Name Class Date 2.1 Graphing Absolute Value Functions
Essential Question: H
ow can you identify the features of the graph of an absolute
value function?
Resource
Locker
Explore 1 Graphing and Analyzing the Parent
Absolute Value Function
Absolute value, written as ⎜​ x⎟​, represents the distance between x and 0 on a number line. As a distance, absolute
value is always positive. For every point on a number line, there is another point on the opposite side of 0 that is the
same distance from 0. For example, both 5 and –5 are five units away from 0. Thus, ⎜​ −5⎟​ = 5 and ⎜​ 5⎟​ = 5.
5 units
-5
5 units
0
5
 ⎧ x x ≥ 0
​. When x is nonnegative,
The absolute value function ​​|x|​​, can be defined piecewise as ⎜​ x⎟​= ⎨ 
​ ​​        
⎩  −x x < 0
the function simply returns the number. When x is negative, the function returns the opposite of x.
AComplete the input-output table for ƒ​ (​ x)​.
f​(x)​
x
–8
x x ≥ 0
 ⎧     
​
​ƒ(​ x)​= ⎜​ x⎟​= ⎨ 
 ⎩​ ​​   −x x < 0 
–4
0
4
© Houghton Mifflin Harcourt Publishing Company
8
B
Plot the points you found on the coordinate grid.
Use the points to complete the graph of the function.
8
y
4
CNow, examine your graph of ƒ(x)​= ⎜​x⎟​and complete the
following statements about the function.
ƒ(x)​ = ​⎜x⎟​is symmetric about the and therefore
x
-8
-4
0
-4
8
-8
is a(n) function.
The domain of ƒ(x)​ = ⎜​x⎟​is .
The range of ƒ(x)​ = ​⎜x⎟​is .
Module 2
4
65
Lesson 1
Reflect
1.
Use the definition of the absolute value function to show that ƒ(x) = ⎜x⎟ is an even function.
Explain 1
Graphing Absolute Value Functions
You can apply general transformations to absolute value functions by changing parameters in the
⎜b
⎟
equation g(x) = a _1 ( x − h ) + k.
Example 1
⎜b
⎟
1 (
Given the function g(x) = a __
x − h) + k, find the vertex of the
function. Use the vertex and two other points to help you graph g(x).
Ag(x) = 4⎜x − 5⎟ − 2
The vertex of the parent absolute value function is at (0, 0).
8
The vertex of g(x) will be the point to which (0, 0) is mapped by g(x).
4
g(x) involves a translation of ƒ(x) 5 units to the right and 2 units down.
(4, 2)
-8
The vertex of g(x) will therefore be at (5, –2).
y
-4
Next, determine the location to which each of the points
(1, 1) and (–1, 1) on ƒ(x) will be mapped.
0
(6, 2)
x
4
8
(5, -2)
-4
-8
Bg(x) = ⎜-_21 (x + 3)⎟ + 1
The vertex of the parent absolute value function is at (0, 0).
g(x) is a translation of ƒ(x)
and
unit
The vertex of g(x) will therefore be at
units to the
(
.
,
).
Next, determine to where the points (2, 2) and (–2, 2) on ƒ(x) will
be mapped.
Module 2
8
66
y
4
x
-8
-4
0
-4
4
8
-8
Lesson 1
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Since a > 1, then g(x), in addition to being a translation, is also a vertical
stretch of ƒ(x) by a factor of 4. The x-coordinate of each point will be shifted 5 units
to the right while the y-coordinate will be stretched by a factor of 4 and then moved
down 2 units. So, (1, 1) moves to (1 + 5, 4 ⋅ ⎜1⎟ - 2) = (6, 2), and (–1, 1) moves to
(-1 + 5, 4 ⋅ ⎜1⎟ - 2) = (4, 2). Now plot the three points and graph g(x).
Since ⎜b⎟ = 2, g(x) is also a
of ƒ(x) and since b is negative,
a
.
