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E. Transformations of a General Function

# E. Transformations of a General Function

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b. The graph of h is a cube root function, shifted right 2, stretched by a factor of 2, then shifted

down 1. This sequence is shown in Figures 2.30 through 2.32 and illustrate how the inflection

point has shifted from (0, 0) to 12, Ϫ12 .

Figure 2.30

y

5

Figure 2.31

3

y ϭ ͙x Ϫ 2

5

Figure 2.32

3

y y ϭ 2͙x

Ϫ2

5

3

y h(x) ϭ 2͙x

Ϫ2Ϫ1

(3, 2)

(3, 1)

(2, 0)

6 x

Inflection

(1, Ϫ1)

Ϫ4

(2, 0)

6

x

Ϫ4

(2, Ϫ1)

(1, Ϫ2)

6

x

(1, Ϫ3)

Ϫ5

Ϫ5

Ϫ5

Shifted right 2

(3, 1)

Ϫ4

Shifted down 1

Stretched by a factor of 2

Now try Exercises 63 through 92

It’s important to note that the transformations can actually be applied to any

function, even those that are new and unfamiliar. Consider the following pattern:

Parent Function

Transformation of Parent Function

y ϭ Ϫ21x Ϫ 32 2 ϩ 1

absolute value: y ϭ 0 x 0

y ϭ Ϫ2 0 x Ϫ 3 0 ϩ 1

3

y ϭ Ϫ2 1

xϪ3ϩ1

cube root: y ϭ 1x

3

general: y ϭ f 1x2

y ϭ Ϫ2f 1x Ϫ 32 ϩ 1

In each case, the transformation involves a horizontal shift 3 units right, a vertical

reflection, a vertical stretch, and a vertical shift up 1. Since the shifts are the same

regardless of the initial function, we can generalize the results to any function f(x).

WORTHY OF NOTE

Since the shape of the initial graph

does not change when translations

or reflections are applied, these are

called rigid transformations.

Stretches and compressions of a

basic graph are called nonrigid

transformations, as the graph is

distended in some way.

vertical reflections,

vertical stretches and compressions

S

y ϭ af 1x Ϯ h2 Ϯ k

S

y ϭ f 1x2

Transformed Function

S

General Function

horizontal shift

h units, opposite

direction of sign

vertical shift

k units, same

direction as sign

Also bear in mind that the graph will be reflected across the y-axis (horizontally)

if x is replaced with Ϫx. This process is illustrated in Example 9 for selected transformations. Remember — if the graph of a function is shifted, the individual points

on the graph are likewise shifted.

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EXAMPLE 9

Graphing Transformations of a General Function

Solution

For g, the graph of f is (1) shifted horizontally 1 unit left (Figure 2.34),

(2) reflected across the x-axis (Figure 2.35), and (3) shifted vertically 2 units down

(Figure 2.36). The final result is that in Figure 2.36.

Given the graph of f(x) shown in Figure 2.33, graph g1x2 ϭ Ϫf 1x ϩ 12 Ϫ 2.

Figure 2.34

Figure 2.33

y

y

5

5

(Ϫ2, 3)

(Ϫ3, 3)

f (x)

(0, 0)

Ϫ5

5

x

Ϫ5

(Ϫ1, 0)

(2, Ϫ3)

5

x

5

x

(1, Ϫ3)

Ϫ5

Ϫ5

Figure 2.36

Figure 2.35

y

y

5

5

(1, 3)

(1, 1)

g (x)

(Ϫ1, 0)

Ϫ5

5

x

Ϫ5

(3, Ϫ2)

(Ϫ1, Ϫ2)

(Ϫ5, Ϫ2)

(Ϫ3, Ϫ3)

Ϫ5

(Ϫ3, Ϫ5)

Ϫ5

Now try Exercises 93 through 96

As noted in Example 9, these shifts and transformation are often combined—

particularly when the toolbox functions are used as real-world models (Section 2.6).

