Polynomial functions are classified by degree. For instance, a constant func- tion has degree 0 and a linear function has degree 1. In this section, y...

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Quadratic Functions and Models

What you should learn • Analyze graphs of quadratic functions. • Write quadratic functions in standard form and use the results to sketch graphs of functions. • Use quadratic functions to model and solve real-life problems.

The Graph of a Quadratic Function In this and the next section, you will study the graphs of polynomial functions. In Section 1.6, you were introduced to the following basic functions. f x ax b

Linear function

f x c

Constant function

f x x

Squaring function

2

These functions are examples of polynomial functions.

Why you should learn it Quadratic functions can be used to model data to analyze consumer behavior. For instance, in Exercise 83 on page 137, you will use a quadratic function to model the revenue earned from manufacturing handheld video games.

Definition of Polynomial Function Let n be a nonnegative integer and let an, an1, . . . , a2, a1, a0 be real numbers with an 0. The function given by f x an x n an1 x n1 . . . a 2 x 2 a1 x a 0 is called a polynomial function of x with degree n. Polynomial functions are classified by degree. For instance, a constant function has degree 0 and a linear function has degree 1. In this section, you will study second-degree polynomial functions, which are called quadratic functions. For instance, each of the following functions is a quadratic function. f x x 2 6x 2 gx 2x 12 3 hx 9 14 x 2 kx 3x 2 4 mx x 2x 1 Note that the squaring function is a simple quadratic function that has degree 2.

Definition of Quadratic Function © John Henley/Corbis

Let a, b, and c be real numbers with a 0. The function given by f x ax 2 bx c

Quadratic function

is called a quadratic function.

The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter.

The graph of a quadratic function is a special type of “U”-shaped curve called a parabola. Parabolas occur in many real-life applications—especially those involving reflective properties of satellite dishes and flashlight reflectors. You will study these properties in Section 10.2.

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All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola, as shown in Figure 2.1. If the leading coefficient is positive, the graph of f x ax 2 bx c is a parabola that opens upward. If the leading coefficient is negative, the graph of f x ax 2 bx c is a parabola that opens downward. y

y

Opens upward

f ( x) = ax 2 + bx + c, a < 0 Vertex is highest point

Axis

Axis Vertex is lowest point

f ( x) = ax 2 + bx + c, a > 0 x

x

Opens downward Leading coefficient is positive. FIGURE 2.1

Leading coefficient is negative.

The simplest type of quadratic function is f x ax 2. Its graph is a parabola whose vertex is (0, 0). If a > 0, the vertex is the point with the minimum y-value on the graph, and if a < 0, the vertex is the point with the maximum y-value on the graph, as shown in Figure 2.2. y

Exploration Graph y ax 2 for a 2, 1, 0.5, 0.5, 1, and 2. How does changing the value of a affect the graph?

3

3

2

2

1 −3

−2

−1

1

f (x) = ax 2, a > 0 x 1

−1

Graph y x h2 for h 4, 2, 2, and 4. How does changing the value of h affect the graph? Graph y x 2 k for k 4, 2, 2, and 4. How does changing the value of k affect the graph?

y

2

3

Minimum: (0, 0)

−3

−2

x

−1

1 −1

−2

−2

−3

−3

Leading coefficient is positive. 2.2

Maximum: (0, 0) 2

3

f (x) = ax 2, a < 0

Leading coefficient is negative.

FIGURE

When sketching the graph of f x ax 2, it is helpful to use the graph of y x 2 as a reference, as discussed in Section 1.7.

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Example 1

Sketching Graphs of Quadratic Functions

a. Compare the graphs of y x 2 and f x 13x 2. b. Compare the graphs of y x 2 and gx 2x 2.

Solution a. Compared with y x 2, each output of f x 13x 2 “shrinks” by a factor of 13, creating the broader parabola shown in Figure 2.3. b. Compared with y x 2, each output of gx 2x 2 “stretches” by a factor of 2, creating the narrower parabola shown in Figure 2.4.

y

y = x2

g (x ) = 2 x 2

y

4

4

3

3

f (x) = 13 x 2

2

2

1

1

y = x2 Emphasize the importance of comparing quadratic functions with the common function y x2, because it is very useful when analyzing functions and preparing to graph them.

