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Section 3.7 Graphing Absolute Value Functions 155
3.7 Graphing Absolute Value Functions
Essential QuestionEssential Question How do the values of a , h , and k affect the
graph of the absolute value function g ( x ) = a ∣^ x − h ∣^ + k?
The parent absolute value function is
f ( x ) = ∣^ x ∣. Parent absolute value function
The graph of f is V-shaped.
Identifying Graphs of Absolute Value Functions
Work with a partner. Match each absolute value function with its graph. Then use a graphing calculator to verify your answers.
a. g ( x ) = −∣^ x − 2 ∣^ b. g ( x ) = ∣^ x − 2 ∣^ + 2 c. g ( x ) = −∣^ x + 2 ∣^ − 2
d. g ( x ) = ∣^ x − 2 ∣^ − 2 e. g ( x ) = 2 ∣^ x − 2 ∣^ f. g ( x ) = −∣^ x + 2 ∣^ + 2
A.
6
− 4
− 6
(^4) B.
6
− 4
− 6
4
C.
6
− 4
− 6
(^4) D.
6
− 4
− 6
4
E.
6
− 4
− 6
(^4) F.
6
− 4
− 6
4
Communicate Your AnswerCommunicate Your Answer
2. How do the values of a , h , and k affect the graph of the absolute value function
g ( x ) = a ∣^ x − h ∣^ + k?
3. Write the equation of the absolute value function whose graph is shown. Use a graphing calculator to verify your equation. 6
− 4
− 6
4
LOOKING FOR
STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.
156 Chapter 3 Graphing Linear Functions
3.7 Lesson^ What You Will LearnWhat You Will Learn
Translate graphs of absolute value functions. Stretch, shrink, and reflect graphs of absolute value functions. Combine transformations of graphs of absolute value functions.
Translating Graphs of Absolute Value Functions
absolute value function, p. 156 vertex, p. 156 vertex form, p. 158 Previous domain range
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
Absolute Value Function
An absolute value function is a function that contains an absolute value expression. The parent absolute
value function is f ( x ) = ∣ x ∣. The graph of f ( x ) = ∣ x ∣ is
V-shaped and symmetric about the y -axis. The vertex is the point where the graph changes direction. The
vertex of the graph of f ( x ) = ∣ x ∣ is (0, 0).
The domain of f ( x ) = ∣ x ∣ is all real numbers. The range is y ≥ 0.
The graphs of all other absolute value functions are transformations of the graph of the
parent function f ( x ) = ∣ x ∣. The transformations presented in Section 3.6 also apply to
absolute value functions.
Graphing g ( x ) = | x |^ + k and g ( x ) = | x – h |
Graph each function. Compare each graph to the graph of f ( x ) = ∣^ x ∣. Describe the
domain and range.
a. g ( x ) = ∣^ x ∣^ + 3 b. m ( x ) = ∣^ x − 2 ∣
SOLUTION
a. Step 1 Make a table of values. b. Step 1 Make a table of values.
x −^2 −^1 0 1 g ( x )^5 4 3 4
x 0 1 2 3 4 m ( x ) 2 1 0 1 2
Step 2 Plot the ordered pairs. Step 2 Plot the ordered pairs. Step 3 Draw the V-shaped graph. Step 3 Draw the V-shaped graph.
The function g is of the form y = f ( x ) + k , where k = 3. So, the graph of g is a vertical translation 3 units up of the graph of f. The domain is all real numbers. The range is y ≥ 3.
The function m is of the form y = f ( x − h ), where h = 2. So, the graph of m is a horizontal translation 2 units right of the graph of f. The domain is all real numbers. The range is y ≥ 0.
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Graph the function. Compare the graph to the graph of f ( x ) = ∣^ x ∣. Describe the
domain and range.
1. h ( x ) = ∣^ x ∣^ − 1 2. n ( x ) = ∣^ x + 4 ∣
x
y
2
4
− 2 2
g ( x ) = x + 3
x
y
3
5
1
2 4
m ( x ) = x − 2
x
y
2
4
− 2 2
f ( x ) = x
vertex
158 Chapter 3 Graphing Linear Functions
CoreCore ConceptConcept
Vertex Form of an Absolute Value Function
An absolute value function written in the form g ( x ) = a ∣^ x − h ∣^ + k , where a ≠ 0,
is in vertex form. The vertex of the graph of g is ( h , k ). Any absolute value function can be written in vertex form, and its graph is symmetric about the line x = h.
Graphing f ( x ) = | x – h |^ + k and g ( x ) = f ( ax )
Graph f ( x ) = ∣^ x + 2 ∣^ − 3 and g ( x ) = ∣^2 x + 2 ∣^ − 3. Compare the graph of g to the
graph of f.
