality for all. And now, we're willing to fight for it.\nTown halls and government buildings saw new\narchitectural relevance in this era.\nRadcliffe Science Library at Oxford University\nCan I Get That In W riting?\nSo, the big pattern we're looking at here is that (thanks again, Gutenberg!) means that the new\nwhen you apply Renaissance ideals to the ideas of writers and philosophers are going to be\ngovernment, you always end up with democratic incredibly diffuse in the culture. And so, the"
	},
	{
		"chunk_id" : "00004648",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "principles. That makes sense, right? People value written word became the reverberating heartbeat\nliberty and independence, and are indeed capable of revolution.[3]\nof governing themselves. I'm sure you also\nLet's try to focus on that notion as we discuss the\ngathered through my not-so-subtle use of\n(literal and philosophical) revolutions in England,\nChekhov's gun (or in this case, Chekhov's pitchfork),\nFrance, and America that define this era. New\nthat if the people wanted a modern government,"
	},
	{
		"chunk_id" : "00004649",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "that if the people wanted a modern government,\npolitical ideologies will be spread through\nthey were going to have to fight for it. Turns out,\nmanifestos that capture the hearts of the citizens.\nkings and nobles weren't willing to just hand over\nBooks spreading the seeds of revolution will\ntheir kingdoms because the people wanted\ninspire people to fight for liberty and equality. And\ndemocracy. So yes, this is going to end in violence[.3]\npolitical documents themselves will strip power"
	},
	{
		"chunk_id" : "00004650",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "political documents themselves will strip power\nBut there's something else going on here as well: from the hands of centuries-old bloodlines.\nanother pattern seeping into the mix. And that is a\nEngland\nnew cultural value for reading and writing; which\nwill ultimately have as much influence on these At the tail end of the Renaissance, English political\noutcomes as the bloodied rakes of revolution. By philosophers Thomas Hobbes and John Locke"
	},
	{
		"chunk_id" : "00004651",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the 18th century, average people are reading and began theorizing about ideals that belonged in\nwriting (yes, we can thank the Renaissance for government. While the two disagreed on a number\nthat). And what's more, it's a normal part of our of items, they both relished the idea that a\nlives, regardless of socio-economic conditions. government's power should be maintained by a\nThat, coupled with the ready availability of books contract rather than a person.[ 4 ] A contract is an"
	},
	{
		"chunk_id" : "00004652",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "agreement signed by two parties, usually outlining\nthe services and/or payment that each party should\ncommit to the other. The very special thing about\ncontracts is that they also include some kind of\nconsequence (called teeth) that each party must face\nif they are unable to uphold their end of the bargain.\nHobbes and Locke believed that this was needed\nbetween a government and its people. This idea is\ncalled a social contract, and would ensure that the\npeople followed the rules and paid their taxes while"
	},
	{
		"chunk_id" : "00004653",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "the government protected us against abuse and\nguaranteed us certain rights.[ 4]\nWith the right teeth, this type of document could\nboth establish the government and act as the force\nLibraries increased access to new ideas\nthat holds all parties accountable to the same set of\nagreements. When a contract does that: we have a\nspecial name for it: a constitution. Sounds like a\npretty solid idea, right? Especially for people trapped\nin medieval systems of government. Well, it caught"
	},
	{
		"chunk_id" : "00004654",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "on, becoming the standard ideal of the era. Oddly,\nenough, the English already had one in place.\nWay back in 1215, a charter called the Magna Carta\nwas signed between King John and a group of rebel\nbarons that established certain protections for\ncitizens as well as limitations of royal power.[ 5 ] In the\nEnlightenment, political leaders realized that such\nconstitutional documents could be more powerful\nthan any judge or legislative body or king, simply\nbecause they established agreements between the"
	},
	{
		"chunk_id" : "00004655",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "because they established agreements between the\ngovernment and the governed (and built-in rules for\nKing John , thrilled to sign the Magna Carta\nwhat to do if either party breaks the agreement).\nSo there's no English Revolution, but there were a\nseries of sneaky, sneaky political actions that took\nplace in the late 1600's that would ultimately place\npower into the hands of contracts rather than\nofficials. In 1688, members of English Parliament\n(the body of officials that makes the laws) became"
	},
	{
		"chunk_id" : "00004656",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "very frustrated with the rule of King James II, who\nattempted to strip them of their power by trying to\nhave the courts repeal the Acts of Parliament (all\nover a religious dispute, of course). Several members\nof parliament got in contact with a Dutch Prince,\nWilliam of Orange, and started organizing a\ntransfer of power by force.[ 6]\nLet's be explicit here, this was English parliament\nWestminster is home to English Parliament inviting a foreign leader to take over England. William"
	},
	{
		"chunk_id" : "00004657",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "started moving his armies into position to fight alongside regular\nEnglish citizens, whom the Parliament promised would join in the\neffort. But they didn't need to. After a couple of very minor\nskirmishes, James got spooked and fled. Because it was a\nrelatively bloodless transfer of power, the British refer to this as\nthe Glorious Revolution.[6]\nIt was decided that William and his wife Mary would take the\nthrone. Now here's the real kicker: in order for William and Mary"
	},
	{
		"chunk_id" : "00004658",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to become the new regents, Parliament had to draft a document\nfor them to sign to transfer the power. What they drafted was a\nconstitutional document called the English Bill of Rights, which\nlimited the power of the king and protected certain rights of\ncitizens. It even gave parliament the right to remove a king if they\ndeemed it necessary. Signing would mean that the government's\npower would now be held by the people, who elected the\nparliament.[ 6 ] Needless to say, they signed.\nAmerica"
	},
	{
		"chunk_id" : "00004659",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "America\nWell, that was all well-and-good for citizens of England, but\nunfortunately, those living in British colonies were subject to the\nrule of English law (not to mention taxes) without the benefits of\nrepresentation. By 1732, thirteen such colonies had been\nestablished on the American continent. Americans were starting\nto feel like a people without a country: obligated to support the\ncrown but unable to contribute to the terms of their governance."
