Exercises

Describe the following characteristics of the graph of each parent function: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing. (Example 1)

1. f (x) = !x" 2. f (x) = 1 _ x 3. f (x) = x 3

4. f (x) = x 4 5. f (x) = c 6. f (x) = x

Use the graph of f (x) = √ " x to graph each function. (Example 2)

7. g(x) = √ $$$ x - 4 8. g(x) = √ $$$ x + 3

9. g(x) = √ $$$ x + 6 - 4 10. g(x) = √ $$$ x - 7 + 3

Use the graph of f (x) = 1 _ x to graph each function. (Example 2)

11. g(x) = 1 _ x + 4 12. g(x) = 1 _

x - 6

13. g(x) = 1 _ x - 6 + 8 14. g(x) = 1 _

x + 7 - 4

Describe how the graphs of f (x) = !x" and g(x) are related. Then write an equation for g(x). (Example 3)

15. 16.

y

xg(x)

y

x

g(x)

17. 18.

y

x

g(x)

y

xg(x)

19 PROFIT An automobile company experienced an unexpected two-month delay on manufacturing of a new car. The projected profit of the car sales before the delay p(x) is shown below. Describe how the graph of p(x) and the graph of a projection including the delay d(x) are related. Then write an equation for d(x). (Example 3)

Projected First-Quarter ProfitProjected First-Quarter Profit

Prof

it (th

ousa

nds

of d

olla

rs)

0

40

80

120

Months After January1 2 3 x

y

p(x) = 10x 3 - 70x 2 +150x - 2

Describe how the graphs of f (x) = |x| and g(x) are related. Then write an equation for g(x). (Example 3)

20. 21. y

x

−4

−8

−4

8

4

4 8 12

g(x)

y

x

−4

−8

−8 −4

8

4

4 8

g(x)

22. 23. y

x

−4

−8

−8 −4

8

4

4 8

g(x)

y

x

−4

−8

−8 −4

8

4

4 8

g(x)

Identify the parent function f (x) of g(x), and describe how the graphs of g(x) and f (x) are related. Then graph f (x) and g(x) on the same axes. (Example 4)

24. g(x) = 3|x| - 4 25. g(x) = 3 √ $$$ x + 8

26. g(x) = 4 _ x + 1

27. g(x) = 2!x - 6"

28. g(x) = -5!x - 2" 29. g(x) = -2|x + 5|

30. g(x) = 1 _ 6x + 7 31. g(x) =

√ $$$ x + 3 _

4

Graph each function. (Example 5)

32. f (x) = ⎧

⎨ ⎩

- x 2 if x < -2 3 if -2 ≤ x < 7 (x - 5) 2 + 2 if x ≥ 7

33. g(x) = ⎧

⎨ ⎩

x + 4 if x < -6 1 _ x if -6 ≤ x < 4 6 if x ≥ 4

34. f (x) = ⎧

⎨ ⎩

4 if x < -5 x 3 if -2 ≤ x ≤ 2 √ $$$ x + 3 if x > 3

35. h(x) = ⎧

⎨ ⎩

|x - 5| if x < -3 4x - 3 if -1 ≤ x < 3 √ $ x if x ≥ 4

36. g(x) = ⎧

⎨ ⎩

2 if x < -4 x 4 - 3 x 3 + 5 if -1 ≤ x < 1 !x" + 1 if x ≥ 3

37. f (x) = ⎧

⎨ ⎩

-3x -1 if x ≤ -1 0.5x + 5 if -1 < x ≤ 3 -|x – 5| + 3 if x > 3

52 | Lesson 1-5 | Parent Functions and Transformations

0045_0055_AM_S_C01_L05_664291.indd 520045_0055_AM_S_C01_L05_664291.indd 52 4/30/12 6:22 PM4/30/12 6:22 PM

38. POSTAGE The cost of a first-class postage stamp in the U.S. from 1988 to 2008 is shown in the table below. Use the data to graph a step function. (Example 5)

Year Price (¢)

1988 25

1991 29

1995 32

1999 33

2001 34

2002 37

2006 39

2007 41

2008 42

39. BUSINESS A no-contract cell phone company charges a flat rate for daily access and $0.10 for each minute. The cost of the plan can be modeled by c(x) = 1.99 + 0.1!x", where x is the number of minutes used. (Example 6)

a. Describe the transformation(s) of the parent function f (x) = !x" used to graph c(x).

b. The company offers another plan in which the daily access rate is $2.49, and the per-minute rate is $0.05. What function c(x) can be used to describe the second plan?

c. Graph both functions on the same graphing calculator screen.

d. Would the cost of the plans ever equal each other? If so, at how many minutes?