On a graphing calculator we again define Y1 as needed, then define Y2 as any desired

combination of shifts, stretches, and/or reflections. For Y1 ϭ X2, we’ll define Y2 as

Ϫ2 Y1 1X ϩ 52 ϩ 3 (Figure 2.37), and expect that the graph of Y2 will be that of Y1

shifted left 5 units, reflected across the x-axis, stretched vertically, and shifted up

three units. This shows the new vertex should be at 1Ϫ5, 32 , which is confirmed in

Figure 2.38 along with the other transformations.

Figure 2.38

Figure 2.37

10

Ϫ10

10

Ϫ10

Try this exploration again using Y1 ϭ abs1X2 .

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Using the general equation y ϭ af 1x Ϯ h2 Ϯ k, we can identify the vertex, initial

point, or inflection point of any toolbox function and sketch its graph. Given the graph

of a toolbox function, we can likewise identify these points and reconstruct its equation. We first identify the function family and the location (h, k) of any characteristic

point. By selecting one other point (x, y) on the graph, we then use the general equation as a formula (substituting h, k, and the x- and y-values of the second point) to solve

for a and complete the equation.

EXAMPLE 10

Writing the Equation of a Function Given Its Graph

Find the equation of the function f(x) shown in the figure.

Solution

The function f belongs to the absolute value family. The

vertex (h, k) is at (1, 2). For an additional point, choose

the x-intercept (Ϫ3, 0) and work as follows:

y ϭ aͿx Ϫ hͿ ϩ k

0 ϭ aͿ 1Ϫ32 Ϫ 1Ϳ ϩ 2

E. You’ve just seen how

we can apply transformations

on a general function f(x)

0 ϭ 4a ϩ 2

Ϫ2 ϭ 4a

1

Ϫ ϭa

2

general equation (function is

shifted right and up)

substitute 1 for h and 2 for k,

substitute ؊3 for x and 0 for y

simplify

y

5

f(x)

Ϫ5

5

x

subtract 2

Ϫ5

solve for a

The equation for f is y ϭ Ϫ12 0 x Ϫ 1 0 ϩ 2.

Now try Exercises 97 through 102

2.2 EXERCISES

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

1. After a vertical

, points on the graph are

farther from the x-axis. After a vertical

,

points on the graph are closer to the x-axis.

3. The vertex of h1x2 ϭ 31x ϩ 52 2 Ϫ 9 is at

and the graph opens

.

5. Given the graph of a general function f (x), discuss/

explain how the graph of F1x2 ϭ Ϫ2f 1x ϩ 12 Ϫ 3

can be obtained. If (0, 5), (6, 7), and 1Ϫ9, Ϫ42 are

on the graph of f, where do they end up on the

graph of F?

2. Transformations that change only the location of a

graph and not its shape or form, include

and

.

4. The inflection point of f 1x2 ϭ Ϫ21x Ϫ 42 3 ϩ 11 is

at

and the end-behavior is

,

.

6. Discuss/Explain why the shift of f 1x2 ϭ x2 ϩ 3 is a

vertical shift of 3 units in the positive direction, while

the shift of g1x2 ϭ 1x ϩ 32 2 is a horizontal shift

3 units in the negative direction. Include several

examples along with a table of values for each.

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By carefully inspecting each graph given, (a) identify the

function family; (b) describe or identify the end-behavior,

vertex, intervals where the function is increasing or

decreasing, maximum or minimum value(s) and x- and

y-intercepts; and (c) determine the domain and range.

Assume required features have integer values.

