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an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points (0, 1) and (3, 2). We can use the coordinates of the two points to find the slope. SECTION 2.1 linear Functions 133 m = y2 − y1 _ x2 − x1 2 − 1 _ 3 − 0 1 __ = 3 = Now we...
through the points (−1, 3) and (0, 0). Then rewrite it in the slope-intercept form. Writing and Interpreting an equation for a linear Function Now that we have written equations for linear functions in both the slope-intercept form and the point-slope form, we can choose which method to use based on the information we...
-axis to identify the y-intercept by visual inspection. 4. Substitute the slope and y-intercept into the slope-intercept form of a line equation. Example 7 Writing an Equation for a Linear Function Write an equation for a linear function given a graph of f shown in Figure 10. y f 642 8 10 x –10 –8 –6 –4 10 8 6 4 2 –2 –...
is represented by So his monthly cost would be $5,000. C(100) = 1250 + 37.5(100) = 5000 Example 9 Writing an Equation for a Linear Function Given Two Points If f is a linear function, with f (3) = −2, and f (8) = 1, find an equation for the function in slope-intercept form. Solution We can write the given points using...
months. How many songs will he own in a year? Solution The initial value for this function is 200 because he currently owns 200 songs, so N(0) = 200, which means that b = 200. The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that m = 15. We can substit...
represents Ilya’s income when n = 0, or when no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold. We can now write the final equation. I(n) = 80n + 520 Our final interpretation is that Ilya’s base salary is $520 per week and he earn...
resource for additional instruction and practice with linear functions. • linear Functions (http://openstaxcollege.org/l/linearfunctions) SECTION 2.1 section exercises 139 2.1 SeCTIOn exeRCISeS VeRBAl 1. Terry is skiing down a steep hill. Terry’s elevation, E(t), in feet after t seconds is given by E(t) = 3000 − 70t. ...
1 __ x − 3 2 24. m(x) = − 3 __ x + 3 8 22. p(x) = 1 __ x − 5 4 23. n(x) = − 1 __ x − 2 3 For the following exercises, find the slope of the line that passes through the two given points. 25. (2, 4) and (4, 10) 26. (1, 5) and (4, 11) 27. (−1, 4) and (5, 2) 28. (8, −2) and (4, 6) 29. (6, 11) and (−4, 3) 140 CHAPTER 2 li...
5 –4 –3 –2 43. 321 4 5 6 x –5 –4 –3 –2 46. 321 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 –6 y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 y 6 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 –6 y 6 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 –6 y 5 4 3 2 1 –1 0 –1 –2 –3 –4 –5 321 4 5 6 x 321 4 5 x 321 4 5 6 x SECTION 2.1 section exercises 141 nUMeRIC...
decimal places. w −10 5.5 67.5 b k 30 −26 a −44 Table 3 59. Table 4 shows the input, p, and output, q, for a linear function q. a. Fill in the missing values of the table. b. Write the linear function k. p q 0.5 400 0.8 700 12 a b 1,000,000 Table 4 1 __ 60. Graph the linear function f on a domain of [−10, 10] for the ...
gym membership with five personal training sessions costs $260. What is cost per session? 70. A clothing business finds there is a linear relationship between the number of shirts, n, it can sell and the price, p, it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of...
. 76. When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of C, the Celsius temperature, F (C). a. Find the rate of change of Fahrenheit temperature for each...
Evaluating the function for an input value of 1 yields an output value of 2, which is represented by the point (1, 2). Evaluating the function for an input value of 2 yields an output value of 4, which is represented by the point (2, 4). Choosing three points is often advisable because if all three points do not fall ...
the y-intercept, we can set x = 0 in the equation. The other characteristic of the linear function is its slope m, which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another wa...
vertical line parallel to the y-axis does not have a y-intercept, but it is not a function.) How To… Given the equation for a linear function, graph the function using the y-intercept and slope. 1. Evaluate the function at an input value of zero to find the y-intercept. 2. Identify the slope as the rate of change of t...