	},
	{
		"chunk_id" : "00004660",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Unfortunately for them, these colonies were each separate\nentities, and no single colony had enough influence to sway the\nsituation (my foreshadowing is on point today).\nIn the following years, a series of escalating provocations\nbetween colonists and the British government would lead to a\nstate of political unrest. As the British continued to issue laws and\nIndependence Hall, Philadelphia\nproclamations that made life harder for people in the colonies,\nYale University, Connecticut\nSkyline in Paris, France"
	},
	{
		"chunk_id" : "00004661",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Skyline in Paris, France\nAmericans would stage protests urging people to consider taking a representative democracy,\nand spread dissent throughout arms against the British. The which would ultimately establish\nthe territory. A tax on tea, for book, titled Common Sense, this form of government as a\nexample, led to the destruction contained advanced ideas about benchmark for societies seeking\nof 342 chests of British tea, government, further promoting to limit the powers of their"
	},
	{
		"chunk_id" : "00004662",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "which was dumped in the the notion of a social contract governments while protecting\nBoston Harbor. [7] throughout the colonies.[ 8] the rights of the citizens.\nPerhaps the most important\nThe situation continued to France\nfacet about Common Sense is\nescalate throughout the 1770's,\nthat it was written in plain With the American Revolution\nleading Americans to form a\nlanguage; a fact that made fresh in the minds of French\nContinental Congress of\nrevolutionary ideals accessible citizens, many people began to"
	},
	{
		"chunk_id" : "00004663",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "representatives from each\nto regular citizens, who became question the authority that\ncolony in order to develop a\ninspired to join the cause. maintained medieval power and\nmore united response. The\nwealth distributions in a place so\nnotion of declaring Fast forward to the American\nrenowned for progressive\nindependence remained Revolution and we find more\nthinking. And I do mean\nunpopular for most of this time, change being facilitated by\nmedieval. The class structure in"
	},
	{
		"chunk_id" : "00004664",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "medieval. The class structure in\nbecause it certainly meant a war constitutional documents. The\nFrance at the time, known as the\nagainst the world's most Declaration of Independence\nEstate System, was established\npowerful empire. [7] of 1776 established our status\nin the early middle ages, and\nas a country (although the bulk\nBut a book would change that divided people into three\nof the document contains insults\nsentiment. In 1775, an activist classes. Nobles and high church\nto the king), and at the end of"
	},
	{
		"chunk_id" : "00004665",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to the king), and at the end of\nnamed Thomas Paine officials (about 2% of the\nthe war (spoiler alert: we won)\npublished a short book outlining population) made up the first\nAmerica wrote the first modern\nthe unfair political situation and second estates, while\nconstitution. Naturally, we chose [2]\nfaced by the colonies, and everyone else was in the third.\nThis included artisans, doctors,\nlawyers, as well as laborers\nnormally associated with the\nworking classes of society.\nFrance allied with American"
	},
	{
		"chunk_id" : "00004666",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "France allied with American\nduring its revolution, which left\nthe French in a great deal of\ndebt. That, coupled with some\nunfortunately-timed drought,\nled to major food shortages for\nthe third estate (the aristocracy\nremained well-fed while most of\nthe population was starving).\nResentment grew, but no clear\nmovement solicited action.\nUntil, once again, a book would\nstoke the fires of revolution. In\n1789, Emanuel Sieys published\na scathing treatise about the\nclass system entitled What is"
	},
	{
		"chunk_id" : "00004667",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "class system entitled What is\nthe Third Estate? The book\nillustrated abuses of power in\nthe upper classes, and\nsuggested that the third estate\nwas a complete nation in and of\nitself, and had no need for the\n\"dead weight\"\" of the privileged"
	},
	{
		"chunk_id" : "00004668",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "references to waging war with\ngardening tools) and would Universities and Libraries like\nquickly impel King Louis XVI to the French Mazarine became\nflee. But France would remain in\nan new architectural focal\na state of political unrest for the\npoint, representing shifting\nnext decade as leaders worked\nto establish stability. Like the cultural values in the Age of\nAmericans, the French turned to Enlightenment.\na representative democracy, and\nwould utilize a constitution to\niron it into place.[ 10]"
	},
	{
		"chunk_id" : "00004669",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "iron it into place.[ 10]\nLibrary of Congress, Washington, DC.\nOne Last Thing\nDemocracy would soon become the standard form revolutionary change, and contracts into the pillars\nof modern government, and constitutions were that would hold the weight of nations. Suddenly,\nseen as the most reliable method of establishing knowledge was a cultural value; and moreover,\nfair, stable structures of power. It makes sense one that everyone had equal access to. In the"
	},
	{
		"chunk_id" : "00004670",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "that when we started to apply Renaissance Enlightenment, the acquisition of knowledge could\nthinking to government, we ended up borrowing be a tool to improve your station in life.\nfrom the Romans. What's interesting is where the Universities and libraries opened, paving new\nreal power ended up: in the written word. avenues for success in society. It brought upon a\ntime when pens were mightier than swords, and\nThe Enlightenment turned books into the tools of\neverybody could afford a pen."
