Skill: Identify vertical stretches and compressions

A vertical scaling changes only the $y$-values of a function.

If

$$g(x)=a f(x),$$

then each point $(x,y)$ on $f$ becomes

$$(x,ay)$$

on $g$.

• If $a>1$, the graph has a vertical stretch: points move farther from the $x$-axis.
• If $0<a<1$, the graph has a vertical compression: points move closer to the $x$-axis.
• The $x$-values stay the same, so the zeros of the function do not change.

Example: Let

$$f(x)=x^2-4$$

and

$$g(x)=\frac{1}{2}f(x).$$

Step 1: Identify the scale factor.

Compare $g(x)=\frac12 f(x)$ with $g(x)=af(x)$. The scale factor is

$$a=\frac12.$$

Step 2: Classify the transformation.

Since

$$0<\frac12<1,$$

the graph is vertically compressed by a factor of $\frac12$.

Step 3: Apply the scale factor to points.

For $f(x)=x^2-4$:

• When $x=0$, $f(0)=-4$, so $(0,-4)$ becomes

$$\left(0,\frac12(-4)\right)=(0,-2).$$

• When $x=3$, $f(3)=5$, so $(3,5)$ becomes

$$\left(3,\frac12(5)\right)=\left(3,\frac52\right).$$

The $x$-intercepts remain the same. Since $f(x)=0$ at $x=-2$ and $x=2$, $g(x)$ also has zeros at $x=-2$ and $x=2$.

Therefore, $g(x)=\frac12 f(x)$ is a vertical compression toward the $x$-axis by a factor of $\frac12$.