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21.1.2 1.1.2 California Standards: 1.0:1.0:1.0:1.0:1.0: Students identify Students identify Students identify Students identify and Students identify use the arithmetic properties integggggererererers and s and s and inte inte subsets of subsets of subsets of inte of subsets of s and inte subsets of s and rrrrraaaaatio... |
rprprprproper umbersssss to umber ties of n n n n number umber ties of ties of oper oper operties of umber ties of oper demonstrate whether assertions are true or false. What it means for you: You’ll deal with simple expressions that contain numbers and unknowns. Key words: expression numeric expression algebraic expre... |
degrees Fahrenheit (°F) to degrees Celsius (°C). What is 131 °F in °C? (F – 32) is used to convert temperatures from ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up A coefficient is just the number in front of a variable. It doesn’t matter whether an expression includes just nu... |
l Multipl Multiplication has a property that’s similar to the property of addition: Every real number also has a multiplicative inverse (or “reciprocal”). When you multiply a number by its multiplicative inverse, the result is 1 — the multiplicative identity. However, there’s an important exception: zero has no recipro... |
set of negative real numbers is closed under addition — so if you add two negative real numbers, you always get another negative real number. 3. Positive or Negative or Zero if a and b have opposite signs. In this case, the sign of (a + b) is the same as the sign of the addend with the larger absolute value. For examp... |
per oper operties of umber ties of oper demonstrate whether assertions are true or false. What it means for you: You’ll see why it’s important to have a set of rules about the order in which you have to deal with operations. Key words grouping symbols parentheses brackets braces exponents tions tions Operaaaaations Ope... |
+ (a × c) So a factor outside parentheses multiplies every term inside. Example Example Example Example Example 11111 Expand: a) 3(x + y) b) x(3 + y) c) 3[(x + y) + z] Solution Solution Solution Solution Solution a) 3(x + y) = 3x + 3y b) x(3 + y) = x·3 + xy = 3x + xy (using the commutative law of multiplication) c) 3[(... |
Solution Solution Solution Solution Solution (x + y)(x – y) = (x + y)(x + (–y)) Definition of Definition of subtr subtr action action Definition of subtr subtraction action Definition of Definition of subtr action = (x + y)x + (x + y)(–y) Distrib Distrib Distributiutiutiutiutivvvvve lae lae lae lae lawwwww Distrib Dist... |
xponents xponents xponents What it means for you: You’ll look more closely at the rules of square roots. Key words: square root radical radicand principal square root minor square root Check it out: Technically, the square root of a number p can be written as p2 , but you don’t usually write the 2. Square Re Re Re Re R... |
enominator greatest common factor simplify actions actions alent Frrrrractions alent F alent F EquiEquiEquiEquiEquivvvvvalent F actions actions alent F alent Frrrrractions EquiEquiEquiEquiEquivvvvvalent F actions actions alent F alent F actions alent F actions Here’s another Algebra I Topic that you’ve seen in earlier ... |
1 = 12 actorsssss actor actor Cancel all the common f Cancel all the common f Cancel all the common factor actor Cancel all the common f Cancel all the common f Guided Practice In Exercises 1–6, divide and simplify. 22. 4 9 ÷ 16 15 25. 28. 48 7 24 32 ÷ ÷ 16 35 9 56 23. 26. 29. 5 21 13 27 40 72 ÷ ÷ 25 14 2 63 ÷ 24 36 24... |
ntiontiontion, ultiplica ultiplica action, mmmmmultiplica action action subtr subtr subtraction action subtr ultiplica subtr and dididididivision vision vision vision. They basically vision say that you can do the same thing to both sides of an equation. For a formal definition of each, refer to Topic 2.2.1. A lot of A... |