−2 FIGURE

x

−1

1

2

2.3

−2 FIGURE

x

−1

1

2

2.4

Now try Exercise 9. In Example 1, note that the coefficient a determines how widely the parabola given by f x ax 2 opens. If a is small, the parabola opens more widely than if a is large. Recall from Section 1.7 that the graphs of y f x ± c, y f x ± c, y f x, and y f x are rigid transformations of the graph of y f x. For instance, in Figure 2.5, notice how the graph of y x 2 can be transformed to produce the graphs of f x x 2 1 and gx x 22 3.

y

2

g(x) = (x + 2) − 3 y

2

3

(0, 1) y = x2

2

f(x) = − x 2 + 1

−2

y = x2

1

x 2 −1

−4

−3

1

2

−2

−2

(−2, −3)

Reflection in x-axis followed by an upward shift of one unit FIGURE 2.5

x

−1

−3

Left shift of two units followed by a downward shift of three units

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The Standard Form of a Quadratic Function The standard form of a quadratic function identifies four basic transformations of the graph of y x 2.

a. The factor a produces a vertical stretch or shrink. b. If a < 0, the graph is reflected in the x-axis. c. The factor x h2 represents a horizontal shift of h units. d. The term k represents a vertical shift of k units.

To prepare for rewriting a quadraticfunction in standard form, review the process of completing the square for an algebraic expression, paying special attention to problems in which a 1.

The standard form of a quadratic function is f x ax h 2 k. This form is especially convenient for sketching a parabola because it identifies the vertex of the parabola as h, k.

Standard Form of a Quadratic Function The quadratic function given by f x ax h 2 k,

a0

is in standard form. The graph of f is a parabola whose axis is the vertical line x h and whose vertex is the point h, k. If a > 0, the parabola opens upward, and if a < 0, the parabola opens downward. To graph a parabola, it is helpful to begin by writing the quadratic function in standard form using the process of completing the square, as illustrated in Example 2. In this example, notice that when completing the square, you add and subtract the square of half the coefficient of x within the parentheses instead of adding the value to each side of the equation as is done in Appendix A.5.

Example 2

Graphing a Parabola in Standard Form

Sketch the graph of f x 2x 2 8x 7 and identify the vertex and the axis of the parabola.

Solution Begin by writing the quadratic function in standard form. Notice that the first step in completing the square is to factor out any coefficient of x2 that is not 1. f x 2x 2 8x 7 Write original function. 2x 2 4x 7

Factor 2 out of x-terms.

2x 2 4x 4 4 7

Add and subtract 4 within parentheses.

422

f (x) = 2(x + 2)2 − 1

After adding and subtracting 4 within the parentheses, you must now regroup the terms to form a perfect square trinomial. The 4 can be removed from inside the parentheses; however, because of the 2 outside of the parentheses, you must multiply 4 by 2, as shown below. f x 2x 2 4x 4 24 7 Regroup terms.

y 4 3

2x 2 4x 4 8 7

Simplify.

2

2x 2 1

Write in standard form.

1

−3

−1

(−2, −1) FIGURE

2.6

x = −2

2

y = 2x 2 x 1

From this form, you can see that the graph of f is a parabola that opens upward and has its vertex at 2, 1. This corresponds to a left shift of two units and a downward shift of one unit relative to the graph of y 2x 2, as shown in Figure 2.6. In the figure, you can see that the axis of the parabola is the vertical line through the vertex, x 2. Now try Exercise 13.

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To find the x-intercepts of the graph of f x ax 2 bx c, you must solve the equation ax 2 bx c 0. If ax 2 bx c does not factor, you can use the Quadratic Formula to find the x-intercepts. Remember, however, that a parabola may not have x-intercepts.

Example 3

Finding the Vertex and x-Intercepts of a Parabola

Sketch the graph of f x x 2 6x 8 and identify the vertex and x-intercepts.

