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Compress or Stretch Horizontally

Another transformation that can be applied to a function is stretching or compressing a function horizontally. $\\$ We can multiply a function by a positive constant $c$ to stretch and compress a function horizontally. $\\$
Given the graph of $f(x)$, $f(cx)$ will be the graph of $f(x)$ stretched horizontally if the value of $c$ is greater than $0$ but less than $1$ ($0<c<1$). $f(cx)$ will be the graph of $f(x)$ compressed horizontally if the value of $c$ is greater than $1$ ($c>1$).

Example 1

Graph of $g(x)= (\dfrac{1}{3}x)^2$ $\\$ The graph of $g(x)= (\dfrac{1}{3}x)^2$ will be a horizontal stretch of $f(x)=x^2$ by $3$ units. $\\$ The red graph is the graph of $f(x)=x^2$ and the blue graph is the graph of $g(x)=(\dfrac{1}{3}x)^2$

Example 2

Graph of $g(x)=(3x)^2$ $\\$ The graph $g(x)=(3x)^2$ will be a horizontal compression of $f(x)=x^2$ by $3$ units. $\\$ The red graph is the graph of $f(x)=x^2$ and the blue graph is the graph of $g(x)=(3x)^2$

(Video) Horizontal Stretches and Compressions

by mathlablady1

This video by mathlablady1 explains what happens with a horizontal stretch or compression.

Summary

When we are looking at a horizontal stretch or compression, what we are seeing is the result of a value being multiplied by the $x$ value.

$$f(ax)$$

We are multiplying each $x$-coordinate by $a$. The function will stretch horizontally if $0<a<1$, and the function will compress horizontally if $a>1$.

Example

Let's look at the graphs of the parent function, $f(x)=x^2$, which is in purple, and the transformed function, $f(x)=(2x)^2$, which is in green.

As we can see, the green (the transformed) graphed looks much skinner than the original parent function. That's because this is a horizontal compression! It kind of seems as though the two legs of the quadratic function are being pushed toward the $y$-axis.

Now, let's look at the graph of $f(x)=(\frac{1}{2} x)^2$.

In purple, we have the graph of the parent function, $f(x)=x^2$, and in green we have the graph of the function $f(x)=(\frac{1}{2} x)^2$.

Horizontal Dilation

In context

In our series of function transformations, we learned about vertical translations, horizontal translations, vertical reflections, and horizontal reflections. In the last lesson, we learned about vertical stretches and compressions.

The next in the series of function transformations is the horizontal dilation! Think of it as a graph being stretch or compressed by a factor.

When we horizontally stretch or compress a function, we are essentially moving it closer or further from the $y$-axis. Because we are changing the $x$-values, we multiply the $x$ value in the equation by a constant. If the constant is a fraction, the function will be horizontally stretched. If the constant is $>1$, the function will be compressed.

$$f(cx)$$

You are multiplying the inputs. As a result, you change the outputs of the function.

Let's look at the graph of the function of the parent quadratic equation. We know what that looks like.

Let's say we want to stretch the function. This is the graph of $f(x)=\left(\frac{1}{4}x\right)^{2}$ in blue!

For some people, it may help you to understand this concept better if you choose a point on the parent graph and see what happens when you apply a horizontal dilation to it.

As we can see, multiplying the $x$ value inside the parentheses by a fraction will stretch the function graph.

Now, let's look at a horizontal compression in green.

In green, we have the graph of the function $f(x)=\left(4x\right)^{2}$. This is a horizontal compression.

We have finished both lessons on the stretching of the graphs! We're well on our way to master transforming more complex functions into each other.