	},
	{
		"chunk_id" : "00004671",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Op en H i st or y\nOpen Source History | Ch. 7: Industrial Age\nLicense + Usage Rights\nAlex Greengaard, MFA\nThis textbook is open. That is to say the information herein (as well as the document itself) is free and\navailable to anyone. You can study it, you can copy it, you can distribute it. You can even print it and sell it if\nyou want. The important part is that it exists, which, well, now it does. Do enjoy.\nFire It Up!\nIn the last chapter, we discussed\nThe Enlightenment (1700-1850),"
	},
	{
		"chunk_id" : "00004672",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "The Enlightenment (1700-1850),\na movement characterized by\nsweeping changes to\ngovernment systems in Western\nCivilization (a weird term to\ndescribe Europe and America,\nwho collectively have a great\ndeal of influence). It's another\nheart-warming slice of history,\nas its changes were brought on\nby a free-thinking populous, who\ndecided that they were smart\nand savvy enough to govern\nthemselves.\nIt took some creative gardening,\nbut people brought major\nreform (or complete rebuilding"
	},
	{
		"chunk_id" : "00004673",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "reform (or complete rebuilding\nfrom the ground-up) to England,\nFrance, and the United States;\nwho suddenly existed after\nwinning their revolution. As\noppressive aristocracies and\nMedieval power structures were\ncrumbling into oblivion, society\nhad already begun to undergo a\nseries of economic changes\nwhich, like the Enlightenment,\nwould chew up the world we\nknew and spit out a new\n(soggier) one.\nWe call the period between 1760\nand 1930 the Industrial Age, as\nits patterns of change focus"
	},
	{
		"chunk_id" : "00004674",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "its patterns of change focus\nchiefly on processes involved in\nmanufacturing goods. While this\nsounds a bit innocuous on the\nsurface, these changes fostered\nmore irrevocable transformation\nto our physical and social\nlandscape than we had\n[1]\nexperienced since the Neolithic.\nThis was the birth of the modern\nworld, and, just like the first\ntime, it all started with a seed.\nCutting Cor ner s\nThe Industrial Age is characterized by changes that made manufacturing"
	},
	{
		"chunk_id" : "00004675",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "processes more efficient. Seems good, until it killed skilled labor.\nThe Gloomy Garden of Eden\nOh, I'm supposed to tell you what changed, and then we can get\nthe the British part. In this era, Western Civilization shifted from a\nlargely rural configuration (small, sparsely populated villages,\nspread out, and driven by an agricultural economy) to a much\nmore urban one (large cities, heavily populated, with a\nmanufactured goods-based economy). Work moved from farms"
	},
	{
		"chunk_id" : "00004676",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "to factories, a number of modern conveniences entered the fray,\nand skilled trade work was replaced by unskilled labor.\nCosmetically, we see a lot more machines, horses are replaced by\ncars, candles are replaced by electric light, cities get bigger and\nbuildings get taller.\nAs you may have guessed from my tone, the changes in labor are\nmore informative than the changes in decor. Economists believe\nthese changes were made possible by a sufficient abundance of"
	},
	{
		"chunk_id" : "00004677",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "The Factors of Production, which are land, capital, and labor.\nThis makes sense to me, as you need a place to do business, raw\nmaterials to make into material goods, and people to actually do\nthe making. By 1850, Britain had all of these in great abundance,\n[2]\nand so that's where the Industrial Revolution began.\nI used to teach out of a textbook that just left it at that. As though\none day the British were like \"You know"
	},
	{
		"chunk_id" : "00004678",
		"source" : "Open_History_Ch_6_Enlightenment.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Production, let's do an Industrial Revolution. But it has to be\nbefore tea.\" As you know by this point"
	},
	{
		"chunk_id" : "00004679",
		"source" : "Open_History_Ch_5_Renaissance.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "toward it is plain and simple greed. Anyone else getting dj vu?\nA newly-industrial skyline in Paris, France.\nChicago was one of the first cities to build skyscrapers.\nSomething, Something, Lightbulb\nOkay, one of the reasons I was terrified to write Thomas Edison didn't invent the light bulb. There\nthis chapter is because this era is especially were twenty two people responsible for its\nsusceptible to \"List History.\"\" That's where"
	},
	{
		"chunk_id" : "00004680",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "of examining a pattern of change, you just list underlying technological development. There were\nstuff. Seriously, look for this in normie textbooks. working light bulbs before Edison was born, and\nThe Industrial Revolution chapters are these long many on display in Robert Houdin's house when\nlists of inventions, and we're supposed to, like, nod Edison was four years old. What Edison did do was\nour heads because we're so impressed by robot figure out a way to make the light bulb cheaper. [3]"
	},
	{
		"chunk_id" : "00004681",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "horses (for clarity, I'm referring to cars here). I find He also took credit for hundreds of other people's\nthis convention so prevalent and so useless. The inventions, electrocuted elephants, and exposed\ntough part is, this was an era of rapid innovation, his assistant (Clarence Dally) to enough radiation\nmuch of which can be strong evidence for a case to give him terminal cancer. All in pursuit of profit.[4]\nabout the nature of this change. So I am going to\nEfficiency & Innovation"
	},
	{
		"chunk_id" : "00004682",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Efficiency & Innovation\nhave to mention a few inventions.\nI thought that was going to be off-topic, but we're\nThe other thing I want to avoid is unchecked praise\nactually right on track. Greed made all this\nfor the inventors because their inventions changed\ninnovation possible. It moved businesses towards\nthe world. Most innovators weren't trying to make\nefficiency, an effort to make the most out of\nlife more convenient or improve people's living\navailable resources. This was meant to reduce a"
	},
	{
		"chunk_id" : "00004683",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "available resources. This was meant to reduce a\nconditions. They were trying to figure out how to\nbusinesses costs (money that must be spent on\nmake more money for a business, primarily\nthe factors of production in order to stay in\nthrough finding unique ways to cut costs. This is a\nbusiness) and thereby maximize profits (the\nperfect segue into my thesis, but first, I need to\nmoney left over when you subtract costs).\ntake a moment to soft cancel Thomas Edison."