7. f 1x2 ϭ x2 ϩ 4x

15. r 1x2 ϭ Ϫ314 Ϫ x ϩ 3 16. f 1x2 ϭ 21x ϩ 1 Ϫ 4

y

y

5

5

Ϫ5

5 x

8. g1x2 ϭ Ϫx2 ϩ 2x

y

5 x

f(x)

Ϫ5

Ϫ5

y

5

Ϫ5

r(x)

5

17. g1x2 ϭ 2 14 Ϫ x

18. h1x2 ϭ Ϫ21x ϩ 1 ϩ 4

y

Ϫ5

5 x

Ϫ5

y

5

5

5 x

g(x)

h(x)

Ϫ5

Ϫ5

9. p1x2 ϭ x2 Ϫ 2x Ϫ 3

Ϫ5

5 x

Ϫ5

5 x

10. q1x2 ϭ Ϫx2 ϩ 2x ϩ 8

y

Ϫ5

Ϫ5

y

5

10

Ϫ5

5 x

Ϫ10

10 x

Ϫ5

Ϫ10

11. f 1x2 ϭ x2 Ϫ 4x Ϫ 5

12. g1x2 ϭ x2 ϩ 6x ϩ 5

y

For each graph given, (a) identify the function family;

(b) describe or identify the end-behavior, vertex,

intervals where the function is increasing or decreasing,

maximum or minimum value(s) and x- and y-intercepts;

and (c) determine the domain and range. Assume

required features have integer values.

19. p1x2 ϭ 2Ϳx ϩ 1Ϳ Ϫ 4

10

20. q1x2 ϭ Ϫ3Ϳx Ϫ 2Ϳ ϩ 3

y

y

y

5

10

5

q(x)

Ϫ10

10 x

Ϫ10

10 x

Ϫ10

Ϫ5

5 x

Ϫ5

For each graph given, (a) identify the function family;

(b) describe or identify the end-behavior, initial point,

intervals where the function is increasing or decreasing,

and x- and y-intercepts; and (c) determine the domain

and range. Assume required features have integer values.

5 x

Ϫ5

Ϫ5

Ϫ10

13. p1x2 ϭ 2 1x ϩ 4 Ϫ 2

p(x)

21. r 1x2 ϭ Ϫ2Ϳx ϩ 1Ϳ ϩ 6 22. f 1x2 ϭ 3Ϳx Ϫ 2Ϳ Ϫ 6

y

y

4

6

r(x)

Ϫ5

Ϫ5

5 x

5 x

f(x)

14. q1x2 ϭ Ϫ2 1x ϩ 4 ϩ 2

Ϫ6

Ϫ4

y

y

5

5

23. g1x2 ϭ Ϫ3ͿxͿ ϩ 6

p(x)

24. h1x2 ϭ 2Ϳx ϩ 1Ϳ

y

y

6

6

Ϫ5

5 x

Ϫ5

5 x

q(x)

g(x)

Ϫ5

h(x)

Ϫ5

Ϫ5

5 x

Ϫ4

Ϫ5

5 x

Ϫ4

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For each graph given, (a) identify the function family;

(b) describe or identify the end-behavior, inflection

point, and x- and y-intercepts; and (c) determine the

domain and range. Assume required features have

integer values. Be sure to note the scaling of each axis.

25. f 1x2 ϭ Ϫ1x Ϫ 12 3

26. g1x2 ϭ 1x ϩ 12 3

y

5

f(x)

Ϫ5

5 x

5 x

Ϫ5

3

28. p1x2 ϭ Ϫ 2x ϩ 1

y

Ϫ5

5 x

5 x

Ϫ5

Ϫ5

29. q1x2 ϭ 2x Ϫ 1 Ϫ 1

3

30. r 1x2 ϭ Ϫ 2x ϩ 1 Ϫ1

3

y

y

Ϫ5

Ϫ5

5 x

q(x)

Ϫ5

5 x

r(x)

y

5

32.

f(x)

42. t1x2 ϭ 0 x 0 Ϫ 3

45. Y1 ϭ 0 x 0 , Y2 ϭ 0 x Ϫ 4 0

46. h1x2 ϭ x3, H1x2 ϭ 1x Ϫ 42 3

Sketch each graph by hand using transformations of a

parent function (without a table of values).

47. p1x2 ϭ 1x Ϫ 32 2

48. q1x2 ϭ 1x Ϫ 1

51. g1x2 ϭ Ϫ 0 x 0

52. j1x2 ϭ Ϫ 1x

3

53. f 1x2 ϭ 2

Ϫx

3

50. f 1x2 ϭ 1

xϩ2

54. g1x2 ϭ 1Ϫx2 3

Use a graphing calculator to graph the functions given

in the same window. Comment on what you observe.