rule. Key words: inductive reasoning deductive reasoning counterexample e and e and Inductivvvvve and Inducti Inducti e and e and Inducti Inducti e and e and Inductivvvvve and Inducti Inducti e and e and Inducti Inducti easoning easoning Deductivvvvve Re Re Re Re Reasoning Deducti Deducti easoning easoning Deducti Ded... |
true. Section 1.4 Section 1.4 Section 1.4 — Mathematical Logic Section 1.4 Section 1.4 6161616161 Check it out: See Topic 1.2.3 for more about absolute value equations. Check it out: If |a| < |b|, then: 0 < a < b, or b < a < 0 or a < 0 < b, or b < 0 < a. tement is Nevvvvver er er er er TTTTTrrrrrueueueueue tement is Ne... |
fourth term in the algebraic expression 8x2 + 2xy – 6y + 9xy3 – 4? LikLikLikLikLike e e e e TTTTTerererererms Can Be Combined ms Can Be Combined ms Can Be Combined ms Can Be Combined ms Can Be Combined Like terms are terms with identical variables that have identical exponents. The terms 4x2 and –2x2 are like terms bec... |
's age. Madeline has 5(2x + 3) dolls. Write and simplify an algebraic expression showing the total number of dolls in Mia and Madeline's collection. 17. Ruby, Sara, and Keisha are counting stamps. If x represents Ruby's age, she has 4(x – 4) stamps, Sara has 2(8 – x) stamps, and Keisha has 8(7 + 2x) stamps. Write and s... |
dition and Subtr dition and Subtr tions action in Equations dition and Subtraction in Equa tions action in Equa dition and Subtr Addition Property of Equality For any real numbers a, b, and c, if a = b, then a + c = b + c. Subtraction Property of Equality For any real numbers a, b, and c, if a = b, then a – c = b – c. ... |
ominators: 1. 1 9 x + 13 – In Exercises 3–6, solve the equations for the unknown variable. 3. 1 2 x – 1 = x 5. 1 10 . 1 10 x – 3 = 1 4 x 6 or More Fe Fe Fe Fe Frrrrractions actions actions o or Mor k Out the LCM fCM fCM fCM fCM for or or or or TTTTTwwwwwo or Mor o or Mor k Out the L ou Can WWWWWororororork Out the L k ... |
mple Example 33333 Solve 0.015x = 0.2 – 0.025x. Solution Solution Solution Solution Solution Multiply both sides by 1000 this time. 1000(0.015x) = 1000(0.2 – 0.025x) 15x = 200 – 25x 40x = 200 x = 5 If your longest decimal has 3 decimal places, multiply by 103 = 1000. If the longest decimal has 4 decimal places, multipl... |
California Standards: 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify 4.0: Students simplify essions befororororore solving eeeeexprxprxprxprxpressions bef e solving e solving essions bef essions bef e solving e solving essions bef tions tions linear equa linear equa tions a... |
or each ste tion for eac justification f h ste or eac tion f justifica justifica What it means for you: You’ll solve word problems that refer to patterns of integers. Key words: consecutive integer common difference When you’re solving word problems, the most important thing to do is to write down what you know. Then ... |
in the question to write an equation that you can go on to solve. Example Example Example Example Example 11111 Charles is 7 years older than Jorge. In 20 years’ time, the sum of their ages will be 81 years. How old is each one now? Solution Solution Solution Solution Solution The first thing to do is write expression... |
te Do hen Solve an Equa e an Equa hen Solv ou Kno rite Do Motion tasks normally involve two objects with different speeds. Example Example Example Example Example 11111 Tim drives along a road at 70 km/h. Josh leaves from the same point an hour later and follows exactly the same route. If Josh drives at 90 km/h, how lo... |
es $500 in interest on her investment. 108108108108108 Section 2.6 Section 2.6 Section 2.6 — Investment and Mixture Tasks Section 2.6 Section 2.6 Guided Practice 1. A banker invested $7000 at an annual interest rate of 8%. What would be the return on the investment at the end of the year? 2. A banker invested $5000 at ... |
xtur The interest problems in Topic 2.6.1 were actually mixture problems because you had to add together returns from investments with different interest rates. In this Topic you’ll see some mixture problems that don’t involve money. olume” olume” Amount per Unit Mass/V Amount per Unit Mass/V asks Use “ Some TTTTTasks ... |