40. GOLF The path of a drive can be modeled by the function shown, where g(x) is the vertical distance in feet of the ball from the ground and x is the horizontal distance in feet such that x = 0 corresponds to the initial point. (Example 6)

Drive PathDrive Path

x

y

Vert

ical D

istan

ce (f

t)

0

8

16

Horizontal Distance (ft)100 200 300 400

g(x) = 0.176x - 0.0004x 2

a. Describe the transformation(s) of the parent function f (x) = x 2 used to graph g(x).

b. If a second golfer hits a similar shot 30 feet farther down the fairway from the first player, what function h(x) can be used to describe the second golfer’s shot?

c. Graph both golfers’ shots on the same graphing calculator screen.

d. If both golfers hit their shots at the same time, at what horizontal and vertical distances will the shots cross paths?

Use the graph of f (x) to graph g(x) = | f (x)| and h(x) = f (|x|). (Example 7)

41. f (x) = 2 _ x 42. f (x) = √ $$$ x - 4

43. f (x) = x 4 - x 3 - 4 x 2 44. f (x) = 1 _ 2 x 3 + 2 x 2 - 8x - 2

45. f (x) = 1 _ x - 3 + 5 46. f (x) = √ $$$ x + 2 - 6

47. TRANSPORTATION In New York City, the standard cost for taxi fare is shown. One unit is equal to a distance of 0.2 mile or a time of 60 seconds when the car is not in motion.

a. Write a greatest integer function f (x) that would represent the cost for units of cab fare, where x > 0. Round to the nearest unit.

b. Graph the function.

c. How would the graph of f (x) change if the fare for the first unit increased to $3.70? Graph the new function.

48. PHYSICS The potential energy in joules of a spring that has been stretched or compressed is given by p(x) = c x 2 _ 2 , where c is the spring constant and x is the distance from the equilibrium position. When x is negative, the spring is compressed, and when x is positive, the spring is stretched.

a. Describe the transformation(s) of the parent function f (x) = x 2 used to graph p(x).

b. The graph of the potential energy for a second spring passes through the point (3, 315). Find the spring constant for the spring and write the function for the potential energy.

Write and graph the function with the given parent function and characteristics.

49 f (x) = 1 _ x ; expanded vertically by a factor of 2; translated 7 units to the left and 5 units up

50. f (x) = !x"; expanded vertically by a factor of 3; reflected in the x-axis; translated 4 units down

PHYSICS The distance an object travels as a function of time is given by f (t) = 1 _

2 a t 2 + v 0 t + x 0 , where a is the

acceleration, v 0 is the initial velocity, and x 0 is the initial position of the object. Describe the transformations of the parent function f (t) = t 2 used to graph f (t) for each of the following.

51. a = 2, v 0 = 2, x 0 = 0 52. a = 2, v 0 = 0, x 0 = 10

53. a = 4, v 0 = 8, x 0 = 1 54. a = 3, v 0 = 5, x 0 = 3

ion f (x) that would

$2.50 per trip plus$0.40 per unit

53connectED.mcgraw-hill.com

0045_0055_AM_S_C01_L05_664291.indd 530045_0055_AM_S_C01_L05_664291.indd 53 4/30/12 6:22 PM4/30/12 6:22 PM

Write an equation for each g(x).

55. 56.

y

−4

−8

−4−8

8

4

4 8 x

(4, 6)

g(x)

f (x)

y

−8

−16

−4−8

16

8

4 8 x(3, -4)

g(x)f (x)

57. 58. y

−4

−8

−4−8

8

4

4 8 x

(0, 2)

g(x)

f (x)

y

xg(x)

f (x)

59. SHOPPING The management of a new shopping mall originally predicted that attendance in thousands would follow f (x)= √ " 7x for the first 60 days of operation, where x is the number of days after opening and x = 1 corresponds with opening day. Write g(x) in terms of f (x) for each situation below.

a. Attendance was consistently 12% higher than expected. b. The opening was delayed 30 days due to construction. c. Attendance was consistently 450 less than expected.

Identify the parent function f (x) of g(x), and describe the transformation of f (x) used to graph g(x).

60. 61.

y

−4

−12

−4−8 4 8 x

y = g(x)

(-2, -5)

y

−4

−8

−4−8

8

4

4 8 xy = g(x)

(3, 5)

62. 63.

y

−4

−8

−4−8

8

4

4 8 xy = g(x)

(4, -5)

y

−4

−8

8

4

6020 80x

y = g(x)

(3, 5)(7, 3)

Use f (x) to graph g(x).

64. g(x) = 0.25f (x) + 4

65. g(x) = 3f (x) - 6

66. g(x) = f (x - 5) + 3

67. g(x) = -2f (x) + 1

y

x

−4

−8

−4−8

8

4

4 8

f (x)

Use f (x) = 8 _ √ """ x + 6

- 4 to graph each function.