Ϫ5

For Exercises 31–34, identify and state the characteristic

features of each graph, including (as applicable) the

function family, end-behavior, vertex, axis of symmetry,

point of inflection, initial point, maximum and minimum

value(s), x- and y-intercepts, and the domain and range.

31.

40. g1x2 ϭ 1x Ϫ 4

Use a graphing calculator to graph the functions given

in the same window. Comment on what you observe.

49. h1x2 ϭ Ϳx ϩ 3Ϳ

5

5

39. f 1x2 ϭ x3 Ϫ 2

44. f 1x2 ϭ 1x, g1x2 ϭ 1x ϩ 4

p(x)

h(x)

Ϫ5

37. p1x2 ϭ ͿxͿ, q1x2 ϭ ͿxͿ Ϫ 5, r 1x2 ϭ ͿxͿ ϩ 2

43. p1x2 ϭ x2, q1x2 ϭ 1x ϩ 52 2

y

5

5

h1x2 ϭ 1x Ϫ 3

3

3

g1x2 ϭ 2

x Ϫ 3, h1x2 ϭ 2

xϩ4

41. h1x2 ϭ x2 ϩ 3

Ϫ5

27. h1x2 ϭ x3 ϩ 1

3

36. f 1x2 ϭ 2

x,

g1x2 ϭ 1x ϩ 2,

Sketch each graph by hand using transformations of a

parent function (without a table of values).

g(x)

Ϫ5

35. f 1x2 ϭ 1x,

38. p1x2 ϭ x2, q1x2 ϭ x2 Ϫ 7, r 1x2 ϭ x2 ϩ 3

y

5

Use a graphing calculator to graph the functions given

in the same window. Comment on what you observe.

y

5

55. p1x2 ϭ x2, q1x2 ϭ 3x2, r 1x2 ϭ 15x2

56. f 1x2 ϭ 1Ϫx, g1x2 ϭ 41Ϫx,

h1x2 ϭ 14 1Ϫx

57. Y1 ϭ 0 x 0 , Y2 ϭ 3 0 x 0 , Y3 ϭ 13 0 x 0

58. u1x2 ϭ x3, v1x2 ϭ 8x3, w1x2 ϭ 15x3

g(x)

Sketch each graph by hand using transformations of a

parent function (without a table of values).

Ϫ5

Ϫ5

5 x

5 x

59. f 1x2 ϭ 4 2x

3

61. p1x2 ϭ 13x3

y

5

Ϫ5

34.

f(x)

5 x

Ϫ5

62. q1x2 ϭ 34 1x

Ϫ5

Ϫ5

33.

60. g1x2 ϭ Ϫ2 0x 0

y

5

Ϫ5

Use the characteristics of each function family to match

a given function to its corresponding graph. The graphs

are not scaled — make your selection based on a careful

comparison.

g(x)

5 x

Ϫ5

63. f 1x2 ϭ 12x3

64. f 1x2 ϭ Ϫ2

3 x ϩ 2

3

65. f 1x2 ϭ Ϫ1x Ϫ 32 2 ϩ 2 66. f 1x2 ϭ Ϫ 1

xϪ1Ϫ1

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67. f 1x2 ϭ Ϳx ϩ 4Ϳ ϩ 1

68. f 1x2 ϭ Ϫ 1x ϩ 6

71. f 1x2 ϭ 1x Ϫ 42 2 Ϫ 3

72. f 1x2 ϭ Ϳx Ϫ 2Ϳ Ϫ 5

Graph each function using shifts of a parent function

and a few characteristic points. Clearly state and indicate

the transformations used and identify the location of all

vertices, initial points, and/or inflection points.

69. f 1x2 ϭ Ϫ 1x ϩ 6 Ϫ 1 70. f 1x2 ϭ x ϩ 1

73. f 1x2 ϭ 1x ϩ 3 Ϫ 1

y

a.

