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4. Functions & Linear Relationships
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United States of AmericaTennessee
Algebra 1

4.09 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.

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x
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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.

Outcomes

A1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

A1.F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.*

A1.F.IF.C.7

Graph functions expressed algebraically and show key features of the graph by hand and using technology.*

A1.F.BF.B.2

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given graphs.

A1.MP1

Make sense of problems and persevere in solving them.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP6

Attend to precision.

A1.MP8

Look for and express regularity in repeated reasoning.

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