	},
	{
		"chunk_id" : "00004684",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "take a moment to soft cancel Thomas Edison.\nHistorically, one of the most popular cost-cutting\nstrategies was slavery, as it reduced labor costs. It\nwas also morally wrong. Terribly, terribly wrong. This\ndid not stop all businesses from using it, however,\nwhich I'm pointing out because it supports my thesis\nabout greed being a factor that drives industry. What\ndid eventually stop them was that slavery was\nbecoming illegal in many countries at the dawn of\nthe Industrial Age. The United States banned it in"
	},
	{
		"chunk_id" : "00004685",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "1865, for example.\nFor the record, I'm not calling cause-and-effect here.\nIndustry had already become prevalent by that\npoint. In fact the North used it to help win the Civil\nWar that ended slavery in the US. But there is a\nCotton fiber requires a lot of cleaning.\npattern. By 1850, businesses are becoming obsessed\nwith cutting costs. Increasingly, they were able to do\nso via inventions that increased efficiency. It worked.\nAnd so, it diffused like wildfire. The process of many"
	},
	{
		"chunk_id" : "00004686",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "businesses and countries rapidly adopting efficient\nnew techniques and practices is called (by historians)\nIndustrialization.\nThe innovations that moved this process along most\nreadily were the ones that could really make a dent\nin one (or more) of the factors of production. I would\nargue that these solicited significantly more change\nin this era than modern conveniences or exciting\nfeats of technology.\nTo support this claim, I'd like to take a second to"
	},
	{
		"chunk_id" : "00004687",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "reflect on the textile (that means clothing) industry.\nThe old process of making clothing required skilled\nAutomated loom.\nlabor, people who took years learning a trade in all\nits complexities.[ 5 ] You had to clean your fiber\n(typically cotton), and then spin it into cloth with a\nloom or spinning wheel. Then each piece of cloth\nneeded to be hand cut and hand sewn into\ngarments. The process could take days from start to\nfinish to produce a single garment.\nHowever, Eli Whitney's Cotton Gin could clean the"
	},
	{
		"chunk_id" : "00004688",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "However, Eli Whitney's Cotton Gin could clean the\nfiber much more efficiently, cutting hours away from\nthe process. James Hargreaves's Spinning Jenny\nwas a multi-spindle spinning wheel, which could\ndouble or tipple the amount of cloth you could\nmake. And Elias Howe's Sewing Machine could put\nthe pieces together much more quickly. Combined,\nthese inventions could significantly reduce the time\n[6]\nSewing machines could stich much faster. it took to make a garment. And time is money.\nThe Bargain Basement"
	},
	{
		"chunk_id" : "00004689",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "The Bargain Basement\nHere's the score so far: in countries rife with resources, there was\na massive influx of innovation that started really cooking in the\n1850's. The inventors were largely driven by a desire to maximize\nprofits (some people call this greed). And the easiest way to do\nthat was to make processes more efficient. As soon as that\nstarted happening, the landscape changed rapidly, from rural\nagrarian societies to urban manufacturing societies."
	},
	{
		"chunk_id" : "00004690",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "My examples from the last section showed innovations that\nmade the textile industry more efficient, mostly by reducing the\namount of time it took to produce a garment. This type of\nthinking was instrumental to a number of industries. However,\nthere are a few innovations from this era that were so useful that\nthey could be applied to any industry. I'd like to point out two of\nthese because I posit that they are responsible for more change\nthan any human idea since we hashtag invented farming."