r the ula f s a Fororororormmmmmula f ula f s a F TTTTTherherherherhere’e’e’e’e’s a F s a F h Substance Each Substance Amount of Eac or the Amount of ula for the h Substance Eac Amount of or the ula f s a F Multiplying the volume by the concentration gives you the amount of substance in each solution. And since the tot... |
working at different rates. Example Example Example Example Example 11111 John takes 1 hour to deliver 100 newspapers, and David takes 90 minutes to deliver 100 newspapers. How long would it take them to deliver 100 newspapers between them? Assume that they work independently, that they both start at the same time, and... |
or or x TTTTThen solv hen solv hen solv hen solv x = 2 2 5 Therefore painting the wall would take Jesse and Melinda 2 2 5 hours = 2 hours and 24 minutes. Example Example Example Example Example 33333 Liza can dig a garden in 7 hours alone. If Marisa helps her, they finish all the digging in just 3 hours, working indepe... |
a alues..... alues alues bsolute v bsolute v olving a olving a bsolute values olving absolute v alues bsolute v olving a What it means for you: You’ll solve equations involving absolute values. Key words: absolute value Absolute VVVVValuealuealuealuealue Absolute Absolute e on e on MorMorMorMorMore on e on Absolute Ab... |
ing with any real-life problem is to write down all the math carefully before you start solving. estigaaaaationtiontiontiontion — Wildlife Park Trains estigestig estig pter 2 Invvvvvestig pter 2 In ChaChaChaChaChapter 2 In pter 2 In pter 2 In 135135135135135 Chapter 3 Single Variable Linear Inequalities Section 3.1 Ine... |
f inequalities Subtr oper action pr inequalities ty of x £ 6 Graph: –1 0 1 2 3 4 5 6 Solution in interval notation: (–•••••, 6] So the maximum integer value of x is 6. opertiestiestiestiesties oper oper ou to Use Both Pr lems Need YYYYYou to Use Both Pr ou to Use Both Pr lems Need Some Proboboboboblems Need lems Need S... |
s inequalities Multiplica tion pr oper ty of inequalities Guided Practice Solve each inequality in Exercises 27–37. 27. 30. 33. ≤1 x 7 − 10 − 4 > −d 3 − ≥j 8 7 28. 31. 34. − <1 c 9 − 1 4 − < −k 11 a 5 29. 11 < < − 12 32. − ≤ 4 1 35 36. − 1 9 1 ⋅ > 4 x( 4 − 13 3 ) 37. 1 + ⋅ 5 10 1 ≥ − g 6 4 144144144144144 Section 3.1 S... |
10 > 7(t + 3) + 4t 28. 5(a – 5) – 3 > 4(a + 8) + 7a ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up If you ever get stuck when you’re solving inequalities with more than one step, refer to the checklist on the previous page. Just take it one step at a time, as if you were deali... |
d be cheaper to use. Set up and solve an inequality that supports this recommendation. 2. Jack is doing a sponsored swim to raise money for charity. His mom sponsors him $10, plus $1 for every length of the pool he completes. His uncle sponsors him just $1.50 for every length he completes. How many lengths will Jack ha... |
ndpoint interval Check it out: The solution interval is the open interval (–4, 18). On the graph, the endpoints –4 and 18 aren’t included. You last saw absolute value equations in Section 2.8 — now you’re going to see inequalities involving absolute values. As with normal inequalities, you can have conjunctions and dis... |
at has a height of 83 cm. How wide could the picture be so that it fits in a rectangular prism shaped box that satisfies the shipping company’s requirements? Write an inequality to represent the possible width of the picture. width 83 c m Open-ended Extension 1) You work for an organization that produces pocket diction... |
Guided Practice In Exercises 1–6, name the quadrant or axis where each point is located. Justify your answers. 1. (1, 0) 4. (0, –5) In Exercises 7–12, a > 0, k > 0, m < 0, v < 0, and p Œ R. Name the quadrant or axis where each point is located. Justify your answers. 7. (a, k) 10. (p, 0) 3. (–4, –1) 6. (–p, p) 8. (m, k)... |
