Identify vertical stretches and compressions

Vertical scaling means multiplying every output of a function by a scale factor: $g(x)=a f(x)$ , mapping each point $(x,y)$ to $(x,ay)$ . For $a>1$ , the graph stretches away from the $x$ -axis; for $0<a<1$ , it compresses toward it, while $x$ -values and zeros remain unchanged. Negative scale factors and their accompanying reflections are outside this scope.

Detailed Explanation: Identify vertical stretches and compressions

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

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 $\frac12$ .

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Function Transformations (Definition) - Double Definition (Values) to Transformation

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Identify vertical stretches and compressions

Detailed Explanation: Identify vertical stretches and compressions

Learn by doing: Identify vertical stretches and compressions

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Function Transformations (Definition) - Double Definition (Values) to Transformation