	},
	{
		"chunk_id" : "00004691",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Eli Whitney gets a lot of credit for being the cotton gin guy. In his\ntime, he was known more for manufacturing muskets. Muskets\nhave a lot of working parts that fit together and each one needed\nto be made by hand. That is, until Whitney had the fabulous idea\nof, well, not doing that. Interchangeable parts are components\nto a larger device that are all made identically, often from a single\ncast. Once Whitney applied this to his muskets, they could be\nmade much more quickly and for a lot less money.[7]"
	},
	{
		"chunk_id" : "00004692",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "It also occurred to him that if the parts were being manufactured\nfrom identical casts, the assembly of a musket wouldn't require\nknowledge of how a musket works. You could just create a guide\nfor a worker to follow (think Legos), and they could assemble it,\neasy squeezy, lemon peasy. Think on that for a second. With this\nmethod, the worker doesn't need to know how a musket works.\nThis era changed the nature of labor.\nSpecialized knowledge is expensive. Without it, labor is cheap."
	},
	{
		"chunk_id" : "00004693",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "The factory system was highly efficient but\noffered tough working conditions.\nNatural History Museum, London.\nMany manufacturers borrowed The Briar Patch decided the value of that labor\nthis idea, including Henry Ford power; and most of them\nPerfecting the process of\nin his auto plants. Ford was one decided that it was pretty low\nmanufacturing wasn't the only\nof the more efficiency-obsessed for unskilled work (which was all\ndriving factor in the move to the"
	},
	{
		"chunk_id" : "00004694",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "driving factor in the move to the\nof the Industrial Age innovators, they were offering). This is, of\nfactory system. Workers,\nand would often visit other course, a problem. Ideally, a\nstanding on an assembly line,\nfactories to get ideas. He got worker could simply quit and try\nfitting a single interchangeable\none of his biggest ideas while a business with a more\npart into a device as it moved\nwatching workers take competitive wage. But, as\nalong a conveyor belt, required"
	},
	{
		"chunk_id" : "00004695",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "along a conveyor belt, required\nsomething apart, rather than industry grew, more factories\nlittle to no skill or training. This\nput it together. were adopting the Whitney/Ford\nmeant that they were paid very\nmethod, and were thereby\nAt a butcher facility he had little.\noffering equally low wages. [9]\nvisited, workers were lined up,\nMany more workers were\neach removing a different part This paradigm unfortunately\nneeded for the new factory\nof an animal as it passed by on a encourages wealth inequality"
	},
	{
		"chunk_id" : "00004696",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "system, so job availability was\nconveyor belt. Ford envisioned and actively prevents the worker\nvery high, especially in urban\nthis in reverse, and started from attaining social mobility\ncenters. But the listings were for\nimplementing assembly lines (the ability to move up in\nunskilled labor, which came\ninto the workflow of his society). This is not unique in\nwith low wages and often long\nfactories. This, combined with history, and many such schemes\nhours and poor conditions.[ 5]"
	},
	{
		"chunk_id" : "00004697",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "hours and poor conditions.[ 5]\ninterchangeable parts, could cut have cropped up over time.\ncosts process time to When a worker takes a job, they Feudalism, slavery,\nunprecedented levels, enabling are selling their labor power to sharecropping, and now the\nbusinesses to make staggering the business, which is their factory system have all had the\nnew profit margins. This was the ability and willingness to work. effect of making sure the rich"
	},
	{
		"chunk_id" : "00004698",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "start of a massive shift.[8] At this time, the businesses stay rich and the poor stay poor.\nOne Last Thing\nFor real, this time. It's the last\nchapter. And that's because\nwe're still in this particular\ncritical period. These patterns of\nbehavior (business innovation\ndriven by profit maximization,\nrampant wealth inequality) are\nstill the most prevalent driving\nforces of society. I'm sure we'd\nlike to think that the internet\nand space travel are spicy\nenough to call this a new era."
	},
	{
		"chunk_id" : "00004699",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "enough to call this a new era.\nBut, these are symptoms of the\nsame forces that were in the\ndriver's seat at the dawn of\nindustrialization.\nIt's not necessarily a bad thing. I\nrealize I ended the last section\nwith some gloomy discourse,\nbut, of all the systems meant to\nkeep the poor from having\npower, this one is actually the\nweakest. By 1900, many workers\nrealized that if they could stick\ntogether and form a single\nunified voice, their collective\npower would be enough to\nfacilitate change."
	},
	{
		"chunk_id" : "00004700",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "power would be enough to\nfacilitate change.\nWorkers began forming unions,\ngroups that banded together\nand demanded fair wages and\nconditions. If they didn't get\nthem, they would go on strike,\nand stop working altogether. It\nworked. And it still works. This is\nbecause, collectively, labor\npower is the most valuable\nStrikes, boycotts, and protests\ncommodity. This movement\nremain the most powerful tools\ntoday is known as labor reform,\nan effort to improve conditions regular people have for standing"
	},
	{
		"chunk_id" : "00004701",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "SAT Test Dates 2025-2026\nTest Date Registration Deadline\nAugust 23, 2025 August 8, 2025\nSeptember 13, 2025 August 29, 2025\nOctober 4, 2025 September 19, 2025\nNovember 8, 2025 October 24, 2025\nDecember 6, 2025 November 21, 2025\nMarch 14, 2026 February 27, 2026\nMay 2, 2026 April 17, 2026\nJune 6, 2026 May 22, 2026\n*Fee waivers are available to students that qualify-contact the Guidance office\nTo register visit: https://satsuite.collegeboard.org/"
	},
	{
		"chunk_id" : "00004702",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nTelling a Story With\nGraphs\nNo Tags\nAlignments to Content Standards: F-IF.B.4\nStudent View\nTask\nEach of the following graphs tells a story about some aspect of the\nweather: temperature (in degrees Fahrenheit), solar radiation (in watts\nper square meters), and cumulative rainfall (in inches)) measured by\nsensors in Santa Rosa, CA in February 2012. Note that the vertical"
	},
	{
		"chunk_id" : "00004703",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "gridlines represent the start of the day whose date is given.\na. Give a verbal description of the function represented in each graph.\nWhat does each function tell you about the weather in Santa Rosa?\nb. Tell a more detailed story using information across several graphs.\nWhat are the connections between the graphs?\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nIM Commentary\nIn this task students are given graphs of quantities related to weather."
