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
quadratic: y ϭ x2
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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DEVELOPING YOUR SKILLS
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) ؍43r3
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
actual velocity. Are the answers
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
your graph?
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?
MAINTAINING YOUR SKILLS
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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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
x ϭ 10 add 7
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
add 9
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
ᮣ