Absolute Value Functions and Graphs

Absolute Value Functions

5-1

Objectives

❏

Graph an absolute value function and identify the key features of the graph

❏

Calculate and interpret the rate of change of an absolute value function over a
specified interval.

Graph the Absolute Value Function

What are the features of the graph of f(x)=|x|?

x

f(x)=|x|

(x, f(x))

-2

2

(-2,2)

-1

1

(-1,1)

0

0

(0,0)

1

1

(1,1)

2

2

(2,2)

The graph has a vertex where the axis of symmetry
intersects the graph. It represents the minimum value
of the image

The graph has an axis of symmetry, which intersects the vertex
and divides the graph into two sections, or pieces, that are the
image of each other under a reflection

Graph g(x)= 2|x| and
compare it to f(x)=|x|

x

g(x)=2|x|

(x, g(x))

-2

-1

0

1

2

Graph g(x)= 2|x| and
compare it to f(x)=|x|

x

g(x)=2|x|

(x, g(x))

-2

-1

0

1

2

​4 (-2,4)

2 (-1, 2)
0 (0,0)
2 (1,2)
4 (2,4)

Multiple Choice

Do both graphs have the same Domain and Range?

• Yes

• No

Graph h(x)= -1|x| and
compare it to f(x)=|x|

x

h(x)=-1|x|

(x,h(x))

-2

-1

0

1

2

Graph h(x)= -1|x| and
compare it to f(x)=|x|

x

h(x)=-1|x|

(x,h(x))

-2

-1

0

1

2

​-2 (-2,-2)
-1 (-1,-1)
0 (0,0)
-1 (1,-1)
-2 (2,-2)

Multiple Choice

What is the range for the graph of h(x)=-1|x|?

• y<0

• y $$\le$$ 0

Jay rides in a boat from his home to his friends home in a
neighboring state. The graph of the function d(t)=30|t-1.5|
shows the distance of the boat in miles from the state line
at t hours. Assume the graph shows Jay’s entire trip

What is the rate of change over the interval 2≤t≤2.5?

Use the slope
formula to determine
the rate of change
from t=2 to t=2.5
Use the points (2,15)
and (2.5, 30)

The boat is traveling 30m/h
away from the border