	},
	{
		"chunk_id" : "00004704",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "The purpose of the task is to show that graphs are more than a\ncollection of coordinate points, that they can tell a story about the\nvariables that are involved and together they can paint a very complete\npicture of a situation, in this case the weather. Features in one graph,\nlike maximum and minimum points correspond to features in another\ngraph, for example on a rainy day the solar radiation is very low and the\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>"
	},
	{
		"chunk_id" : "00004705",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "cumulative rainfall graph is increasing with a large slope.\nSome of the quantities shown are very familiar to students, such as\ntemperature, where others might be less familiar, such as solar\nradiation. We can think about this quantity as the maximum amount of\npower that a solar panel can absorb. Depending on the experience of\nthe students, teachers might want to discuss the idea of cumulative\nrainfall, i.e. the total amount of rain that has fallen since the beginning\nof the season."
	},
	{
		"chunk_id" : "00004706",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "of the season.\nAll the presented data come from the website of the California\nDepartment of Water Resources and can be found at\nhttp://cdec.water.ca.gov/\nTechnically, the graphs only show some of the values of the functions\nthey are meant to represent. A bivariate data plot is a representation of\na function in the same way that a table is a representation of a function;\nwhile it has some gaps in information, there is an underlying function"
	},
	{
		"chunk_id" : "00004707",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "that the bivarate data plot is assumed to sample. (In this case, the data\npoints are joined by lines which means we are interpolating between\nour given values.) So the tasks implicitly expect students to answer the\nquestion about the temperature (or solar radiation, or precipitation)\nthat is a function of time based on the information about it provided by\nthe sampled data. Given the qualitative nature of the tasks, this does\nnot present a problem.\nSolution\na."
	},
	{
		"chunk_id" : "00004708",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "not present a problem.\nSolution\na.\nAll graphs show functions that have the same independent variable,\nt\nnamely the time , measured in days. All graphs have different domains,\nbut they do overlap. They all show domains for different time periods in\nFebruary of 2012. The independent variables are different in each\ngraph. All graphs show a different weather feature in Santa Rosa, CA.\nT\nThe first graph shows temperature, , in degrees Fahrenheit, as a"
	},
	{
		"chunk_id" : "00004709",
		"source" : "Open_History_Ch_7_Industrial_Age.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "history",
		"content" : "function of time (by date and time) over a one-week period starting at\nmidnight on February 6, 2012. On five days the temperature rose into\nthe high 50s to low 60s during the day and fell to the high 40s to low\n50s during the night. The maximum temperature during the given time\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nperiod was 69 deg. F and it occurred in the early afternoon of February\n9. The minimum temperature was 37 degrees F and it occurred in the"
	},
	{
		"chunk_id" : "00004710",
		"source" : "SAT_Test_Dates_2025.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "standardized_tests",
		"content" : "early morning of February 12. February 7 and 10 were special insofar\nthat the temperature did not change much throughout the entire day.\nEspecially on February 7 the temperature stayed in the low 50s all day\nlong.\nThe second graph shows solar radiation in watts per square meter, as a\nfunction of time for 10 days starting on February 6, 2012. We can think\nof solar radiation as the power that a square meter of solar panel\nproduces. This function shows some definite regularity. Every day the"
	},
	{
		"chunk_id" : "00004711",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "function values are zero for a certain time interval. This corresponds to\nthe hours when it is dark and a solar panel would not produce any\npower. The function increases in the morning, reaches a peak in the\nmiddle of the day and decreases in the evening. On most days the\nmaximum value is between 550 and 650 watts per square meter. Again,\nFebruary 7 and February 10 are the exception. During those two days\nthe maximum solar radiation was just over 50 and just below 250 watts\nper square meter, respectively."