Solution f x x 2 6x 8

Write original function.

x 6x 8

Factor 1 out of x-terms.

x 2 6x 9 9 8

Add and subtract 9 within parentheses.

2

622 y

f(x) = − (x −

3)2

+1

2

(3, 1) (4, 0) x

−1

1

3

Regroup terms.

x 32 1

Write in standard form.

From this form, you can see that f is a parabola that opens downward with vertex 3, 1. The x-intercepts of the graph are determined as follows.

1

(2, 0)

x 2 6x 9 9 8

5

−1

x 2 6x 8 0 x 2x 4 0

−2

y = − x2

−3 −4 FIGURE

Factor out 1. Factor.

x20

x2

Set 1st factor equal to 0.

x40

x4

Set 2nd factor equal to 0.

So, the x-intercepts are 2, 0 and 4, 0, as shown in Figure 2.7. Now try Exercise 19.

2.7

Example 4

y

Write the standard form of the equation of the parabola whose vertex is 1, 2 and that passes through the point 0, 0, as shown in Figure 2.8.

(1, 2) 2

Writing the Equation of a Parabola

y = f(x)

Solution Because the vertex of the parabola is at h, k 1, 2, the equation has the form 1

f x ax 12 2.

Substitute for h and k in standard form.

Because the parabola passes through the point 0, 0, it follows that f 0 0. So, (0, 0)

x 1

FIGURE

2.8

0 a0 12 2

a 2

Substitute 0 for x; solve for a.

which implies that the equation in standard form is f x 2x 12 2. Now try Exercise 43.

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Applications Many applications involve finding the maximum or minimum value of a quadratic function. You can find the maximum or minimum value of a quadratic function by locating the vertex of the graph of the function.

You may wish to mention that much of beginning calculus deals with finding maxima and minima.

Vertex of a Parabola

Activities 1. Describe the effect of a on the graph of f x ax 2 bx c. 2. Rewrite the quadratic function f x x 2 4x 1 in standard form. Answer: f x x 22 3 3. Write the standard form of the equation of the parabola that has vertex 3, 1 and passes through the point 0, 2. Answer: f x 13 x 32 1 4. Fit a quadratic model to the points 0, 2, 2, 0, and 0.5, 0. Answer: y 2x 2 3x 2

The vertex of the graph of f x ax 2 bx c is b . 2a b 2. If a < 0, has a maximum at x . 2a

Example 5

Height (in feet)

f(x) = − 0.0032x 2 + x + 3

Solution From the given function, you can see that a 0.0032 and b 1. Because the function has a maximum when x b2a, you can conclude that the baseball reaches its maximum height when it is x feet from home plate, where x is

80 60

(156.25, 81.125)

x

20 x 100

200

300

Distance (in feet) FIGURE

The Maximum Height of a Baseball

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 with respect to the ground. The path of the baseball is given by the function f x 0.0032x 2 x 3, where f x is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the baseball?

Baseball

40

.

1. If a > 0, has a minimum at x

y 100

b b , f 2a 2a

b b 1 x 156.25 feet. 2a 2a 20.0032

At this distance, the maximum height is f 156.25 0.0032156.25 2 156.25 3 81.125 feet. The path of the baseball is shown in Figure 2.9. Now try Exercise 77.

2.9

Example 6

Minimizing Cost

A small local soft-drink manufacturer has daily production costs of C 70,000 120x 0.075x2, where C is the total cost (in dollars) and x is the number of units produced. How many units should be produced each day to yield a minimum cost?

Solution Use the fact that the function has a minimum when x b2a. From the given function you can see that a 0.075 and b 120. So, producing x

b 120 800 units 2a 2(0.075

each day will yield a minimum cost. Now try Exercise 83.

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Polynomial and Rational Functions The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help.