74. f 1x2 ϭ Ϫ1x ϩ 32 2 ϩ 5

y

b.

x

x

75. f 1x2 ϭ 1x ϩ 2 Ϫ 1

76. g1x2 ϭ 1x Ϫ 3 ϩ 2

79. p1x2 ϭ 1x ϩ 32 3 Ϫ 1

80. q1x2 ϭ 1x Ϫ 22 3 ϩ 1

83. f 1x2 ϭ ϪͿx ϩ 3Ϳ Ϫ 2

84. g1x2 ϭ ϪͿx Ϫ 4Ϳ Ϫ 2

77. h1x2 ϭ Ϫ1x ϩ 32 2 Ϫ 2 78. H1x2 ϭ Ϫ1x Ϫ 22 2 ϩ 5

3

81. s1x2 ϭ 1

xϩ1Ϫ2

3

82. t1x2 ϭ 1

xϪ3ϩ1

85. h1x2 ϭ Ϫ21x ϩ 12 2 Ϫ 3 86. H1x2 ϭ 12Ϳx ϩ 2Ϳ Ϫ 3

c.

d.

y

3

87. p1x2 ϭ Ϫ13 1x ϩ 22 3 Ϫ 1 88. q1x2 ϭ 4 1

xϩ1ϩ2

y

89. u1x2 ϭ Ϫ2 1Ϫx Ϫ 1 ϩ 3 90. v1x2 ϭ 3 1Ϫx ϩ 2 Ϫ 1

x

x

91. h1x2 ϭ 15 1x Ϫ 32 2 ϩ 1

92. H1x2 ϭ Ϫ2Ϳx Ϫ 3Ϳ ϩ 4

Apply the transformations indicated for the graph of the

general functions given.

e.

f.

y

93.

y

y

5

94.

f(x)

y

5

g(x)

(Ϫ1, 4)

(Ϫ4, 4)

(3, 2)

(Ϫ1, 2)

x

x

Ϫ5

Ϫ5

5 x

5 x

(Ϫ4, Ϫ2)

Ϫ5

Ϫ5

g.

h.

y

y

a. f 1x Ϫ 22

b. Ϫf 1x2 Ϫ 3

c. 12 f 1x ϩ 12

d. f 1Ϫx2 ϩ 1

x

x

95.

i.

y

j.

y

5

(2, Ϫ2)

a.

b.

c.

d.

96.

h(x)

g1x2 Ϫ 2

Ϫg1x2 ϩ 3

2g1x ϩ 12

1

2 g1x Ϫ 12 ϩ 2

y

5

y

(Ϫ1, 3)

(2, 0)

(Ϫ1, 0)

Ϫ5

x

5 x

Ϫ5

y

l.

x

Ϫ5

y

x

a.

b.

c.

d.

(1, Ϫ3)

(2, Ϫ4)

h1x2 ϩ 3

Ϫh1x Ϫ 22

h1x Ϫ 22 Ϫ 1

1

4 h1x2 ϩ 5

5 x

(Ϫ2, 0)

x

(Ϫ4, Ϫ4)

k.

H(x)

Ϫ5

a.

b.

c.

d.

H1x Ϫ 32

ϪH1x2 ϩ 1

2H1x Ϫ 32

1

3 H1x Ϫ 22 ϩ 1

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Use the graph given and the points indicated to determine the equation of the function shown using the general form

y ‫ ؍‬af(x ؎ h) ؎ k.

97.

98.

y

5

99.

y

(Ϫ5, 6)

y

5

(6, 4.5)

5

p(x)

g(x)

(2, 0)

Ϫ5

5 x

f(x)

Ϫ5

5 x

Ϫ3(Ϫ3, 0)

(0, Ϫ4)

Ϫ5

100.