tete -Coor -Coor e the Same tical Line Havvvvve the Same e the Same tical Line Ha oints on a VVVVVererererertical Line Ha tical Line Ha oints on a PPPPPoints on a oints on a -Coor e the Same tical Line Ha oints on a The x-coordinate tells you how far to the left or right of the y-axis a point is. Points with the same x... |
3x + y = –6 11. 2x – y = –14 13. 8x + 4y = 24 15. 3x – 9y = –27 6. y – x = 10 8. 5x + y = –12 10. –10x + y = 21 12. 6x + 2y = 18 14. 12x – 4y = 8 16. 2x – 8y = 16 Round Up Round Up It’s easy to make a mistake when working out y-values, so choose x-values that will make the algebra easy (for example, 0 and 1). And it’s ... |
rough them. Graphing a Line Find the x-intercept — let y = 0, then solve the equation for x. Find the y-intercept — let x = 0, then solve the equation for y. Draw a set of axes and plot the two intercepts. Draw a straight line through the points. Check your line by plotting a third point. Example 1 Draw the graph of 5x... |
right, or large and negative if it goes down from left to right). Section 4.3 Section 4.3 Section 4.3 — Slope Section 4.3 Section 4.3 193193193193193 oint-Slope Fororororormmmmmulaulaulaulaula oint-Slope F oint-Slope F PPPPPoint-Slope F oint-Slope F PPPPPoint-Slope F oint-Slope Fororororormmmmmulaulaulaulaula oint-Slop... |
orget: See Topic 4.3.1 for more on calculating the slope of a line. Check it out: It’s possible that all the points lie on the same line — meaning there is actually only one line. To be absolutely accurate, you should check that this isn’t the case before you say the lines are parallel. You can do this by finding the e... |
how that the line through the points (5, –3) and (–8, 1) is perpendicular to the line through (4, 6) and (8, 19). 11. Show that the line through (0, 6) and (5, 1) is perpendicular to the line through (4, 8) and (–1, 3). 12. Show that the line through (4, 3) and (2, 2) is perpendicular to the line through (1, 3) and (3,... |
o a a line per a line per gigigigigivvvvven line tha t passes t passes en line tha en line tha en line that passes t passes en line tha t passes en point. en point. ough a givvvvven point. ough a gi ough a gi thrthrough a gi thrthr en point. en point. ough a gi thr What it means for you: You’ll learn how to tell whethe... |
–5 –4 –3 –2 So at (0, 2), you get 2 = 2(0) + 1, that is, 2 = 1, which is a false statement. 0 –1 –1 –2 –3 –4 –5 –6 1 2 3 4 5 6 x-axis Since 2 > 1, a “>” sign is needed to make it a true statement. So (0, 2) satisfies the inequality y > 2x + 1. Therefore the inequality that defines the shaded region is y > 2x + 1. Guide... |
les of MorMorMorMorMore Examples of e Examples of gions e Examples of Gr gions phing R Gr e Examples of Graphing regions isn’t always straightforward, so here are a couple more examples and some more practice exercises. Example Example Example Example Example 33333 Graph the solution set of 4y – 3x ≥ 12. Solution Solut... |
the shaded region shown on the right. Solution Solution Solution Solution Solution First line: Two points on this line are (0, 1) and (3, 2). –4 –3 –2 m1 = − 2 1 − 3 0 = 1 3 y-axis Line 2 Line 1 x-axis 1 –1 –2 –3 –4 –5 y – y1 = m(x – x1) fi y – 2 = 1 3 (x – 3 224224224224224 Section 4.5 Section 4.5 Section 4.5 — Inequal... |
method’s been used in Example 1. Example Example Example Example Example 11111 Solve this system of equations by graphing: 2x – 3y = 7 –2x + y = –1 Solution Solution Solution Solution Solution Step 1: Graph both equations in the same coordinate plane. Line of first equation: 2x – 3y = 7 3y = 2x – – y 1– 3– The line goe... |
ue statement Therefore x = 2, y = 3 is the solution of the system of equations. 234234234234234 Section 5.1 Section 5.1 Section 5.1 — Systems of Equations Section 5.1 Section 5.1 The Same Example Using the Substitution Method The Same Example Using the Substitution Method The Same Example Using the Substitution Method ... |
