Topics
4. Functions & Linear Relationships
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United States of AmericaFL
Grade 912

4.10 Transforming absolute value functions

Lesson
Practice
Lesson

Concept summary

Absolute value functions can be transformed in the same ways as linear functions. Recall the types of transformations that we have looked at:

Translation

A transformation in which every point in the graph of the function is shifted the same distance in the same direction.

Reflection

A transformation that flips a function across a line of reflection, producing a mirror image of the original function.

Vertical compression

A transformation that pushes all of the y-values of a function towards the x-axis.

Vertical stretch

A transformation that pulls all of the y-values of a function away from the x-axis.

The parent function of the absolute value function family is the function y = \left|x\right|. Other linear absolute value functions can be obtained by transformations of this parent function.

Worked examples

Example 1

A graph of a function f\left(x\right) is shown below.

-10
-8
-6
-4
-2
2
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x
-10
-8
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y
a

Sketch a graph of the function g\left(x\right) = -f\left(x\right).

Approach

The function g\left(x\right) is formed by changing the sign of the function values of f\left(x\right). So the x-intercept will stay the same, and every point above the x-axis will change to a point the same distance below the x-axis.

Solution

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x
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y
b

Describe the transformation from f\left(x\right) to g\left(x\right).

Solution

The transformation from f\left(x\right) to g\left(x\right) is a reflection across the x-axis.

Example 2

Consider the following table of values for two absolute value functions f\left(x\right) and g\left(x\right):

x0123456
f(x63036912
g(x)12963036
a

Describe the transformation from f\left(x\right) to g\left(x\right).

Solution

Looking at the values in the table for f\left(x\right), we can see that it has a vertex at \left(2, 0\right) and has a rate of change of 3 on either side.

Comparing this to the value for g\left(x\right) we can see that it has the same rate of change, but has its vertex at \left(4, 0\right) instead.

So we can get from f\left(x\right) to g\left(x\right) by translating 2 units to the right.

Reflection

Since absolute value functions are symmetric about a vertical axis, it is also possible to describe this transformation as a reflection across the line x = 3.

b

Write a function that describes the relationship between f\left(x\right) and g\left(x\right).

Solution

We can write "a translation by 2 units to the right" as g\left(x\right) = f\left(x - 2\right).

Reflection

Since it is also possible to describe this transformation as a reflection across the line x = 3, we can also express this transformation as g\left(x\right) = f\left(6 - x\right).

Outcomes

MA.912.AR.4.3

Given a table, equation or written description of an absolute value function, graph that function and determine its key features.

MA.912.F.2.1

Identify the effect on the graph or table of a given function after replacing f(x) by f(x)+k,kf(x), f(kx) and f(x+k) for specific values of k.

MA.912.F.2.3

Given the graph or table of f(x) and the graph or table of f(x)+k,kf(x), f(kx) and f(x+k), state the type of transformation and find the value of the real number k.

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