Ϫ4

Ϫ3

(0, Ϫ4)

101.

y

(Ϫ4, 5) 5

x

5

102.

y

5

y

(3, 7)

7

(1, 4)

f(x)

h(x)

r(x)

Ϫ4

5

x

Ϫ8

(5, Ϫ1)

Ϫ5

Ϫ3

Ϫ5

7 x

Ϫ3

(0, Ϫ2)

WORKING WITH FORMULAS

103. Volume of a sphere: V(r) ‫ ؍‬43␲r3

The volume of a sphere is given by the function

shown, where V(r) is the volume in cubic units and

r is the radius. Note this function belongs to the

cubic family of functions. (a) Approximate the

value of 43␲ to one decimal place, then graph the

function on the interval [0, 3]. (b) From your

graph, estimate the volume of a sphere with radius

2.5 in., then compute the actual volume. Are the

results close? (c) For V ϭ 43 ␲r3, solve for r in terms

of V.

2 x

(Ϫ4, 0)

104. Fluid motion: V(h) ‫ ؍‬؊4 1h ؉ 20

Suppose the velocity of a fluid flowing from an

open tank (no top) through an opening in its side is

given by the function shown, where V(h) is the

velocity of the fluid (in feet per second) at water

height h (in feet). Note this function belongs to the

square root family of functions. An open tank is

25 ft deep and filled to the brim with fluid. (a) Use

a table of values to graph the

25 ft

function on the interval [0, 25].

(b) From your graph, estimate the

velocity of the fluid when the

water level is 7 ft, then find the

close? (c) If the fluid velocity is

5 ft/sec, how high is the water in the tank?

APPLICATIONS

105. Gravity, distance, time: After being released, the

time it takes an object to fall x ft is given by the

function T1x2 ϭ 14 1x, where T(x) is in seconds.

(a) Describe the transformation applied to obtain

the graph of T from the graph of y ϭ 1x, then

sketch the graph of T for x ʦ 30, 100 4 . (b) How

long would it take an object to hit the ground if it

were dropped from a height of 81 ft?

106. Stopping distance: In certain weather conditions,

accident investigators will use the function

v1x2 ϭ 4.9 1x to estimate the speed of a car (in

miles per hour) that has been involved in an

accident, based on the length of the skid marks x

(in feet). (a) Describe the transformation applied to

obtain the graph of v from the graph of y ϭ 1x,

then sketch the graph of v for x ʦ 30, 4004 . (b) If the

skid marks were 225 ft long, how fast was the car

traveling? Is this point on your graph?

107. Wind power: The power P generated by a certain

8 3

v

wind turbine is given by the function P1v2 ϭ 125

where P(v) is the power in watts at wind velocity v

(in miles per hour). (a) Describe the transformation

applied to obtain the graph of P from the graph of

y ϭ v3, then sketch the graph of P for v ʦ 30, 254

(scale the axes appropriately). (b) How much

power is being generated when the wind is blowing

at 15 mph?

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108. Wind power: If the power P (in watts) being

generated by a wind turbine is known, the velocity

of the wind can be determined using the function

3

v1P2 ϭ 52 2

P. (a) Describe the transformation

applied to obtain the graph of v from the graph of

3

yϭ 2

P, then sketch the graph of v for P ʦ 3 0, 512 4

(scale the axes appropriately). (b) How fast is the

wind blowing if 343W of power is being generated?

Is this point on your graph?

109. Distance rolled due to gravity: The distance a ball

rolls down an inclined plane is given by the function

d1t2 ϭ 2t2, where d(t) represents the distance

in feet after t sec. (a) Describe the transformation

applied to obtain the graph of d from the graph

of y ϭ t2, then sketch the graph of d for

t ʦ 3 0, 3 4 . (b) How far has the ball rolled after

2.5 sec?

110. Acceleration due to gravity: The velocity of a steel

ball bearing as it rolls down an inclined plane is

given by the function v1t2 ϭ 4t, where v(t)

represents the velocity in feet per second after t sec.