(x) = x by m stretches the graph of f by a factor of m units if m > 1 and compresses the graph of f by a factor of m units if 0 < m < 1. This means the larger the absolute value of m, the steeper the slope. y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 1 2 3 4 5 6 x Figure 4 Vertical stretches and compressions and...
7 –6 –5 –4 –3 –2 4 3 2 1 –1 –1 –2 –3 –4 y = x1 2 x 5 6 7 21 3 4 y 5 4 3 2 1 –1 –1 –2 –3 –4 –7 –6 –5 –4 –3 –2 y = x1 2 1 y = x − 3 2 x 21 3 4 5 6 7 Figure 6 The function, y =x compressed by a factor of 2. 1__ Figure 7 The function y = __1 2 x, shifted down 3 units. –5 Try It #3 Graph f (x) = 4 + 2x, using t ransformatio...
the y-intercept and slope into the slope-intercept form of a line. Example 4 Matching Linear Functions to Their Graphs Match each equation of the linear functions with one of the lines in Figure 9. a. f (x) = 2x + 3 b. g(x) = 2x − 3 c. h(x) = −2x + 3 d. j(x) = 1 __ x + 3 2 –7 –6 –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 ...
2 graphs oF linear Functions 149 Finding the x-intercept of a Line Figure 10 So far, we have been finding the y-intercepts of a function: the point at which the graph of the function crosses the y-axis. A function may also have an x-intercept, which is the x-coordinate of the point where the graph of the function cross...
x − 4. 4 Describing Horizontal and Vertical Lines There are two special cases of lines on a graph—horizontal and vertical lines. A horizontal line indicates a constant output, or y-value. In Figure 13, we see that the output has a value of 2 for every input value. The change in outputs between any two points, therefor...
For any x-value, the y-value is −4, so the equation is y = −4. Example 7 Writing the Equation of a Vertical Line Write the equation of the line graphed in Figure 16. y 10 8 6 4 2 –2 –2 –4 –6 –8 –10 –10 –8 –6 –4 f 42 6 8 10 x Figure 16 Solution The constant x-value is 7, so the equation is x = 7. Determining Whether li...
the sign. As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor perpendicular. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular. The product of the slopes is −1. f (x)...
graph shows that the lines f (x) = 2x + 3 and j(x) = 2x − 6 are parallel, and the lines g(x) = 1 __ x − 4 and 2 h(x) = −2x + 2 are perpendicular. Figure 19 Writing the equation of a line Parallel or Perpendicular to a Given line If we know the equation of a line, we can use what we know about slope to write the equati...
form, we can substitute m = 3, x = 3, and f (x) = 0 into the slope-intercept form to find the y-intercept. g(x) = 3x + b 0 = 3(3) + b b = −9 The line parallel to f (x) that passes through (3, 0) is g(x) = 3x − 9. Analysis We can confirm that the two lines are parallel by graphing them. Figure 20 shows that the two lin...
2 is perpendicular to f (x) = 2x + 4 and passes through the point (4, 0). Be aware that perpendicular 2 lines may not look obviously perpendicular on a graphing calculator unless we use the square zoom feature. Q & A… A horizontal line has a slope of zero and a vertical line has an undefined slope. These two lines are...
points. 2. Find the negative reciprocal of the slope. 3. Use the slope-intercept form or point-slope form to write the equation by substituting the known values. 4. Simplify. Example 11 Finding the Equation of a Line Perpendicular to a Given Line Passing through a Point A line passes through the points (−2, 6) and (4,...
= 5 − t 4t = 9 t = 9 __ 4 9 __ This tells us the lines intersect when the input is. 4 We can then find the output value of the intersection point by evaluating either function at this input.  = 5 − 9 9 __ __ j  4 4 11 __ 4 = 9 __ These lines intersect at the point , 4 11 __ . 4 Analysis Looking at Figure 22, this ...
coordinate pair of the point of intersection, set the two equations equal, and solve for x. C(x) = R(x) 250,000 + 120x = 140x 250,000 = 20x 12,500 = x x = 12,500 To find y, evaluate either the revenue or the cost function at 12,500. R(x) = 140(12,500) = $1,750,000 The break-even point is (12,500, 1,750,000). Analysis ...
