2.7: Use Absolute Value Functions and
Transformations
Objectives:
1. To graph an absolute value function by
performing transformations on the parent
2. To apply transformations to graphing any
function
Building the Absolute Value Function
The absolute value
function is defined by
f (x) = |x|.
The graph of the
absolute value function
is similar to the linear
parent function, except
it must always be
positive.
4
2
-5
5
-2
-4
Building the Absolute Value Function
The absolute value
function is defined by
f (x) = |x|.
So we just take the
negative portion of the
graph and reflect it
across the x-axis
making that part
positive.
4
2
-5
5
-2
-4
Building the Absolute Value Function
The absolute value
function is defined by
f (x) = |x|.
This is the absolute value
parent function.
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2
-5
5
fx = x
-2
-4
Parent Function
• V-shape
• It is symmetric about the y-axis
• The vertex is the minimum point on the
graph
Translation
A translation is a
transformation that
shifts a graph
horizontally or
vertically, but doesn’t
change the overall
shape or orientation.
Translation
The graph of
y = |x – h| + k
is the graph of y = |x|
translated h horizontal
units and y vertical
units.
• The new vertex is at
(h, k)
Stretching and Shrinking
The graph of y = a|x| is graph of y = |x|
vertically stretched or shrunk depending on
the |a|.
The value of a acts like the slope.
Reflection
The graph of y = a|x| is graph of y = |x|
reflected across the x-axis when a < 0.
4
fx = - x
2
2
-5
4
5
-5
5
fx = x
-2
-2
-4
-4
Multiple Transformations
In general, the graph of an absolute value
function of the form y = a|x – h| + k can
involve translations, reflections, stretches
or shrinks.
To graph an absolute value function, start by
identifying the vertex.
Graphing Absolute Value Functions
Graphing y = a|x – h| + k is easy:
1.Plot the vertex (h, k). (note…if +h inside that means h is
negtive, if – h inside that means h is positive)
2.Use the a value as slope to plot more
points. Remember you have to do positive and negative slope to
get points on both sides of the V
3.Connect the dots in a V-shape.
Exercise 2
Graph the following functions without making
a table.
1. y = |x – 2| + 3 This graph will go right 2 and up 3 so from the
origin go right 2 and up 3. This is the vertex (2, 3). Now from that
point use the positive and negative slope (a = 1 here) to get more
points.
4
2
-5
5
fx = x
-2
-4
Exercise 2
Graph the following functions without making
a table.
1.y = (1/2)|x| This function does not have an
“h” or “k” so the vertex is (0, 0). Since a =
½ the slope is ½. Go up 1 and right 2 then
up one and left 2.
4
2
-5
5
fx = x
-2
-4
Exercise 2
Graph the following functions without making
a table.
1.f (x) = -3|x + 1| – 2 This graph will go left 1 and down two
so the vertex will be (-1, -2). Since “a” is negative the graph will
open down. Since the value of “a” is 3 the slope will be 3 and -3 (just
remember to go down.)
4
2
-5
5
fx = x
-2
-4
Transformations in General
You can perform transformations on the graph
of any function in manner similar to
transformations on the absolute value
function.