utions A Dependent System Has Infinitely Many Solutions A Dependent System Has Infinitely Many Solutions A Dependent System Has Infinitely Many Solutions Some systems of two linear equations have an infinite number of solutions — in other words, there are an infinite number of points (x, y) that satisfy both of the equ... |
ble to solve a system of two linear inequalities in two variables and to sketch the solution sets. What it means for you: You’ll learn about the elimination method and then use it to solve systems of linear equations in two variables. Key words: system of linear equations elimination method Check it out: The size (abso... |
e problems involving systems of linear equations. Key words: system of linear equations substitution method elimination method Check it out: 8.50c is the amount spent on CDs. 12.50d is the amount spent on DVDs. Section 5.3 tions tions Equa Equa Systems of Systems of tions Equations Systems of Equa tions Equa Systems of... |
g Systems of Using Systems of Example Example Example Example Example 11111 The sum of two integers is 53. The larger number is 7 less than three times the smaller one. Find the numbers. Solution Solution Solution Solution Solution First form a system of two equations. Let x = smaller number y = larger number The sum o... |
resa’s piggy bank has a total of 100 dimes and nickels. If the piggy bank has a total of $7.50, how many of each coin does she have? 4. Anthony invested $1000 in two stocks. Stock A increased in value by 20%, while Stock B decreased in value by 10%. If Anthony ended the year with $1160 worth of stocks, how much money d... |
ng Ev BrBrBrBrBreaking Ev g Business g Business en in an Eg en in an Eg eaking Ev eaking Ev en in an Egg Business eaking Even in an Eg g Business en in an Eg eaking Ev g Business Even though this Investigation looks tough because it’s about money, it’s just systems of equations. Part 1: You decide to set up an egg-sell... |
2 for more on additive inverses. Guided Practice Find the opposites of the following polynomials. 1. 2x + 1 3. x² + 5x – 2 5. 3x2 + 4x – 8 7. 4x4 – 16 9. 5x4 – 6x2 + 7 11. –0.9x3 – 0.8x2 – 0.4x – 1.0 2. –5x – 1 4. 3x² – 2x + 3 6. –8x2 – 4x + 4 8. 8x3 – 6x2 + 6x – 8 10. –2x4 + 3x3 – 2x2 12. –1.4x3 – 0.8x2 – 1 2 x 266 Se... |
+ 3) – 4(3x – 1) + 3(x – 1). 8. Simplify (8x3 – 5x2 – 2x – 7) – (6x3 – 3x2 + x – 5) – (–3x2 + 3x – 2). 9. Simplify –2(3x3 – 2x) – 3(–x3 + 7) – (2x2 – 5x3 – 5). 10. Find the sum of the opposites of: –2x3 + 3x2 – 5x + 1 and 3x3 – 2x2 + 3x – 3 11. Find the difference between the opposites of: –2x2 – 3x + 5 and 3x2 + 2x –... |
6) inches. The width of the middle rectangle is 3x + 2 – 2x = (x + 2) inches. x Area of space = area of large rectangle – area of small rectangle = (5x + 6)(3x + 2) – (3x + 6)(x + 2) = 15x2 + 10x + 18x + 12 – (3x2 + 6x + 6x + 12) = 15x2 + 28x + 12 – 3x2 – 12x – 12 = (12x2 + 16x) in2 Guided Practice 1. Find the area of... |
these expressions. 1. ab 2. a2 3. 2x + 4b 4. 3x + 1 5. 8x3 – 16x2 + 4 7. 2x2y4 8. 2 Section 6.3 Section 6.3 Section 6.3 — Dividing Polynomials Section 6.3 Section 6.3 285285285285285 ocals as Negggggaaaaatititititivvvvve Exponents e Exponents e Exponents ocals as Ne ocals as Ne ecipr ecipr rite R ou Can WWWWWrite R rit... |
ng Division Method — f visions xact Di or None vision Method — f Long Di Long Di The long division method for dividing polynomials is really similar to the long division method for integers. The aim is to find out how many groups of the divisor can be subtracted from the dividend. Example Example Example Example Exampl... |
of a prism is (2x3 + x2 – 3x) m3. If the area of the base is (x2 – x) m2, what is the height of the prism? 5. A rectangular prism has volume (b3 + 9b2 + 26b + 24) m3, width (b + 2) m, and length (b + 4) m. Find its height. 6. The volume of a prism is (144s3 + 108s2 – 4s – 3) m3. If the area of the base is (36s2 – 1) m... |
rs: 1. 8 4. 16 3. 15 6. 24 2. 10 5. 11 Write down each monomial as a product of the smallest possible factors: 7. 3x 10. 5xy 9. 6p 12. 20mn 8. 7z 11. 12uv h Monomial h Monomial Eac Eac visor of visor of he GCF is a Di TTTTThe GCF is a Di he GCF is a Di h Monomial Each Monomial visor of Eac he GCF is a Divisor of h Mono... |