	},
	{
		"chunk_id" : "00004712",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "per square meter, respectively.\nThe third graph shows the cumulative amount of rainfall in inches as a\nfunction of time; this is the total amount of rain that has fallen since the\nseason started. With the information given, we dont really know\nwhen the beginning of the season was. This function is increasing on\nthe entire time interval shown (February 1 through February 17, 2012),\nwhich makes sense, since we are keeping track of the total amount of"
	},
	{
		"chunk_id" : "00004713",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "rainfall. We can see that the function is increasing slowly from February\n1 until February 7 and then the graph becomes much steeper. The\ncumulative amount of rain increased much more on February 7 than on\nany other day.\nb.\nAfter analyzing all the graphs, it becomes clear what the weather was\nlike in early February of 2012 in Santa Rosa. Most days it was sunny\nwith temperatures reaching the mid 60 during the day and the mid 40\nduring the night. On February 7 it rained, but not very hard. We see that"
	},
	{
		"chunk_id" : "00004714",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "the cumulative rain graph is steeper during that time, but it only\nincreases by 0.2 inches, so the rain cant have been very heavy. Also,\nsince the solar radiation numbers were very low, this shows that there\nwas not much sunshine during the day, which we would expect for a\nrainy day.\nOn FebrEungaargye 1 y0o uitr wstausd ecnlotsu wdiyth a enfdfe cctoivoel edris dtaunrcien gle athrnei ndga rye sbouutr cneos.t ACCESS RESOURCES>>\nespecially rainy. A cooler air system moved into the area after February"
	},
	{
		"chunk_id" : "00004715",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "10 since daytime temperatures reach highs in the low 60s to high 50s\nand the nighttime low temperatures even drop into the 30s. We\ncant say anything about the temperatures after February 13, but\nthe solar radiation and rain graphs suggest continuing sunny days.\nThere must have been a little bit of rain after February 7 as the\ncumulative rainfall continues to increase slightly, but it wasn't very\nmuch and seems to have been spread out over a number of days which"
	},
	{
		"chunk_id" : "00004716",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "is consistent with the information about solar radiation and\ntemperature on those days.\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a\nCreative Commons Attribution-NonCommercial-\nShareAlike 4.0 International License."
	},
	{
		"chunk_id" : "00004717",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "Engage your students with effective distance learning resources. ACCESS RESOURCES>>\nTemperature\nConversions\nNo Tags\nAlignments to Content Standards: F-BF.A.1.c F-BF.B.4.a\nStudent View\nTask\nf\nLet be the function that assigns to a temperature in degrees Celsius\ng\nits equivalent in degrees Fahrenheit. Let be the function that assigns\nto a temperature in degrees Kelvin its equivalent in degrees Celsius.\nx f(g(x))\na. Explain what and represent in terms of temperatures, or"
	},
	{
		"chunk_id" : "00004718",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "explain why there is no reasonable representation.\nx g(f(x))\nb. Explain what and represent in terms of temperatures, or\nexplain why there is no reasonable representation.\nf(x) = 9x+32 g(x) = x273\nc. Given that and , find an expression\n5\nf(g(x))\nfor .\nh\nd. Find an expression for the function which assigns to a temperature\nin degrees Fahrenheit its equivalent in degrees Kelvin.\nIM Commentary\nUnit conversion problems provide a rich source of examples both for"
	},
	{
		"chunk_id" : "00004719",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "composition of functions (when several successive conversions are\nrequired) and inverses (units can always be converted in either of two\ng(x)\ndirections). Note that the conversion function is an approximation:\ng(x) = x273.15\nThe exact conversion formula is given by .\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nSolutions\nSolution: 1\n\n\n\n\n\n\n\nx g(x)\na. Given a temperature in degrees Kelvin, represents its\ng(x)"
	},
	{
		"chunk_id" : "00004720",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "g(x)\nequivalent in degrees Celsius. Given a temperature in degrees\nf(g(x))\nCelsius, represents its equivalent in degrees Fahrenheit.\nx\nCombining these two statements, if represents a temperature in\nf(g(x))\ndegrees Kelvin, then represents its equivalent in degrees\nFahrenheit.\nx f(x)\nb. Given a temperature in degrees Celsius, represents its\ng\nequivalent in degrees Fahrenheit. But since assigns to temperatures\ng(f(x))\nin degrees Kelvin their equivalents in degrees Celsius, has no"
	},
	{
		"chunk_id" : "00004721",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "representation in terms of temperatures.\nf(g(x)) g(x) x\nc. To find an expression for we substitute for in the\nf(x)\nexpression for :\n9 9 9 2297\nf(g(x)) = (g(x))+32 = (x273)+32 = x .\n5 5 5 5\nd. We remember that the function defined by the equation in c) assigns\nto a temperature in degrees Kelvin, its equivalent in degrees\nh\nFahrenheit. We also notice that must do the reverse. Since the\nf(g(x)) x 9 2297\nexpression for multiplies by and then subtracts , an\n5 5\nh(x) 2297 x"
	},
	{
		"chunk_id" : "00004722",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "5 5\nh(x) 2297 x\nexpression for can be obtained by first adding to and then\n5\nEngag9e your students with effective distance learning resources. ACCESS RESOURCES>>\ndividing by . In other words,\n5\n5 2297\nh(x) = (x+ )\n9 5\n5 2297\n= x+ .\n9 9\n\nSolution: 2\nA second way to solve part d) of the problem is to\n1) develop an expression that will convert a temperature in degrees\nFahrenheit to its equivalent in degrees Celsius,\n2) develop an expression that will convert a temperature in degrees"
	},
	{
		"chunk_id" : "00004723",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "Celsius to its equivalent in degrees Kelvin, and finally\nh(x)\n3) use these expressions to develop an expression for .\nx\nIf is a temperature in degrees Celsius then\na(x) = x+273\nx\ngives its equivalent in degrees Kelvin. If is a temperature in degrees\nFahrenheit then\n5\nb(x) = (x32).\n9\nx\ngives its equivalent in degrees Celsius. So, if represents a temperature\na(b(x))\nin degrees Fahrenheit, represents its equivalent in degrees\nKelvin. So we find\n5\nh(x) = a(b(x)) = ( (x32))+273\n9\n5 2\n= x+255 .\n9 9"
	},
	{
		"chunk_id" : "00004724",
		"source" : "Telling a Story With Graphs.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "graphs",
		"content" : "9\n5 2\n= x+255 .\n9 9\n255 2 = 2297\nSince this is the same answer found in solution 1.\n9 9\nEngage your students with effective distance learning resources. ACCESS RESOURCES>>\nTypeset May 4, 2016 at 18:58:52.\nLicensed by Illustrative Mathematics\nunder a\nCreative Commons Attribution-NonCommercial-\nShareAlike 4.0 International License."