-intercepts of each equation 12. f (x) = −x + 2 15. k(x) = −5x + 1 13. g(x) = 2x + 4 16. −2x + 5y = 20 8. 3y + 4x = 12 −6y = 8x + 1 11. y = 3 __ x + 1 4 −3x + 4y = 1 14. h(x) = 3x − 5 17. 7x + 2y = 56 For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Li...
) = −2x − 1 29. Find the point at which the line f (x) = 2x + 5 intersects the line g(x) = −x. intersects the line g(x) = −3x − 5. 30. Use algebra to find the point at which the line 73 __. 10 intersects h(x) = 9 __ x + 4 f (x) = − 4 __ x + 5 274 ___ 25 31. Use algebra to find the point at which the line f (x) = 7 __ x...
3 49. p(t) = −2 + 3t 52. r(x) = 4 y x _ __ = 1 − 55. 4 3 58. 3y = 12 59. If g(x) is the transformation of f (x) = x after a 3 __, a shift right by 2, and vertical compression by 4 a shift down by 4 a. Write an equation for g(x). b. What is the slope of this line? c. Find the y-intercept of this line. 60. If g(x) is th...
to the line g(x) = −0.01x + 2.01 through the point (1, 2). For the following exercises, use the functions f (x) = −0.1x + 200 and g(x) = 20x + 0.1. 72. Find the point of intersection of the lines f and g. 73. Where is f (x) greater than g(x)? Where is g(x) greater than f (x)? ReAl-WORlD APPlICATIOnS 74. A car rental c...
typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them: 1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system. 2. Carefully read the problem to id...
the output to zero, and solve for the input. 0 = −400t + 3500 t = 3500 ____ 400 = 8.75 The x-intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks. When modeling any real-life scenario with functions, there is typ...
= 4 The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan. Using a Given Input and Output to Build a Model Many real-world applications are not as direct as the ones we just considered. Instead they require us ...
the year 2004 would correspond to t = 0, giving the point (0, 6200). Notice that through our clever choice of variable definition, we have “given” ourselves the y-intercept of the function. The year 2009 would correspond to t = 5, giving the point (5, 8100). The two coordinate pairs are (0, 6200) and (5, 8100). Recall...
the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful. Example 2 Using a Diagram to Model Distance Walked Anna and Emanuel start at the s...
, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure. We can then define a third variable, D, to be the measurement of the distance between Anna and Emanuel. Showing the variables ...
4. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates (30, 10), and Eastborough at (20, 0). Agritown (30, 10) (0, 0) Westborough (20, 0) Eastborough 20 miles Figure 4 Using this point along w...
with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 5. Exactly one solution x g Infinitely many solutions y f (a) y y x g f x No solutions (c) (b) Figure ...
the intersection, and then see where the K(d) function is smaller. These graphs are sketched in Figure 6, with K(d) in red. $ 120 110 100 90 80 70 60 50 40 30 20 10 0 K(d) = 0.59d + 20 (100, 80) M(d) = 0.63d + 16 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 d Figure 6 To find the intersection, we set the e...
x, and the line perpendicular to 7 f (x) that passes through the origin. x-axis, the line g (x) = 2, the line f (x) = 3x, and the line parallel to f (x) passing through (6, 1). For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,...
your answer. For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted. 25. Find the linear function that models the number of people inflicted with the co...
will be (or were) equal in what year? (The answer might be absurd.) For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 3. Assume that the house values are changing linearly. Year 1950 2000 Indiana $37,700 $94,300 Table 3 Alabama $27,100 $85,100 42. In...
the number of monthly minutes used. 48. A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly co...
is constant, when will the world’s oil reserves be depleted? 174 CHAPTER 2 linear Functions 53. You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you hav...
model to make predictions. 2.4 FITTInG lIneAR MODelS TO DATA A professor is attempting to identify trends among final exam scores. His class has a mixture of students, so he wonders if there is any relationship between age and final exam scores. One way for him to analyze the scores is by creating a diagram that relat...