Exercises

VOCABULARY CHECK: Fill in the blanks. 1. A polynomial function of degree n and leading coefficient an is a function of the form f x an x n an1 x n1 . . . a1x a0 an 0 where n is a ________ ________ and a1 are ________ numbers. 2. A ________ function is a second-degree polynomial function, and its graph is called a ________. 3. The graph of a quadratic function is symmetric about its ________. 4. If the graph of a quadratic function opens upward, then its leading coefficient is ________ and the vertex of the graph is a ________. 5. If the graph of a quadratic function opens downward, then its leading coefficient is ________ and the vertex of the graph is a ________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 8, match the quadratic function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), (f), (g), and (h).] y

(a)

y

(b)

6

6

4

4

2

2 x

−4

−4

2

(−1, −2)

(c)

x

−2

−4

(e)

−2

x −2

4

(c) hx

6

8

(c)

(c)

(b) gx x 2 1

1

(d) kx x 2 3 (b) gx 3x2 1

2 hx 13 x 3 f x 12x 22 1 2 gx 12x 1 3 1 hx 2x 22 1

(d) kx x 32

y

In Exercises 13–28, sketch the graph of the quadratic function without using a graphing utility. Identify the vertex, axis of symmetry, and x -intercept(s).

(2, 4)

2

(3, −2)

x

−2

2

(h)

6

y 4

13. f x x 2 5

14. hx 25 x 2

15. f x

1 16. f x 16 4 x 2

(0, 3)

17. f x x 52 6

18. f x x 62 3

19. hx x 2 8x 16

20. gx x 2 2x 1

21. f x x x

1 22. f x x 2 3x 4

25. hx −4 x 6

x −2 −4

4

5 4

23. f x x 2 2x 5

4

(2, 0)

1 2 2x

2

6

4

x2

11. (a) f x x 1

−6

6

y

2

(d) kx 3x 2

3 2 2x

2

−6

−2

1 (b) gx 8 x 2

1 9. (a) f x 2 x 2

12. (a)

4

−4

2

8. f x x 42

(b)

x

(g)

7. f x x 3 2

−4

(f)

−2

6. f x x 1 2 2

(d) kx 2x 1 2 4

y

4

5. f x 4 (x 2)2

10. (a) f x

2

−2

2 2

4

x

2 −6

2

(4, 0)

4

4. f x 3 x 2

2

(c) hx x 2 3

6

(− 4, 0)

3. f x

In Exercises 9–12, graph each function. Compare the graph of each function with the graph of y x2.

y

(d)

2. f x x 42

x2

2

(0, −2)

y

1. f x x 22

4

4x 2

4x 21

26. f x 2x 2 x 1 1 27. f x 4x 2 2x 12 1 28. f x 3x 2 3x 6

24. f x x 2 4x 1

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Section 2.1 In Exercises 29–36, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x -intercepts. Then check your results algebraically by writing the quadratic function in standard form. 29. f x x 2 2x 3

30. f x x 2 x 30

31. gx

32. f x x 2 10x 14

x2

8x 11

33. f x 2x 2 16x 31 1 2 2 x

35. gx

4x 2

50. Vertex: 52, 34 ; point: 2, 4

51. Vertex: 52, 0; point: 72, 16 3 3 52. Vertex: 6, 6; point: 61 10 , 2

34. f x 4x 2 24x 41

Graphical Reasoning In Exercises 53–56, determine the x -intercept(s) of the graph visually. Then find the x -intercepts algebraically to confirm your results.

36. f x 35x 2 6x 5

53. y x 2 16

54. y x 2 6x 9

y

In Exercises 37– 42, find the standard form of the quadratic function. y

37.

−8

(0, 1) (1, 0)

−2

2

2

6

(1, 0)

8

8 −4

(−1, 0)

8

y x

y

38.

6 4 2

x

4 x 2

−4

(0, 1) 2

(−1, 4) (−3, 0)

y 2

6 2

2

y

(−2, 2) (−3, 0)

x

8

2

6

x

−6 −4

In Exercises 57–64, use a graphing utility to graph the quadratic function. Find the x -intercepts of the graph and compare them with the solutions of the corresponding quadratic equation when f x 0.

y

42.

2

57. f x x 2 4x

(2, 0)

58. f x 2 x 2 10x

4

−6

−2

59. f x x 2 9x 18

(3, 2)

2

(−1, 0)

−2

2

(−2, −1)

41.

