Identify horizontal stretches and compressions

A horizontal stretch or compression changes the input scale while preserving output values: for $g(x)=f(bx)$ with $b>0$ , each point $(x,y)$ on $f$ maps to $(x/b,y)$ ; $b>1$ produces a horizontal compression by factor $1/b$ , while $0<b<1$ produces a stretch by factor $1/b$ . This relationship is interpreted across equations, graphs, and point mappings, emphasizing that an input multiplier acts reciprocally and is distinct from a vertical scale factor; negative factors and more generalized parameter analysis are outside this scope.

Detailed Explanation: 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.

Learn by doing: Identify horizontal stretches and compressions

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Function Transformations - Mapping Notation - Single Transformation to Action

Earned ?