The x-coordinate will be reflected in the y-axis and
of
, then moved
units to the
The y-coordinate will move
So, (2, 2) becomes
becomes
(
,
(
unit.
,
)
by a factor
)=(
.
)
, and (–2, 2)
,
. Now plot the three points and use them to sketch g(x).
Your Turn
2.
1 (x + 6) + 4, find the vertex and two other points
Given g(x) = -_
⎟
5⎜
and use them to help you graph g(x).
y
16
8
x
-16 -8
0
8
-8
16
-16
Explain 2
Writing Absolute Value Functions from a Graph
⎜
⎟
1 (x − h) + k has values other than 1 for
If an absolute value function in the form g(x) = a _
b
both a and b, you can rewrite that function so that the value of at least one of a or b is 1.
a (x − h) = _
a (x − h) .
1 (x − h) = _
When a and b are positive: a _
b
b
b
⎜
⎟ ⎜
⎟
⎜
⎟
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When a is negative and b is positive, you can move the opposite of a inside the absolute value
⎜⎟
⎜⎟
⎜_⎟
1 = −1(2) _
1 = −1 2 .
expression. This leaves −1 outside the absolute value symbol: −2 _
b
b
b
When b is negative, you can rewrite the equation without a negative sign, because of the
1 (x − h) = a _
1 (x − h) . This case has now been
properties of absolute value: a _
−b
b
reduced to one of the other two cases.
⎜
⎟ ⎜
⎟
Given the graph of an absolute value function, write the
function in the form g(x) = a 1 (x − h) + k.
b
Let a = 1.
Example 2
A
⎜_
⎟
The vertex of g(x) is at (2, 5). This means that h = 2 and k = 5.
The value of a is given: a = 1.
⎜
⎟
1 (x - 2) + 5.
Substitute these values into g(x), giving g(x) = _
b
8
4
y
(2, 5)
(6, 6)
x
-8
-4
0
-4
4
8
-8
Choose a point on g(x) like (6, 6), Substitute these values into g(x), and
solve for b.
Module 2
67
Lesson 1
⎜_  ⎟
⎜ ⎟
6 = ​ ​  1  ​(6 - 2) ​ + 5
b
Simplify.6 = ​ _
​  1  ​(  4) ​ + 5
b
4
Subtract 5 from each side.
1 = ​ ​ _ ​  ​
b
4
4  ​ 
Rewrite the absolute value as two equations. 1 = ​ _  ​ or 1 = -​ _
b
b
Substitute.
⎜⎟
b = 4 or b = -4
Solve for b.
Based on the problem conditions, only consider b = 4. Substitute into g(x) to find the
equation for the graph.
⎜
⎟
g(x) = ​ _
​  1 ​ (x - 2) ​+ 5
4
BLet b = 1.
The vertex of g(x) is at . This means that h =
k=
The value of b is given: b = 1.
|​​ ​ 
Substitute these values into g(x), giving g(x) = a​|​|​ ​x  ​​
Now, choose a point on g(x) with integer coordinates, ​ 0,
Substitute these values into g(x) and solve for a.
and
)
.
4
(0, 3)
-8
-4
​.
x
0
4
-4
8
-8
|​​  ​
​|​  ​+
​|​
|​​ ​ 
g(x) = a​|​|​ x ​ ​​
Substitute.
= a​⎜0 - 1⎟​ + 6
Simplify.
= a​⎜-1⎟​ + 6
Solve for a.
= a
.
Your Turn
3.
Given the graph of an absolute value function, write the function in the
​  1b  ​(x − h) ​ + k.
form g(x) = a​ __
⎜
⎟
8
y
4
x
-8
Module 2
a
68
-4
0
-4
4
8
-8
Lesson 1
© Houghton Mifflin Harcourt Publishing Company
Therefore g(x) =
(
|​​  ​
​|​  ​+
​|​
y
8 (1, 6)
Explain 3
Modeling with Absolute Value Functions
Light travels in a straight line and can be modeled by a linear function. When
light is reflected off a mirror, it travels in a straight line in a different direction.