2

−4

−8

−6 −4

−4

x

−6 −4

8 −4

2 −2

x

−4

(0, 3) x

−2

6

56. y 2x 2 5x 3

y

40.

(1, 0) −4

4

y

4

y

39.

55. y x 2 4x 5

−6

x

−2

135

Quadratic Functions and Models

x 2

4

6

60. f x x 2 8x 20 61. f x 2x 2 7x 30 62. f x 4x 2 25x 21

In Exercises 43–52, write the standard form of the equation of the parabola that has the indicated vertex and whose graph passes through the given point. 43. Vertex: 2, 5; point: 0, 9 44. Vertex: 4, 1; point: 2, 3 45. Vertex: 3, 4; point: 1, 2 46. Vertex: 2, 3; point: 0, 2 47. Vertex: 5, 12; point: 7, 15 48. Vertex: 2, 2; point: 1, 0 49. Vertex:

14, 32

; point: 2, 0

1 63. f x 2x 2 6x 7

64. f x 10x 2 12x 45 7

In Exercises 65–70, find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given x -intercepts. (There are many correct answers.) 65. 1, 0, 3, 0

66. 5, 0, 5, 0

67. 0, 0, 10, 0

68. 4, 0, 8, 0

69. 3, 0,

12,

0

5 70. 2, 0, 2, 0

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In Exercises 71–74, find two positive real numbers whose product is a maximum. 71. The sum is 110.

(c) Use the result of part (b) to write the area A of the rectangular region as a function of x. What dimensions will produce a maximum area of the rectangle? 77. Path of a Diver

72. The sum is S. 73. The sum of the first and twice the second is 24. 74. The sum of the first and three times the second is 42. 75. Numerical, Graphical, and Analytical Analysis A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).

24 4 y x 2 x 12 9 9 where y is the height (in feet) and x is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver? 78. Height of a Ball ball is given by y

y x

The path of a diver is given by

The height y (in feet) of a punted foot-

16 2 9 x x 1.5 2025 5

where x is the horizontal distance (in feet) from the point at which the ball is punted (see figure).

x

(a) Write the area A of the corral as a function of x. (b) Create a table showing possible values of x and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area. (c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area. (d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area. (e) Compare your results from parts (b), (c), and (d). 76. Geometry An indoor physical fitness room consists of a rectangular region with a semicircle on each end (see figure). The perimeter of the room is to be a 200-meter single-lane running track. x

y x Not drawn to scale

(a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt? 79. Minimum Cost A manufacturer of lighting fixtures has daily production costs of C 800 10x 0.25x 2 where C is the total cost (in dollars) and x is the number of units produced. How many fixtures should be produced each day to yield a minimum cost? 80. Minimum Cost A textile manufacturer has daily production costs of

y

C 100,000 110x 0.045x 2 where C is the total cost (in dollars) and x is the number of units produced. How many units should be produced each day to yield a minimum cost?

(a) Determine the radius of the semicircular ends of the room. Determine the distance, in terms of y, around the inside edge of the two semicircular parts of the track. (b) Use the result of part (a) to write an equation, in terms of x and y, for the distance traveled in one lap around the track. Solve for y.

81. Maximum Profit The profit P (in dollars) for a company that produces antivirus and system utilities software is P 0.0002x 2 140x 250,000 where x is the number of units sold. What sales level will yield a maximum profit?

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Section 2.1 82. Maximum Profit The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model P 230 20x 0.5x 2. What expenditure for advertising will yield a maximum profit? 83. Maximum Revenue The total revenue R earned (in thousands of dollars) from manufacturing handheld video games is given by R p 25p2 1200p where p is the price per unit (in dollars). (a) Find the revenue earned for each price per unit given below. $20 $25 $30 (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 84. Maximum Revenue The total revenue R earned per day (in dollars) from a pet-sitting service is given by R p 12p2 150p where p is the price charged per pet (in dollars). (a) Find the revenue earned for each price per pet given below. $4 $6 $8 (b) Find the price that will yield a maximum revenue. What is the maximum revenue? Explain your results. 85. Graphical Analysis From 1960 to 2003, the per capita consumption C of cigarettes by Americans (age 18 and older) can be modeled by C 4299 1.8t 1.36t 2, 0 ≤ t ≤ 43 where t is the year, with t 0 corresponding to 1960. (Source: Tobacco Outlook Report) (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (c) In 2000, the U.S. population (age 18 and over) was 209,128,094. Of those, about 48,308,590 were smokers. What was the average annual cigarette consumption per smoker in 2000? What was the average daily cigarette consumption per smoker?