(a) Describe the transformation applied to obtain

the graph of v from the graph of y ϭ t, then sketch

the graph of v for t ʦ 30, 3 4 . (b) What is the velocity

of the ball bearing after 2.5 sec? Is this point on

EXTENDING THE CONCEPT

111. Carefully graph the functions f 1x2 ϭ ͿxͿ and

g1x2 ϭ 2 1x on the same coordinate grid. From the

graph, in what interval is the graph of g(x) above

the graph of f (x)? Pick a number (call it h) from this

interval and substitute it in both functions. Is

g1h2 7 f 1h2? In what interval is the graph of g(x)

below the graph of f(x)? Pick a number from this

interval (call it k) and substitute it in both functions.

Is g1k2 6 f 1k2?

217

112. Sketch the graph of f 1x2 ϭ Ϫ2Ϳx Ϫ 3Ϳ ϩ 8 using

transformations of the parent function, then

determine the area of the region in quadrant I that

is beneath the graph and bounded by the vertical

lines x ϭ 0 and x ϭ 6.

113. Sketch the graph of f 1x2 ϭ x2 Ϫ 4, then sketch the

graph of F1x2 ϭ Ϳx2 Ϫ 4Ϳ using your intuition and

the meaning of absolute value (not a table of

values). What happens to the graph?

114. (1.1) Find the distance between the points 1Ϫ13, 92

and 17, Ϫ122, and the slope of the line containing

these points.

115. (R.2) Find the perimeter

of the figure shown.

5x ϩ 2

2x2 ϩ3x

5x

2x2 ϩ3x ϩ 5

1

1

7

2

116. (1.5) Solve for x: x ϩ ϭ x Ϫ .

3

4

2

12

117. (2.1) Without graphing, state intervals where f 1x2c

and f 1x2T for f 1x2 ϭ 1x Ϫ 42 2 ϩ 3.

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College Algebra G&M—

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Absolute Value Functions, Equations, and Inequalities

While the equations x ϩ 1 ϭ 5 and Ϳx ϩ 1Ϳ ϭ 5 are similar in many respects, note the

first has only the solution x ϭ 4, while either x ϭ 4 or x ϭ Ϫ6 will satisfy the second.

The fact there are two solutions shouldn’t surprise us, as it’s a natural result of how

absolute value is defined.

LEARNING OBJECTIVES

In Section 2.3 you will see

how we can:

A. Solve absolute value

equations

A. Solving Absolute Value Equations

B. Solve “less than”

absolute value

inequalities

C. Solve “greater than”

absolute value

inequalities

D. Solve absolute value

equations and

inequalities graphically

E. Solve applications

involving absolute value

The absolute value of a number x can be thought of as its distance from zero on the number line, regardless of direction. This means ͿxͿ ϭ 4 will have two solutions, since there

are two numbers that are four units from zero: x ϭ Ϫ4 and x ϭ 4 (see Figure 2.39).

Exactly 4 units

from zero

Figure 2.39

Ϫ5 Ϫ4

Exactly 4 units

from zero

Ϫ3 Ϫ2 Ϫ1

0

1

2

3

4

5

This basic idea can be extended to include situations where the quantity within

absolute value bars is an algebraic expression, and suggests the following property.

Property of Absolute Value Equations

If X represents an algebraic expression and k is a positive real number,

WORTHY OF NOTE

Note if k 6 0, the equation ͿXͿ ϭ k

has no solutions since the absolute

value of any quantity is always

positive or zero. On a related note,

we can verify that if k ϭ 0, the

equation ͿXͿ ϭ 0 has only the

solution X ϭ 0.

then ͿXͿ ϭ k

implies X ϭ Ϫk or X ϭ k

As the statement of this property suggests, it can only be applied after the absolute

value expression has been isolated on one side.

EXAMPLE 1

Solving an Absolute Value Equation

Solve: Ϫ5Ϳx Ϫ 7Ϳ ϩ 2 ϭ Ϫ13.

Solution

Begin by isolating the absolute value expression.