the data. Then we can extend the line until we can verify the y-intercept. We can approximate the slope of the line by extending it until we can estimate the rise ___. run Example 2 Finding a Line of Best Fit Find a linear function that fits the data in Table 1 by “eyeballing” a line that seems to fit. Solution On a g...
model no longer applies after a certain point, it is sometimes called model breakdown. For example, predicting a cost function for a period of two years may involve examining the data where the input is the time in years and the output is the cost. But if we try to extrapolate a cost when x = 50, that is in 50 years, ...
data from Table 1, what temperature can we predict it is if we counted 20 chirps in 15 seconds? Finding the Line of Best Fit Using a Graphing Utility While eyeballing a line works reasonably well, there are statistical techniques for fitting a line to data that minimize the differences between the line and data values...
irps vs. Temperature 0 10 20 30 40 50 c, Number of Chirps Figure 6 Q & A… Will there ever be a case where two different lines will serve as the best fit for the data? No. There is only one best fit line. 180 CHAPTER 2 linear Functions Distinguishing Between linear and non-linear Models As we saw above with the cricket-...
will also provide you with the correlation coefficient, r = 0.9509. This value is very close to 1, which suggests a strong increasing linear relationship. Note: For some calculators, the Diagnostics must be turned “on” in order to get the correlation coefficient when linear regression is performed: [2nd]>[0]>[alpha][x...
ress) • linear Regression (http://openstaxcollege.org/l/linearregress) 13 http://www.bts.gov/publications/national_transportation_statistics/2005/html/table_04_10.html 182 CHAPTER 2 linear Functions 2.4 SeCTIOn exeRCISeS VeRBAl 1. Describe what it means if there is a model breakdown when using a linear model. 2. What i...
Population 11,500 12,100 12,700 13,000 13,750 2000 2005 1990 1995 Temperature, °F Time, seconds 16 46 18 50 20 54 25 55 30 62 SECTION 2.4 section exercises 183 GRAPHICAl For the following exercises, match each scatterplot with one of the four specified correlations in Figure 9 and Figure 10. (a) (b) Figure 9 14. r = 0...
103 15 24 14 http://www.census.gov/hhes/socdemo/education/data/cps/historical/index.html. Accessed 5/1/2014. SECTION 2.4 section exercises 185 3 21.9 11 15.76 4 22.22 12 13.68 5 22.74 13 14.1 6 22.26 14 14.02 7 20.78 15 11.94 8 17.6 16 12.76 9 16.52 17 11.28 10 18.54 18 9.1 4 44.8 5 43.1 6 38.8 7 39 8 38 9 32.7 10 30....
, year) for specific recorded years: (2500, 2000), (2650, 2001), (3000, 2003), (3500, 2006), (4200, 2010) 37. Use linear regression to determine a function y, 38. Predict when the population will hit 8,000. where the year depends on the population. Round to three decimal places of accuracy. For the following exercises,...
and data values linear function a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line model breakdown when a model no longer applies after a certain point parallel lines two or more lines with the same slope perpendicular lines two lines that intersect at right ...
of a problem indicates whether a linear function is increasing, decreasing, or constant. See Example 2. • The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line. See Example 3 and Example 4. • The slope and ...
in the same manner, with the exception of using the negative reciprocal slope. See Example 10 and Example 11. • A system of linear equations may be solved setting the two equations equal to one another and solving for x. The y-value may be found by evaluating either one of the original equations using this x-value. • ...
equation is linear. 2x + 3y = 7 6x 2 − y = 5 3. Determine whether the function is increasing or 4. Determine whether the function is increasing or decreasing. f (x) = 7x − 2 decreasing. g(x) = −x + 2 5. Given each set of information, find a linear equation that satisfies the given conditions, if possible. Passes throu...
intercepts of the given equation 15. 7x + 9y = −63 16. f (x) = 2x − 1 For 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 lines parallel, perpendicular, or neither? 17. Line 1: Passes through (5, 11) and (10, 1) Line 2: Passes through (−1, 3) ...