Quadratic Functions and Models

137

Model It 86. Data Analysis The numbers y (in thousands) of hairdressers and cosmetologists in the United States for the years 1994 through 2002 are shown in the table. (Source: U.S. Bureau of Labor Statistics)

Year

Number of hairdressers and cosmetologists, y

1994 1995 1996 1997 1998 1999 2000 2001 2002

753 750 737 748 763 784 820 854 908

(a) Use a graphing utility to create a scatter plot of the data. Let x represent the year, with x 4 corresponding to 1994. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. How well does the model fit the data? (d) Use the trace feature of the graphing utility to approximate the year in which the number of hairdressers and cosmetologists was the least. (e) Verify your answer to part (d) algebraically. (f) Use the model to predict the number of hairdressers and cosmetologists in 2008.

87. Wind Drag The number of horsepower y required to overcome wind drag on an automobile is approximated by y 0.002s 2 0.005s 0.029,

0 ≤ s ≤ 100

where s is the speed of the car (in miles per hour). (a) Use a graphing utility to graph the function. (b) Graphically estimate the maximum speed of the car if the power required to overcome wind drag is not to exceed 10 horsepower. Verify your estimate algebraically.

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88. Maximum Fuel Economy A study was done to compare the speed x (in miles per hour) with the mileage y (in miles per gallon) of an automobile. The results are shown in the table. (Source: Federal Highway Administration)

Speed, x

Mileage, y

15 20 25 30 35 40 45 50 55 60 65 70 75

22.3 25.5 27.5 29.0 28.8 30.0 29.9 30.2 30.4 28.8 27.4 25.3 23.3

92. Profit The profit P (in millions of dollars) for a recreational vehicle retailer is modeled by a quadratic function of the form P at 2 bt c where t represents the year. If you were president of the company, which of the models below would you prefer? Explain your reasoning. (a) a is positive and b2a ≤ t. (b) a is positive and t ≤ b2a. (c) a is negative and b2a ≤ t. (d) a is negative and t ≤ b2a. 93. Is it possible for a quadratic equation to have only one x-intercept? Explain. 94. Assume that the function given by f x ax 2 bx c, a 0 has two real zeros. Show that the x-coordinate of the vertex of the graph is the average of the zeros of f. (Hint: Use the Quadratic Formula.)

Skills Review (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. (d) Estimate the speed for which the miles per gallon is greatest.

Synthesis True or False? In Exercises 89 and 90, determine whether the statement is true or false. Justify your answer. 89. The function given by x-intercepts.

f x 12x 2 1 has no

90. The graphs of

In Exercises 95–98, find the equation of the line in slope-intercept form that has the given characteristics. 95. Passes through the points 4, 3 and 2, 1

96. Passes through the point 2, 2 and has a slope of 2 7

97. Passes through the point 0, 3 and is perpendicular to the line 4x 5y 10 98. Passes through the point 8, 4 and is parallel to the line y 3x 2 In Exercises 99–104, let f x 14x 3 and let g x 8x 2. Find the indicated value. 99. f g3 100. g f 2 101. fg 7 4

gf 1.5

f x 4x 2 10x 7

102.

and

103. f g1 104. g f 0

gx 12x 30x 1 2

3

have the same axis of symmetry. 91. Write the quadratic function f x ax 2 bx c in standard form to verify that the vertex occurs at

2ab , f 2ab .

105. Make a Decision To work an extended application analyzing the height of a basketball after it has been dropped, visit this text’s website at college.hmco.com.