Skill: Identify horizontal stretches and compressions

A horizontal transformation changes the input values of a function, so the $y$-values stay the same.

For a function

$$g(x)=f(bx), \qquad b>0,$$

each point $(x,y)$ on the graph of $f$ moves to

$$\left(\frac{x}{b},y\right).$$

The input multiplier $b$ works reciprocally:

• If $b>1$, the graph is horizontally compressed by a factor of $\frac{1}{b}$.
• If $0<b<1$, the graph is horizontally stretched by a factor of $\frac{1}{b}$.

Example

Suppose the graph of $f$ contains the point $(6,4)$. Describe the transformation and find the corresponding point on

$$g(x)=f(2x).$$

Step 1: Identify the input multiplier.

The expression inside $f$ is $2x$, so

$$b=2.$$

Step 2: Determine the type of horizontal change.

Because $b=2>1$, the graph is horizontally compressed by a factor of

$$\frac{1}{2}.$$

Step 3: Apply the point rule.

Divide the $x$-coordinate by $2$, but leave the $y$-coordinate unchanged:

$$(6,4)\longrightarrow \left(\frac{6}{2},4\right)=(3,4).$$

So, the point $(6,4)$ on $f$ becomes $(3,4)$ on $g$. The graph moves closer to the $y$-axis horizontally, while its output value, $4$, remains the same.