d – 1) ⎛ ⎜⎜⎜⎜ ⎝ d – 1)(x2 + x + 1) ⎞ ⎟⎟⎟⎟ ⎠ Example Example Example Example Example 44444 Show that (a + b) is a factor of ac + bc + ad + bd. Solution Solution Solution Solution Solution You’ve been given the factor, so try writing the polynomial as a factored expression. If you can do that, you’ll have shown that (a +... |
3) is a factor of x2 – 2x – 15. 22. If 2n3 – 5 is a factor of 12n5 + 2n4 – 30n2 – 5n, find the other factors. 23. If (8n – 3) is a factor of 8n3 – 3n2 – 8n + 3, find the other factors. 24. If (2x + 5) is a factor of 2x3 + 15x2 + 13x – 30, find the other factors. 25. If (a – 1) is a factor of a3 – 6a2 + 9a – 4, find th... |
negative). 316316316316316 Section 6.6 Section 6.6 Section 6.6 — Factoring Quadratics Section 6.6 Section 6.6 Example Example Example Example Example 66666 Factor 6x² + 5x – 6. Solution Solution Solution Solution Solution In this expression, c is negative, so one of c1 and c2 must be positive and the other must be nega... |
) ft2 The areas of the parallelograms below are the products of two binomials with integer coefficients. Find the dimensions of each parallelogram if a = 10 and b = 5. 19. Parallelogram with area (8a2 – 10ab + 3b2) ft2 20. Parallelogram with area (4a2 – 9b2) ft2 21. Parallelogram with area (12a2 + 11ab – 5b2) ft2 22. P... |
icsticstics actor Quadr actor Quadr es to F es to F o Squar ence of TTTTTwwwwwo Squar o Squar ence of Use Difffffferererererence of ence of Use Dif Use Dif es to Factor Quadr o Squares to F actor Quadr es to F o Squar ence of Use Dif Use Dif A difference of two squares is one term squared minus another term squared: m2... |
y – 6k –10tk. Solution Solution Solution Solution Solution Group 3y + 5ty and –6k – 10tk together in parentheses: (3y + 5ty) + (–6k – 10tk) 3y and 5ty have a common factor of y. –6k and –10tk have a common factor of –2k. Factor out the common factors: (3y + 5ty) + (–6k –10tk) = y(3 + 5t) – 2k(3 + 5t) Now you can see th... |
toring to Solve Quadratic Equations 1) First arrange the terms in the quadratic equation so that you have zero on one side. 2) Then factor the nonzero expression (if possible). 3) Once done, you can use the zero property to find the solutions. Example 4 Solve x2 – 6x – 7 = 0. Solution The right-hand side of the equatio... |
another method for solving quadratic equations — but before you solve any equations, you need to know how completing the square actually works. Writing Perfect Square Trinomials as Perfect Squares An expression such as (x + 1)2 is called a perfect square — because it’s (something)2. In a similar way, an expression suc... |
8 12. x2 + 8x + 5 14. x2 + 3x + 5 16. x2 + 4x + 5 18. x2 – 20x + 20 20. 4x2 + 16x + 4 11. x2 + 6x + 14 13. x2 – 12x + 8 15. x2 – 5x – 7 17. x2 – 6x + 30 19. x2 + 22x + 54 21. 2x2 + 20x + 25 Writing ax2 + bx + c in the Form a(x + k)2 + m This is the most general case you can meet — writing ax2 + bx + c as an expression... |
a gives you the solutions of any quadratic equation in terms of a, b, and c. It’s sometimes called the general solution of the quadratic equation ax2 + bx + c = 0. Don’t forget: See Section 7.2 for a reminder about completing the square. Don’t forget: Be very careful if there are minus signs. Check it out: You can chec... |
graphs above, the line of symmetry is the y-axis. Notice that a bigger value of ΩaΩ results in a steeper (narrower) parabola. For example, the graph of y = 3x2 is steeper than the graph of y = x2. The basic shape of all quadratic graphs (that is, for any quadratic function y = ax2 + bx + c) is very similar to the ones... |
ient of x2 is positive, so this parabola is concave up. Example 1 continued The graph of y = x2 – 6x + 10 = (x – 3)2 + 1: line of symmetry = 3 x y 6 5 4 3 2 1 vertex (3, 1) -1 1 2 3 4 5 6 7 -1 x 8 Section 7.4 — Quadratic Graphs 369 Don’t forget: See Section 7.2 for more about completing the square. Check it out: The co... |
