5.5: Horizontal Stretches & Compressions

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Objectives:

• Recognize that horizontal stretches and compressions correspond with changes to the inputs
• Horizontally stretch and compress a function that is given either explicitly or graphically

Important Items

Definitions:

horizontal stretch, horizontal compression

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Lesson Guide

Notes to the instructor: The main focus on this section should be taking a given function and knowing how the stretches/compressions affect the graph of this function.

Warm-Up

Perhaps remind students again of the transformations they have seen up to this point. Reiterate the ideas of changing inputs and outputs when we apply transformations.

Recognize that horizontal stretches and compressions correspond with changes to the inputs

Have students do Problem 1. Discuss the differences of the graphs.

Observe that the $x$-intercept values change, but the $y$-intercepts stay the same, which makes sense since only the input is being changed.

Horizontally stretch and compress a function that is given either explicitly or graphically

Work with students to fill this part out in their course packet (part (d)).

If $f(x)$ is a function and $k>1$ is a constant, then the graph of 
 * $g(x)=f\left(\frac{1}{k}x\right)$ horizontally stretches the graph of $f(x)$ by a factor of $k$,
* $g(x)=f(kx)$ horizontally compresses the graph of $f(x)$ by a factor of $k$.

If $k<-1$, then the graph of $g(x)$ also involves a reflection of the graph of $f(x)$ about the $y$-axis.

Have students do Problems 2-6.

Have students talk about Problem 7 at their tables. Force each table to make a decision and then appoint someone to write their answer on the board. Use this to lead a discussion.

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