2,000 0 5 10 15 20 25 30 x Figure 1 27. Find the linear function y, where y depends on x, the number of years since 1980. 28. Find and interpret the y-intercept. 192 CHAPTER 2 linear Functions For the following exercise, consider this scenario: In 2004, a school population was 1,700. By 2012 the population had grown t...
the plotted data. y 120 100 80 60 40 20 0 0 2 4 6 8 10 12 x For the following exercises, consider the data in Table 5, which shows the percent of unemployed in a city of people 25 years or older who are college graduates is given below, by year. Year Percent Graduates 2000 6.5 2002 7.0 Table 5 2005 7.4 2007 8.2 2010 9...
, find a linear equation satisfying the conditions, if possible. x-intercept at (−4, 0) and y-intercept at (0, −6) 6. Find the slope of the line in Figure 1. 7. Write an equation for line in Figure 2. y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 21 3 4 5 6 x –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 21 3 4...
3y = −1 19. A car rental company offers two plans for renting 20. Find the area of a triangle bounded by the y-axis, a car. Plan A: $25 per day and $0.10 per mile Plan B: $40 per day with free unlimited mileage How many miles would you need to drive for plan B to save you money? 21. A town’s population increases at a ...
8.5 2002 8.0 Table 4 2005 7.2 2007 6.7 2010 6.4 28. Determine whether the trend appears linear. If so, and assuming the trend continues, find a linear regression model to predict the percent of unemployed in a given year to three decimal places. 29. In what year will the percentage drop below 4%? 30. Based on the set ...
crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications. 197 Polynomial and Rational Functions 198 CHAPTER 3 polynomi...
5i because the principal root of 25 is the positive root. A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written a + bi where a is the real part and bi is the imaginary part. For example, 5 + 2i is a complex number. So, too, is 3 + 4 √ — 3i. 5 +...
the complex number is −2 and the imaginary part is 3i. We plot the ordered pair (−2, 3) to represent the complex number −2 + 3i as shown in Figure 1. i 3 2 1 –3 –2 –1 1 2 3 r Figure 1 complex plane In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in Figur...
2) + (−4 + 5)i = 5 + i Try It #3 Subtract 2 + 5i from 3 − 4i. Multiplying Complex numbers Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately. Multiplying a Complex Numbers by a Real Number Let’s begin by multiplying a comple...
3 ⋅ (−5)) + (4 ⋅ (−5) + 3 ⋅ 2)i = (8 + 15) + (−20 + 6)i = 23 − 14i Try It #5 Multiply (3 − 4i)(2 + 3i). Dividing Complex numbers Division of two complex numbers is more complicated than addition, subtraction, and multiplication because we cannot divide by an imaginary number, meaning that any fraction must have a real...
a complex number a + bi is a − bi. It is found by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged. • When a complex number is multiplied by its complex conjugate, the result is a real number. • When a complex number is added to its complex conjugate, the res...
i − 4i − (−1) Because i2 = −1 = 3 + 22i ______ 17 = 3 __ 17 + 22 ___ i 17 Separate real and imaginary parts. Note that this expresses the quotient in standard form. 204 CHAPTER 3 polynomial and rational Functions Example 8 Substituting a Complex Number into a Polynomial Function Let f(x) = x2 − 5x + 2. Evaluate f (3 + ...
_____ x − 4 Simplifying Powers of i The powers of i are cyclic. Let’s look at what happens when we raise i to increasing powers. i1 = i i2 = −1 i3 = i2 ⋅ i = −1 ⋅ i = −i i4 = i3 ⋅ i = −i ⋅ i = −i2 = −(−1) = 1 i5 = i4 ⋅ i = 1 ⋅ i = i SECTION 3.1 complex numBers 205 We can see that when we get to the fifth power of i, it...
i19 Table 1 Each of these will eventually result in the answer we obtained above but may require several more steps than our earlier method. Access these online resources for additional instruction and practice with complex numbers. • Adding and Subtracting Complex numbers (http://openstaxcollege.org/l/addsubcomplex) ...