ere a = 1, b = 2, and c = 3, so: b2 – 4ac = 22 – 4 × 1 × 3 = 4 – 12 = –8 So the graph of y = x2 + 2x + 3 never intersects the x-axis. Since a > 0, the graph is concave up (u-shaped) and stays above the x-axis. -5 -4 -3 -2 (b) Here a = –2, b = 4, and c = –5 , so: b2 – 4ac = 42 – 4 × (–2) × (–5) = 16 – 40 = –24 y (ab) = ... |
drops her watch. The distance in feet, h, that the watch has fallen after t seconds is given by the equation h = 16t2 + 4t. After how many seconds will the watch have fallen 600 feet? The height in feet of an object projected upwards is modeled by the equation h = 100t – 16t2. 7. How long after being projected is the o... |
e denominator. Section 8.1 Fractions and Fractions and Rational Expressions Rational Expressions In this Topic you’ll find out about the necessary conditions for rational numbers to be defined. Rational Expressions Can Be Written as Fractions A rational expression is any expression that can be written in the form of a ... |
ng and Dividing Rational Expressions Section 8.2 Section 8.2 Example Example Example Example Example 22222 Multiply and simplify Solution Solution Solution Solution Solution Step 1: Factor the numerators and denominators if possible ( ) 33 2 2 a a )( 3 2 a )( − 2 a a )( ( − ⋅ − 2 1 )( ( − + + − ( ( = = a a ) ⋅ ⋅ Step 2... |
r Distrib Distrib opertytytytyty oper oper e pre proper e pre pr Distributiutiutiutiutivvvvve pr Distrib Distrib oper Distrib Distrib CommCommCommCommCommutautautautautatititititivvvvve and e and e and e and e and associa associa associatititititivvvvve pr e pre proper e pre pr oper oper opertiestiestiestiesties associ... |
factor the numerator and denominator and cancel any common factors. 19 410 Section 8.3 — Adding and Subtracting Rational Expressions Independent Practice Simplify each expression. 1 16 + − x 4 5 2 a 3 10 − 11 − 10 ab 5 2y 6. − + xy x − y 1 − + 1 xy 2 y 7 −( 1 ) t 2 9. 2 t 4 x + x 5 15 − − + 1 3 x x 8. 10. 2 y 3 + + 7 y... |
the algebraic problem, it isn’t a correct answer for this example because speed can only have a positive value. Guided Practice 1. On Monday, a distribution company shipped a load of oranges in crates, with a total weight of 124 lb. On Tuesday it shipped another load of oranges, also with a total weight of 124 lb. Howe... |
ollowing restriction on it: A function is a set of ordered pairs (x, y) such that no two ordered pairs in the set have the same x-value but different y-values. That is, each member of the domain maps to a unique member of the range. Example 1 Determine whether each of the following relations is a function or not. Justi... |
. What it means for you: You’ll show whether two functions are equal. Key words: function ordered pair domain range More on Functions More on Functions This Topic’s all about the special rules for telling whether two functions are equal. Equality of Functions A function f is equal to another function g if, and only if,... |
er oper tion) for ultiplication) dition and multiplica operties (of tion) ultiplica dition and m ties (of oper any a, b: a + b = b + a and ab = ba completing the squareeeee the process of changing a quadratic completing the squar completing the squar completing the squar completing the squar expression into a perfect s... |
ressions together tions tions equa equa system of system of tions two or more equations equations system of equa tions equa system of system of T tertermsmsmsmsms the parts that are added to form an expression terter ter trinomial trinomial trinomial a polynomial with three terms trinomial trinomial U sets) sets) union... |
ties 147-148 properties of addition 139, 140, 147, 148 division 143, 145, 147, 148 multiplication 142, 144 subtraction 140 regions defined by 212-215, 217-220 input-output tables 422 inputs of binary operations 15 integer problems 153 197-199, 201-203, 205-207, 209-210 consecutive integer tasks 95-97, 150, graph of a n... |