68. g(x) = 2f (x) + 5 69. g(x) = -3f (x) + 6

70. g(x) = f (4x) - 5 71. g(x) = f (2x + 1) + 8

72. MULTIPLE REPRESENTATIONS In this problem, you will investigate operations with functions. Consider

• f (x) = x 2 + 2x + 7, • g(x) = 4x + 3, and • h(x) = x 2 + 6x + 10.

a. TABULAR Copy and complete the table below for three values for a.

a f(a) g(a) f(a) + g(a) h(a)

b. VERBAL How are f (x), g(x), and h(x) related?

c. ALGEBRAIC Prove the relationship from part b algebraically.

H.O.T. Problems Use Higher-Order Thinking Skills

73. ERROR ANALYSIS Danielle and Miranda are describing the transformation g(x) = #x + 4$. Danielle says that the graph is shifted 4 units to the left, while Miranda says that the graph is shifted 4 units up. Is either of them correct? Explain.

74. REASONING Let f (x) be an odd function. If g(x) is a reflection of f (x) in the x-axis and h(x) is a reflection of g(x) in the y-axis, what is the relationship between f (x) and h(x)? Explain.

75. WRITING IN MATH Explain why order is important when transforming a function with reflections and translations.

REASONING Determine whether the following statements are sometimes, always, or never true. Explain your reasoning.

76. If f (x) is an even function, then f (x) = | f (x)|.

77. If f (x) is an odd function, then f (-x) = - | f (x)|.

78. If f (x) is an even function, then f (-x) = - | f (x)|.

79 CHALLENGE Describe the transformation of f (x) = √ " x if (-2, -6) lies on the curve.

80. REASONING Suppose (a, b) is a point on the graph of f (x). Describe the difference between the transformations of (a, b) when the graph of f (x) is expanded vertically by a factor of 4 and when the graph of f (x) is compressed horizontally by a factor of 4.

81. WRITING IN MATH Use words, graphs, tables, and equations to relate parent functions and transformations. Show this relationship through a specific example.

54 | Lesson 1-5 | Parent Functions and Transformations

0045_0055_AM_S_C01_L05_664291.indd 540045_0055_AM_S_C01_L05_664291.indd 54 4/30/12 6:22 PM4/30/12 6:22 PM

Find the average rate of change of each function on the given interval. (Lesson 1-4)

82. g(x) = -2 x 2 + x - 3; [-1, 3] 83. g(x) = x 2 - 6x + 1; [4, 8] 84. f (x) = -2 x 3 - x 2 + x - 4; [-2, 3]

Use the graph of each function to describe its end behavior. Support the conjecture numerically. (Lesson 1-3)

85. q(x) = - 12 _ x 86. f (x) = 0.5 _

x 2 87. p(x) = x + 2 _

x - 3

Use the graph of each function to estimate its y-intercept and zero(s). Then find these values algebraically. (Lesson 1-2)

88. 89. 90.

x

−4

−8

−4−8

8

4

4 8

y = (x - 4)2 - 3

y

x

−4

−8

−4−8

8

4

4 8

y

y = x 3 - x 2 - 2x

x

−4

−8

−4−8

8

4

4 8

y

y = √x - 2 - 1

91. RETAIL Amparo bought a jacket with a gift certificate she received as a birthday present. The jacket was marked 33% off, and the sales tax in her area is 5.5%. If she paid $45.95 for the jacket, use composition of functions to determine the original price of the jacket. (Lesson 1-1)

92. Points A(2, 5) and B(7,8) lie on line ℓ. What is the standard form of the equation of line ℓ? (Lesson 1-4)

93. DEMOGRAPHICS In July 1990, the population of Georgia was 6,506,416. By July 1997, the population had grown to 7,486,242. (Lesson 1-4)

a. If x represents the year and y represents the population, find the average annual rate of increase of the population. b. Write an equation to model the population change.

Spiral Review

94. SAT/ACT The figure shows the graph of y = g(x), which has a minimum located at (1, -2). What is the maximum value of h(x) = -3g(x) - 1?

y

x

y = g(x)−2

−2

4

2

42

A 0 D 3 B 1 E C 2

95. REVIEW What is the simplified form of 4 x 3 y 2 z -1

_ ( x -2 y 3 z 2 ) 2

?

96. What is the range of y = x 2 + 8 _ 2 ?

F {y | y ≠ ±2 √ $ 2 } G {y | y ≥ 4}

H {y | y ≥ 0}

J {y | y ≤ 0}

97. REVIEW What is the effect on the graph of y = k x 2 as k decreases from 3 to 2?

A The graph of y = 2 x 2 is a reflection of the graph of y = 3 x 2 across the y-axis.

B The graph is rotated 90° about the origin. C The graph becomes narrower. D The graph becomes wider.

Skills Review for Standardized Tests

It cannot be determined from the information given.

55connectED.mcgraw-hill.com

0045_0055_AM_S_C01_L05_664291.indd 550045_0055_AM_S_C01_L05_664291.indd 55 5/25/12 6:38 PM5/25/12 6:38 PM