SOLUTION
Step 1 Make a table of values for each function.
x − 4 − 3 − 2 − 1 0 1 2 f ( x ) − 1 − 2 − 3 − 2 − 1 0 1
x − 2 −1.5 − 1 −0.5 0 0.5 1 g ( x ) − 1 − 2 − 3 − 2 − 1 0 1
Step 2 Plot the ordered pairs. Step 3 Draw the V-shaped graph of each function. Notice that the vertex of the graph of f is (−2, −3) and the graph is symmetric about x = −2.
Note that you can rewrite g as g ( x ) = f (2 x ), which is of the form y = f ( ax ), where a = 2. So, the graph of g is a horizontal shrink of the graph of f by a factor of 1 — 2. The y -intercept is the same for both graphs. The points on the graph of f move halfway closer to the y -axis, resulting in the graph of g. When the input values of f are 2 times the input values of g , the output values of f and g are the same.
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5. Graph f ( x ) = ∣^ x − 1 ∣^ and g ( x ) = ∣^ —^12 x − 1 ∣. Compare the graph of g to the graph of f.
6. Graph f ( x ) = ∣^ x + 2 ∣^ + 2 and g ( x ) = ∣^ − 4 x + 2 ∣^ + 2. Compare the graph of g to
the graph of f.
STUDY TIP
The function g is not in vertex form because the x variable does not have a coefficient of 1.
x
y
2
− 4
− 5 − 3 − 1 1 3
f ( x ) = x + 2 − 3
g ( x ) = 2 x + 2 − 3
Section 3.7 Graphing Absolute Value Functions 159
Graphing g ( x ) = a|x – h| + k
Let g ( x ) = − 2 ∣^ x − 1 ∣^ + 3. (a) Describe the transformations from the graph of
f ( x ) = ∣^ x ∣^ to the graph of g. (b) Graph g.
SOLUTION
a. Step 1 Translate the graph of f horizontally 1 unit right to get the graph of
t ( x ) = ∣^ x − 1 ∣.
Step 2 Stretch the graph of t vertically by a factor of 2 to get the graph of
h ( x ) = 2 ∣^ x − 1 ∣.
Step 3 Reflect the graph of h in the x -axis to get the graph of r ( x ) = − 2 ∣^ x − 1 ∣.
Step 4 Translate the graph of r vertically 3 units up to get the graph of
g ( x ) = − 2 ∣^ x − 1 ∣^ + 3.
b. Method 1
Step 1 Make a table of values. x −^1 0 1 2
g ( x ) −^1 1 3 1 −^1 Step 2 Plot the ordered pairs. Step 3 Draw the V-shaped graph.
x
y 4
2
− 2
− 4 − 2 2 4 g ( x ) = − 2 x − 1 + 3
Method 2 Step 1 Identify and plot the vertex. ( h , k ) = (1, 3)
x
y 4
2
− 2
− 4 − 2 2 4 g ( x ) = − 2 x − 1 + 3
(0, 1)
(1, 3)
(2, 1)
Step 2 Plot another point on the graph, such as (2, 1). Because the graph is symmetric about the line x = 1, you can use symmetry to plot a third point, (0, 1). Step 3 Draw the V-shaped graph.
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7. Let g ( x ) = ∣^ − (^1) — 2 x^ +^2 ∣^ +^ 1. (a) Describe the transformations from the graph
of f ( x ) = ∣^ x ∣^ to the graph of g. (b) Graph g.
Combining Transformations
REMEMBER
You can obtain the graph of y = a ⋅ f ( x – h ) + k from the graph of y = f ( x ) using the steps you learned in Section 3.6.
Section 3.7 Graphing Absolute Value Functions 161
In Exercises 33–40, describe the transformations from
the graph of f ( x ) = ∣^ x ∣^ to the graph of the given function.
Then graph the given function. (See Example 4.)
33. r ( x ) = ∣^ x + 2 ∣^ − 6 34. c ( x ) = ∣^ x + 4 ∣^ + 4
35. d ( x ) = −∣^ x − 3 ∣^ + 5 36. v ( x ) = − 3 ∣^ x + 1 ∣^ + 4
37. m ( x ) = 1 — 2 ∣^ x + 4 ∣^ − 1 38. s ( x ) = ∣^2 x − 2 ∣^ − 3
39. j ( x ) = ∣^ − x + 1 ∣^ − 5 40. n ( x ) = ∣^ −
(^1) —
3 x^ +^1 ∣^ +^2
41. MODELING WITH MATHEMATICS The number of pairs of shoes sold s (in thousands) increases and then decreases as described by the function
s ( t ) = − 2 ∣^ t − 15 ∣^ + 50, where t is the time
(in weeks).
a. Graph the function. b. What is the greatest number of pairs of shoes sold in 1 week?
42. MODELING WITH MATHEMATICS On the pool table shown, you bank the five ball off the side represented by the x -axis. The path of the ball is described by the function p ( x ) = —^4
3 ∣^ x^ −^
(^5) —
x
y
5
(5, 5)
(−5, 0)(0, 0) (5, 0)
(−−5, 5)5, 5)5 5)
a. At what point does the five ball bank off the side? b. Do you make the shot? Explain your reasoning.
