Stretches and Compressions

Changing the Shape

Unlike translations, stretches and compressions change the shape of a graph.

They make it taller, shorter, wider, or narrower.

Vertical Stretch and Compression

$a \cdot f(x)$

Multiply the output by a constant.

• $|a| > 1$ → stretch vertically (taller)
• $|a| < 1$ → compress vertically (shorter)

Example: $f(x) = x^2$

• $2f(x) = 2x^2$ → twice as tall
• $\frac{1}{2}f(x) = \frac{1}{2}x^2$ → half as tall

Every y-value gets multiplied by $a$.

Horizontal Stretch and Compression

$f(bx)$

Multiply the input by a constant.

• $|b| > 1$ → compress horizontally (narrower)
• $|b| < 1$ → stretch horizontally (wider)

Warning: This is backwards again!

$f(2x)$ makes the graph narrower, not wider.

$f(\frac{1}{2}x)$ makes the graph wider, not narrower.

Why is it backwards?

For $f(x) = x^2$, the point $(2, 4)$ is on the graph.

For $g(x) = (2x)^2$, we need $g(1) = (2 \cdot 1)^2 = 4$.

The point moved from $x = 2$ to $x = 1$. The graph got narrower.

Combining with Reflections

If $a$ or $b$ is negative, you also get a reflection:

• $a < 0$ → vertical stretch/compress and reflect across x-axis
• $b < 0$ → horizontal stretch/compress and reflect across y-axis

Example: $-2f(x)$ stretches vertically by 2 and flips upside down.