) − (3 + 2i) 21. (−4 + 4i) − (−6 + 9i) 22. (2 + 3i)(4i) 23. (5 − 2i)(3i) 26. (2 + 3i)(4 − i) 29. (3 + 4i)(3 − 4i) 32. −5 + 3i ______ 2i 24. (6 − 2i)(5) 25. (−2 + 4i)(8) 27. (−1 + 2i)(−2 + 3i) 28. (4 − 2i)(4 + 2i) 30. 3 + 4i _____ 2 33. 6 + 4i _____ i 31. 6 − 2i _____ 3 34. 2 − 3i _____ 4 + 3i SECTION 3.1 section exerci...
+ i 50. 1 __ i11 − 1 __ i21 51. i7(1 + i2) 53. (2 + i)(4 − 2i) ___________ (1 + i) 54. (1 + 3i)(2 − 4i) ____________ (1 + 2i) 56. 3 + 2i _____ 2 + i + (4 + 3i) 57. 4 + i ____ i + 3 − 4i _____ 1 − i 208 CHAPTER 3 polynomial and rational Functions leARnInG OBjeCTIVeS In this section, you will: • • • • Recognize characte...
cept 4 Vertex –6 Figure 2 SECTION 3.2 Quadratic Functions 209 The y-intercept is the point at which the parabola crosses the y-axis. The x-intercepts are the points at which the parabola crosses the x-axis. If they exist, the x-intercepts represent the zeros, or roots, of the quadratic function, the values of x at whic...
4 ____ = −2. This 2(1) also makes sense because we can see from the graph that the vertical line x = −2 divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, (−2, −1). The x-intercepts, tho...
Figure 5, h < 0, so the graph is shifted 2 units to the left. The magnitude of a indicates the stretch of the graph. If ∣a∣ > 1, the point associated with a particular x-value shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But if ∣a∣ < 1, the point associated ...
parabola; this value is k. 2. Substitute the values of the horizontal and vertical shift for h and k. in the function f(x) = a(x − h)2 + k. 3. Substitute the values of any point, other than the vertex, on the graph of the parabola for x and f (x). 4. Solve for the stretch factor, ∣a∣. 5. If the parabola opens up, a > ...
1 = 2 select TBLSET, then use TblStart = − 6 and ΔTbl = 2, and select TABLE. See Table 1. x y −6 5 −4 −1 −2 −3 0 −1 2 5 Table 1 The ordered pairs in the table correspond to points on the graph. Try It #1 A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8 Find an equation for the...
and then in standard form. Finding the Domain and Range of a Quadratic Function Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola wi...
a Quadratic Function Find the domain and range of f (x) = −5x2 + 9x − 1. Solution As with any quadratic function, the domain is all real numbers. Because a is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x-value of the vertex. h = − ...
area? Solution Let’s use a diagram such as Figure 10 to record the given information. It is also helpful to introduce a temporary variable, W, to represent the width of the garden and the length of the fence section parallel to the backyard fence. Garden L W Backyard Figure 10 a. We know we have only 80 feet of fence ...
maximum. 1. Write a quadratic equation for revenue. 2. Find the vertex of the quadratic equation. 3. Determine the y-value of the vertex. Example 6 Finding Maximum Revenue The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For exa...
8 The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function. maximum revenue = −2,500(31.8)2 + 159,000(31.8) = 2,528,100 Analysis This could also be solved by graphing the quadratic as in Figure 12. We...
f(x) = 0. 0 = 3x2 + 5x − 2 f(0) = 3(0)2 + 5(0) − 2 = −2 In this case, the quadratic can be factored easily, providing the simplest method for solution. 1 __, 0  and (−2, 0). So the x-intercepts are at  3 0 = (3x − 1)(x + 2) 0 = 3x − 1 1 __ x = 3 0 = x + 2 or x = −2 218 CHAPTER 3 polynomial and rational Functions Ana...