43. USING TRANSFORMATIONS The points A ( −
(^1) —
2 , 3^ ),
B (1, 0), and C (−4, −2) lie on the graph of the absolute value function f. Find the coordinates of the points corresponding to A , B , and C on the graph of each function. a. g ( x ) = f ( x ) − 5 b. h ( x ) = f ( x − 3) c. j ( x ) = − f ( x ) d. k ( x ) = 4 f ( x )
44. USING STRUCTURE Explain how the graph of each
function compares to the graph of y = ∣^ x ∣^ for positive
and negative values of k , h , and a.
a. y = ∣^ x ∣^ + k
b. y = ∣^ x − h ∣
c. y = a ∣^ x ∣
d. y = ∣^ ax ∣
ERROR ANALYSIS In Exercises 45 and 46, describe and correct the error in graphing the function.
45.
y = ∣^ x − 1 ∣^ − 3 x
y 2
− 5 − 1 3
46.
y = − 3 ∣^ x ∣
x
y 4
− 2
− 2 2
MATHEMATICAL CONNECTIONS In Exercises 47 and 48, write an absolute value function whose graph forms a square with the given graph.
47.
x
y 3
1
− 3
− 3 3
y = x − 2
48.
x
y
2
6
2 4 6
y = x − 3 + 1
49. WRITING Compare the graphs of p ( x ) = ∣^ x − 6 ∣^ and
q ( x ) = ∣^ x ∣^ − 6.
162 Chapter 3 Graphing Linear Functions
50. HOW DO YOU SEE IT? The object of a computer game is to break bricks by deflecting a ball toward them using a paddle. The graph shows the current path of the ball and the location of the last brick.
BRICK FACTORY
x
y
4
2
0
8
6
0 2 4 6 8 10 12 14
brick
paddle
a. You can move the paddle up, down, left, and right. At what coordinates should you place the paddle to break the last brick? Assume the ball deflects at a right angle. b. You move the paddle to the coordinates in part (a), and the ball is deflected. How can you write an absolute value function that describes the path of the ball?
In Exercises 51–54, graph the function. Then rewrite the absolute value function as two linear functions, one that has the domain x < 0 and one that has the domain x ≥ 0.
51. y = ∣^ x ∣^ 52. y = ∣^ x ∣^ − 3
53. y = −∣^ x ∣^ + 9 54. y = − 4 ∣^ x ∣
In Exercises 55–58, graph and compare the two functions.
55. f ( x ) = ∣ x − 1 ∣ + 2; g ( x ) = 4 ∣ x − 1 ∣ + 8
56. s ( x ) = ∣ 2 x − 5 ∣ − 6; t ( x ) = 1 — 2 ∣ 2 x − 5 ∣ − 3
57. v ( x ) = − 2 ∣ 3 x + 1 ∣ + 4; w ( x ) = 3 ∣ 3 x + 1 ∣ − 6
58. c ( x ) = 4 ∣ x + 3 ∣ − 1; d ( x ) = −^4 — 3 ∣ x + 3 ∣ + 1 — 3
59. REASONING Describe the transformations from the
graph of g ( x ) = − 2 ∣ x + 1 ∣ + 4 to the graph of
h ( x ) = ∣ x ∣. Explain your reasoning.
60. THOUGHT PROVOKING Graph an absolute value function f that represents the route a wide receiver runs in a football game. Let the x -axis represent distance (in yards) across the field horizontally. Let the y -axis represent distance (in yards) down the field. Be sure to limit the domain so the route is realistic.
61. SOLVING BY GRAPHING Graph y = 2 ∣ x + 2 ∣ − 6
and y = −2 in the same coordinate plane. Use the
graph to solve the equation 2∣ x + 2 ∣ − 6 = −2.
Check your solutions.
62. MAKING AN ARGUMENT Let p be a positive constant.
Your friend says that because the graph of y = ∣ x ∣ + p
is a positive vertical translation of the graph of
y = ∣ x ∣, the graph of y = ∣ x + p ∣ is a positive
horizontal translation of the graph of y = ∣ x ∣.
Is your friend correct? Explain.
63. ABSTRACT REASONING Write the vertex of the
absolute value function f ( x ) = ∣ ax − h ∣ + k in terms
of a , h , and k.
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Solve the inequality. (Section 2.4)
64. 8 a − 7 ≤ 2(3 a − 1) 65. −3(2 p + 4) > − 6 p − 5 66. 4(3 h + 1.5) ≥ 6(2 h − 2) 67. −4( x + 6) < 2(2 x − 9)
Find the slope of the line. (Section 3.5) 68.
x
y 3
− 2
− 4 − 2 2
(0, 3)
(−2, −2)
69.
x
y 2
− 2
2 4
(5, 2)
(−1, 0)
70.
x
y 1
− 3
− 5
− 3 1 3
(1, −4)
(−3, 1)
Reviewing what you learned in previous grades and lessons