From physics, the angle at which the light ray comes in is equal to the angle
at which it is reflected away: the angle of incidence is equal to the angle of
reflection. You can use an absolute value function to model this situation.
Example 3
Source
Normal
Angle of
Incidence
Angle of
Reflection
Solve the problem by modeling the situation with an
absolute value function.
Law of Reflection
At a science museum exhibit, a beam of light originates at a point 10 feet off the floor. It is reflected off a mirror
on the floor that is 15 feet from the wall the light originates from. How high off the floor on the opposite wall
does the light hit if the other wall is 8.5 feet from the mirror?
Analyze Information
Identify the important information.
• The model will be of the form g(x) =
.
• The vertex of g(x) is
.
• Another point on g(x) is
.
• The opposite wall is
10 ft
15 ft
feet from the first wall.
8.5 ft
Mirror
Formulate a Plan
Let the base of the first wall be the origin. You want to find the value of g(x) at
x=
, which will give the height of the beam on the opposite wall. To do
so, find the value of the parameters in the transformation of the parent function.
In this situation, let b = 1. The vertex of g(x) will give you the values of
(
).
Use a second point to solve for a. Evaluate g
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Solve
The vertex of g(x) is at
Evaluate g(x) at
Substitute.
Simplify.
Simplify.
Solve for a.
Therefore g(x) =
Module 2
(
)
.
|
, 0 . Substitute, giving g(x) = a||x -
||
|+
.
and solve for a.
|
10 = a||
|
- 15 || +
|
= a ||
10 =
||
|
a
a=
(
. Find g
). g(23.5) =
69
Lesson 1
Justify and Evaluate
The answer of
makes sense because the function is symmetric with
respect to the line
. The distance from this line to the second wall is
a little more than
the distance from the line to the beam’s origin.
Since the beam originates at a height of
at a height of a little over
, it should hit the second wall
.
Your Turn
4.
Two students are passing a ball back and forth, allowing it to bounce once between them. If one student
bounce-passes the ball from a height of 1.4 m and it bounces 3 m away from the student, where should the
second student stand to catch the ball at a height of 1.2 m? Assume the path of the ball is linear over this
short distance.
5.
In the general form of the absolute value function, what does each parameter represent?
6.
Discussion Explain why the vertex of ƒ(x) = ⎜x⎟ remains the same when ƒ(x) is stretched or compressed
but not when it is translated.
7.
Essential Question Check-In What are the features of the graph of an absolute value function?
Module 2
70
Lesson 1
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Elaborate
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Predict what the graph of each given function will look like. Verify your
prediction using a graphing calculator. Then sketch the graph of the function.
1.
g(x) = 5|x − 3|
2.
g(x) = −4|x + 2| + 5
y
4
4
2
2
x
x
-4
-2
0
2
-2
-4
4
-2
⎜
⎟
7 (x − 6) + 4
g(x) = _
5
y
8
4.
© Houghton Mifflin Harcourt Publishing Company
0
-4
⎜
⎟
x
4
-8
8
-4
0
4
-4
8
-8
-8
5.
4
4
x
-4
2
-2
3 (x − 4) + 2
g(x) = _
7
y
8
4
-8
0
-4
-4
3.
y
7 (x − 2) − 3
g(x) = _
⎜
⎟
4
4
y
2
x
-4
-2
0
-2
2
4
-4
Module 2
71
Lesson 1
Graph the given function and identify the domain and range.
6.
g​(x)​ = ⎜​ x⎟​
7.
8
y
g​(x)​ = _
​  4 ​​ ⎜  (x − 5)⎟​+ 7
3
y
16
0
4
-4
-12 -8
8
⎜
⎟
y
8
x
-8
-4
0
0
4
-4
4
-4
8
-8
⎜
⎟
y
⎟
7 ​ (x + 5) ​ − 4
11. g​(x)​ = ​ -​ _
3
y
8
4
4
x
x
-4
12
10. g​(x)​ = ​ _
​  5 ​ (x − 4) ​
7
4
-8
8
-8
⎜
​  3 ​  (x − 2) ​ − 7
g​(x)​ = ​ _
4
8
0
-16
-8
9.