	},
	{
		"chunk_id" : "00004725",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "THE ANATOMY OF WRITING\nFour Main Types of Writing\nthe four types of writing, their\napplications, and some examples Narrative Writing\nDescriptive Writing\nExpository Writing\nPersuasive Writing\nselect a type to learn more about it!\nEXAMPLES\nNarrative Writing\nDescriptive\nNarrative writing tells a story Narrative Essay (non-fiction):\nthat usually includes Goodbye to All That by Joan Didion,\ncharact E e x rs p , o c s o it n o f r l y ict, and a from Slouching Towards Bethlehem\nsetting. The story may be"
	},
	{
		"chunk_id" : "00004726",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "setting. The story may be\nfiction or fact (fiction or non- Autobiography (non-fiction):\nfiction). Narrative of the Life of Frederick\nPersuasive\nDouglass by Frederick Douglass\nWriters use narrative to share\ninformation in a storylike Novel:\nway. It is important to Emma by Jane Austen\nrecognize that narrative style\ncan be used in all types of Short Story:\nwriting but is typically used in Eight Bites by Carmen Maria Machado,\nshort stories, some types of Gulf Coast Magazine\npoetry, history, anecdotes,"
	},
	{
		"chunk_id" : "00004727",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "poetry, history, anecdotes,\nnovels, and novellas.\nReturn to table of contents\nNarrative\nEXAMPLES\nDescriptive Writing\nExpository\nDescriptive writing is used in\nTravel Writing (non-fiction):\nboth fiction and non-fiction. It\nTime Has Stood Still on This Belize\nappeals Pteor stuhaes fivivee senses by\nIsland. How Wonderful.\ndescribing settings,\nby Katie Arnold, Outside Magazine\nemotions, people, things, and\nexperiences in detail, using\nExamples:\nmetaphors, similes, imagery,\nDescriptive Text Examples by Matt"
	},
	{
		"chunk_id" : "00004728",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "Descriptive Text Examples by Matt\nand other literary devices.\nSalter, yourdictionary.com\nIt is common in novels, short\nShort Story:\nstories, poetry, and creative\nBroads by Roxane Gay, Guernica\nessays. Dont be afraid to use\nMagazine\ndescriptive language to make\nanything you write more\nenjoyable for the reader!\nReturn to table of contents\nDescriptive\nEXAMPLES\nExpository Writing\nPersuasive\nThe word expository means\nWikipedia Entry:\nto explain or to expound\nHawaii\nupon. So, expository writing"
	},
	{
		"chunk_id" : "00004729",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "Hawaii\nupon. So, expository writing\nis simply writing that explains\nHow-to:\nand informs. It does not\nNursing Notes: How to Write Them by\nusually include opinion and\nAlex Lukey, nursetogether.com\ntypically includes evidence\nand factual data.\nNews\nCDC highlights increased cancer risk\nThis type of writing is very\nfrom Camp Lejeune water\ncommon and can be found in\ncontamination by Bilyana Garland,\nthe news, business writing,\nABC45 News\nweb articles, recipes, and\nmany more.\nReturn to table of contents"
	},
	{
		"chunk_id" : "00004730",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "many more.\nReturn to table of contents\nExpository\nEXAMPLES\nPersuasive Writing\nPersuasive writing attempts\nPersuasive Essay:\nto convince readers of an\nI Just Called to Say I Love You by\nopinion, belief, or theory.\nJonathan Franzen, MIT Technology\nReview\nPersuasive writing can be\nfound in opinion essays,\nArgumentative Essay:\nreviews and complaints,\nPolitics and the English Language by\nrecommendations, cover\nGeorge Orwell, The Orwell Foundation\nletters, and academic\nargumentative essays.\nArgumentative Essay:"
	},
	{
		"chunk_id" : "00004731",
		"source" : "The_Anatomy_of_Writing_RUHKM5v.pdf",
		"MyUnknownColumn" : "2025-09-10",
		"subject" : "English",
		"content" : "argumentative essays.\nArgumentative Essay:\nLetter from a Birmingham Jail by\nPersuasive writing relies on\nMartin Luther King Jr., African\nevidence to back up opinion\nStudies Center, University of\nor belief and also typically\nPennsylvania\nprovides counterarguments\nwhich it then attempts to\nrefute.\nReturn to table of contents"
	}
]