lationwithheadpain.MaterialsandmethodsThestudyenrolled94females,diagnosedasmigrainewithoutaurafollowingtheInternationalClassificationofHeadacheDisorders[5],whoweresubsequentlyexaminedattheWomen’sHeadacheCentre,DepartmentofGynae-cologyandObstetricsofTurinUniversity.Theywereallincludedinthestudyduringamigraineattackprovid... |
county residents and takes values below hs, hs diploma, some college, or bachelors in each county. This variable seems to be a hybrid: it is a categorical variable but the levels have a natural ordering. A variable with these properties is called an ordinal variable, while a regular categorical variable without this ty... |
reward children who report white.13 (a) Identify the main research question of the study. (b) Who are the subjects in this study, and how many are included? (c) The study’s findings can be summarized as follows: ”Half the students were explicitly told not to cheat and the others were not given any explicit instructions.... |
in all swordfish in the Atlantic Ocean. If we take a sample of 50 swordfish from the Atlantic Ocean, the average mercury content among just those 50 swordfish will be the statistic. Two statistics we will study are the mean (also called the average) and proportion. When we are discussing a population, we label the mean as... |
data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. (a) Identify the population of interest and the sample in this study. (b) Comment on whether or not the results of the study can be generalized to the population, and if the... |
arch for confounding variables, there is no guarantee that all confounding variables can be examined or measured. In the same way, the county data set is an observational study with confounding variables, and its data cannot be used to make causal conclusions. sun exposureuse sunscreenskin cancer? 1.4. OBSERVATIONAL ST... |
le versus systematic sampling while Figure 1.15 provides a graphical representation of stratified, cluster, and multistage sampling. Simple random sampling is probably the most intuitive form of random sampling. Consider the salaries of Major League Baseball (MLB) players, where each player is a member of one of the lea... |
clusters in this example. We wouldn’t want to use the grades as clusters and sample everyone from a couple of the grades. This would create too large a sample and would not give us the nice representation from each grade afforded by the stratified random sample. EXAMPLE 1.27 Suppose we are interested in estimating the m... |
survey. (c) He posts a link to an online survey on Facebook and asks his friends to fill out the survey. (d) He randomly samples 5 classes and asks a random sample of students from those classes to fill out the survey. 1.28 Reading the paper. Below are excerpts from two articles published in the NY Times: (a) An article... |
the explanatory variable on the response. In an experiment with six subjects, even if there is randomization, it is quite possible for the three healthiest people to be in the same treatment group. In a randomized experiment with 100 people, it is virtually impossible for the healthiest 50 people to end up in the same ... |
mized experiment involves randomly assigning the subjects to the different treatment groups. To do this, first number the subjects from 1 to N. Then, randomly choose some of those numbers and assign the corresponding subjects to a treatment group. Do this in such a way that the treatment group sizes are balanced, unless ... |
n generalize the conclusion to the population at large. 1.38 City council survey. A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments, and others a diverse mix... |
stem-and-leaf plots, dot plots, and histograms, to visualize the distribution of a numerical variable. Be able to read off specific information and summary information from these graphs. 4. Identify the shape of a distribution as approximately symmetric, right skewed, or left skewed. Also, identify whether a distribution... |
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