A stretch changes a graph’s proportions rather than its position. Multiplying the function stretches it vertically; multiplying the variable stretches it horizontally — in the opposite direction from what you would expect.
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.
Every output is multiplied by , so the height changes while the horizontal positions stay put. For a sine wave this is exactly the amplitude: runs from −5 to 5, and from −0.4 to 0.4.
This is the reversal to watch for. A larger squeezes the graph rather than widening it, because reaches at the value the parent only reached at — the graph gets there five times sooner.
| 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.
In and the same number produces opposite effects on opposite properties. Reading which side of the function the coefficient sits on is the whole skill.