4
x
x
-4
7 ​​ (x − 2) ​
g​(x)​ = −​ _
⎜
⎟
6
y
8
8
4
-8
8.
-8
8
-4
0
4
-4
8
x
-8
-4
-8
-8
0
4
-4
8
-8
Write the absolute value function in standard form for the given graph. Use a or b as
directed, b > 0.
12. Let a = 1.
8
13. Let b = 1.
y
4
4
-4
0
-4
2
4
8x
-4
-8
Module 2
-2
0
-2
2
© Houghton Mifflin Harcourt Publishing Company
-8
y
4x
-4
72
Lesson 1
14. A rainstorm begins as a drizzle, builds up to a heavy rain, and then
drops back to a drizzle. The rate r (in inches per hour) at which it
rains is given by the function r = −0.5⎜t − 1⎟ + 0.5, where t is the
time (in hours). Graph the function. Determine for how long it rains
and when it rains the hardest.
2
r
1
t
-2
-1
0
1
-1
2
-2
15. While playing pool, a player tries to shoot the eight ball into the corner
pocket as shown. Imagine that a coordinate plane is placed over the
pool table. The eight ball is at 5,__54 and the pocket they are aiming for
is at (10, 5). The player is going to bank the ball off the side at (6, 0).
( )
a. Write an equation for the path of the ball.
5 ft
10 ft
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b. Did the player make the shot? How do you know?
Module 2
73
Lesson 1
16. Sam is sitting in a boat on a lake. She can get burned by from the
sunlight that hits her directly and from sunlight that reflects off the
water. Sunlight reflects off the water at the point (2, 0) and hits Sam
at the point (3.5, 3). Write and graph the function that shows the
path of the sunlight.
y
4
2
x
-4
-2
0
2
-2
4
-4
17. The Transamerica Pyramid is an office building in San Francisco. It stands
853 feet tall and is 145 feet wide at its base. Imagine that a coordinate plane is
placed over a side of the building. In the coordinate plane, each unit represents
one foot. Write an absolute value function whose graph is the V-shaped outline
of the sides of the building, ignoring the “shoulders” of the building.
y = ⎜​ x + 6⎟​ − 4​ y = ⎜​ x − 6⎟​ − 4​ y = ⎜​ x − 6⎟​ + 4​
y
A B C
y
4
8
x
-8
Module 2
-4
0
-4
4
x
-8
-4
0
74
y
x
4
8
4
4
8
-4
0
-4
4
8
Lesson 1
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Alvarez/Getty Images
18. Match each graph with its function.
H.O.T. Focus on Higher Order Thinking
19. Explain the Error Explain why the graph shown is not the graph of
y = ⎜x + 3⎟ + 2. What is the correct equation shown in the graph?
8
y
4
x
-8
-4
0
-4
4
8
-8
20. Multi-Step A golf player is trying to make a hole-in-one on the miniature
golf green shown. Imagine that a coordinate plane is placed over the golf
green. The golf ball is at (2.5, 2) and the hole is at (9.5, 2). The player is going
to bank the ball off the side wall of the green at (6, 8).
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a. Write an equation for the path of the ball.
b. Use the equation in part a to determine if the player makes the shot.
Module 2
75
Lesson 1
Lesson Performance Task
Suppose a musical piece calls for an orchestra to start at fortissimo
(about 90 decibels), decrease steadily in loudness to pianissimo
(about 50 decibels) in four measures, and then increase steadily
back to fortissimo in another four measures.
a.Write a function to represent the sound level s in
decibels as a function of the number of measures m.
b.After how many measures should the orchestra be at the
loudness of mezzo forte (about 70 decibels)?
c. Describe what the graph of this function would look like.
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Alamy Images
Module 2
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Lesson 1