Ϫ5Ϳx Ϫ 7Ϳ ϩ 2 ϭ Ϫ13 original equation

Ϫ5Ϳx Ϫ 7Ϳ ϭ Ϫ15 subtract 2

Ϳx Ϫ 7Ϳ ϭ 3

divide by Ϫ5 (simplified form)

Now consider x Ϫ 7 as the variable expression “X” in the property of absolute

value equations, giving

or

x Ϫ 7 ϭ Ϫ3

x Ϫ 7 ϭ 3 apply the property of absolute value equations

xϭ4

or

Substituting into the original equation verifies the solution set is {4, 10}.

Now try Exercises 7 through 18

CAUTION

218

For equations like those in Example 1, be careful not to treat the absolute value bars as

simple grouping symbols. The equation Ϫ51x Ϫ 72 ϩ 2 ϭ Ϫ13 has only the solution

x ϭ 10, and “misses” the second solution since it yields x Ϫ 7 ϭ 3 in simplified form.

The equation Ϫ5Ϳx Ϫ 7Ϳ ϩ 2 ϭ Ϫ13 simplifies to Ϳx Ϫ 7Ϳ ϭ 3 and there are actually two

solutions. Also note that Ϫ5Ϳx Ϫ 7Ϳ ͿϪ5x ϩ 35Ϳ!

2–32

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Section 2.3 Absolute Value Functions, Equations, and Inequalities

219

If an equation has more than one solution as in Example 1, they cannot be

simultaneously stored using the, X,T,␪,n key to perform a calculator check (in function or “Func” mode, this is the variable X). While there are other ways to “get

around” this (using Y 1 on the home screen, using a TABLE in ASK mode,

enclosing the solutions in braces as in {4, 10}, etc.), we can also store solutions

using the ALPHA keys. To illustrate, we’ll place the solution x ϭ 4 in storage location A,

using 4 STO ALPHA MATH (A). Using this “ STO ALPHA ” sequence we’ll next place the

solution x ϭ 10 in storage location B (Figure 2.40). We can then check both solutions

in turn. Note that after we check the first solution, we can recall the expression

using 2nd

and simply change the A to B (Figure 2.41).

ENTER

Figure 2.40

Figure 2.41

Absolute value equations come in many different forms. Always begin by isolating the absolute value expression, then apply the property of absolute value equations to solve.

EXAMPLE 2

Solving an Absolute Value Equation

Solve:

Solution

2

` 5 Ϫ x ` Ϫ 9 ϭ 8.

3

2

`5 Ϫ x ` Ϫ 9 ϭ 8

3

2

` 5 Ϫ x ` ϭ 17

3

2

5 Ϫ x ϭ Ϫ17

3

2

Ϫ x ϭ Ϫ22

3

x ϭ 33

Check

WORTHY OF NOTE

As illustrated in both Examples

1 and 2, the property we use to

solve absolute value equations can

only be applied after the absolute

value term has been isolated. As

you will see, the same is true for the

properties used to solve absolute

value inequalities.

2

For x ϭ 33: ` 5 Ϫ 1332 `

3

|5 Ϫ 21112 |

05 Ϫ 22 0

0 Ϫ17 0

17

original equation

or

or

or

Ϫ9ϭ8

Ϫ9ϭ8

Ϫ9ϭ8

Ϫ9ϭ8

Ϫ9ϭ8

8 ϭ 8✓

2

5 Ϫ x ϭ 17

3

2

Ϫ x ϭ 12

3

x ϭ Ϫ18

apply the property of absolute

value equations

subtract 5

multiply by Ϫ32

2

1Ϫ182 ` Ϫ 9 ϭ 8

3

| 5 Ϫ 21Ϫ62 | Ϫ 9 ϭ 8

0 5 ϩ 12 0 Ϫ 9 ϭ 8

0 17 0 Ϫ 9 ϭ 8

17 Ϫ 9 ϭ 8

8 ϭ 8✓

For x ϭ Ϫ18: ` 5 Ϫ

Both solutions check. The solution set is 5Ϫ18, 336.

Now try Exercises 19 through 22

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