− h)2 + k h = − b __ 2a = − 4 ___ 2(2) = −1 k = f (−1) = 2(−1)2 + 4(−1) − 4 = −6 So now we can rewrite in standard form. We can now solve for when the output will be zero. f(x) = 2(x + 1)2 − 6 0 = 2(x + 1)2 − 6 6 = 2(x + 1)2 The graph has x-intercepts at (−1 − √ 3, 0) and (−x + 1)2 3 x = −1 ± √ 3, 0). — — 3 SECTION 3....
____ 2 — and −1 ___ 2 − — 7 i √ ____. 2 Example 10 Applying the Vertex and x-Intercepts of a Parabola A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation H(t) = −16t 2 + 80t + 40. a. When does the ball reach the...
H(t) = −16t 2 + 96t + 112. a. When does the rock reach the maximum height? b. What is the maximum height of the rock? c. When does the rock hit the ocean? Access these online resources for additional instruction and practice with quadratic equations. • Graphing Quadratic Functions in General Form (http://openstaxcolle...
f(x) = −x 2 + 4x + 3 1 __ x 2 + 3x + 1 19. f(x) = 2 20. f(x) = − 1 __ x 2 − 2x + 3 3 For the following exercises, determine the domain and range of the quadratic function. 21. f(x) = (x − 3)2 + 2 22. f(x) = −2(x + 3)2 − 6 23. f(x) = x 2 + 6x + 4 24. f(x) = 2x 2 − 4x + 2 25. k(x) = 3x 2 − 6x − 9 For the following exerc...
. (h, k) = (3, 2), (x, y) = (10, 1) 51. (h, k) = (0, 1), (x, y) = (1, 0) 52. (h, k) = (1, 0), (x, y) = (0, 1) 222 CHAPTER 3 polynomial and rational Functions GRAPHICAl For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts. 53. f(x) = x 2 − 2x 54. f(x...
axes f(x) = x 2, f(x) = x 2 + 2 and f(x) = x 2, f(x) = x 2 + 5 and f(x) = x 2 − 3. What appears to be the effect of adding a constant? 72. Graph on the same set of axes f(x) = x 2, f(x) = (x − 2)2, f(x − 3)2, and f(x) = (x + 4)2. What appears to be the effect of adding or subtracting those numbers? 73. The path of an ...
f(x) = −x 2. Vertex is on the y-axis. Vertex is on the y-axis. 83. Contains (4, 3) and has the shape of f(x) = 5x 2. 84. Contains (1, −6) has the shape of f(x) = 3x 2. Vertex Vertex is on the y-axis. has x-coordinate of −1. ReAl-WORlD APPlICATIOnS 85. Find the dimensions of the rectangular corral producing the greates...
estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest? 224 CHAPTER 3 polynomial and rational Functions leARnInG OBjeCTIVeS In this section, you will: • • • • Identify power functions. Identify end be...
power function? No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function. Example 1 Identifying Power Functions Which of the following functions are power functions? f (x) = 1 f (x) ...
infinity. We use the symbol ∞ for positive infinity and −∞ for negative infinity. When we say that “x approaches infinity,” which can be symbolically written as x → ∞, we are describing a behavior; we are saying that x is increasing without bound. With the even-power function, as the input increases or decreases witho...
0 x x x → −∞, f (x) → ∞ and x → ∞, f (x) → ∞ y x → −∞, f (x) → −∞ and x → ∞, f (x) → ∞ y Negative constant k < 0 x x x → −∞, f (x) → −∞ and x → ∞, f (x) → −∞ x → −∞, f (x) → ∞ and x → ∞, f (x) → −∞ Figure 4 SECTION 3.3 power Functions and polynomial Functions 227 How To… Given a power function f (x) = kx n where n is ...
–7 –8 –5 –4 –3 –2 21 3 4 5 x Figure 6 228 CHAPTER 3 polynomial and rational Functions Analysis We can check our work by using the table feature on a graphing utility. x −10 −5 0 5 10 f (x) 1,000,000,000 1,953,125 0 −1,953,125 −1,000,000,000 Table 2 We can see from Table 2 that, when we substitute very small values for...

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