Stretching and Compressing Functions

Stretching and Compressing Functions — Moosa Academy

Translations move a graph and reflections flip it; a stretch changes its proportions. Multiplying the function stretches it vertically, while multiplying the variable stretches it horizontally — and the second of these works in the direction you would not expect.

Concept Vertical stretch

g(x) = a \cdot f(x)

a > 1

— stretched vertically, taller.

0 < a < 1

— compressed vertically, flatter.

Every output is multiplied by $a$ , so the height changes while the horizontal positions stay put. For a sine wave this is exactly the amplitude: $5\sin x$ runs from −5 to 5, and $0.4\sin x$ from −0.4 to 0.4.

Concept Horizontal stretch

g(x) = f(a x)

a > 1

— compressed horizontally; the cycle repeats faster.

0 < a < 1

— stretched horizontally; the cycle repeats slower.

This is the reversal to watch for. A larger $a$ squeezes the graph rather than widening it, because $f(5x)$ reaches at $x = 1$ the value the parent only reached at $x = 5$ — the graph gets there five times sooner.

Note The four cases

Form	a > 1	0 < a < 1	What changes
a · f(x)	Stretch	Compress	Height, or amplitude
f(a x)	Compress	Stretch	Width, or period

The pattern matches the translation rules exactly: whatever acts on the whole function behaves as expected, and whatever acts on the variable inside behaves in reverse.

Example Four sine transformations

5\sin x

— vertical stretch, amplitude 5, running from −5 to 5.

0.4\sin x

— vertical compression, amplitude 0.4.

\sin(5x)

— horizontal compression; five complete cycles fit where one did.

\sin(0.4x)

— horizontal stretch; the cycle takes longer to complete.

⟹ amplitude comes from the outer factor, period from the inner one

In $5\sin x$ and $\sin(5x)$ the same number produces opposite effects on opposite properties. Reading which side of the function the coefficient sits on is the whole skill.

Summary

a · f(x) scales the graph vertically: a > 1 stretches, 0 < a < 1 compresses.
f(ax) scales it horizontally, and in reverse: a > 1 compresses, 0 < a < 1 stretches.
The vertical factor sets the amplitude; the horizontal one sets the period.
Acting on the whole function behaves as expected; acting on the variable inside behaves in reverse.