Stretches and Compressions of Functions
Simple Explanation
A vertical stretch/compression, y = a·f(x), scales the graph's height by a factor of a (taller if |a|>1, flatter if 0<|a|<1). A horizontal stretch/compression, y = f(bx), scales its width — but in the OPPOSITE sense: a larger b actually compresses the graph horizontally, not stretches it.
Why Do We Need It?
The horizontal case is genuinely counter-intuitive — recognizing that b works "backwards" compared to a is essential to avoid a very common and persistent error.
See It
An upward-opening parabola and a wider, flatter version of the same shape
Formula
Stretching or Compressing a Function
y = a·f(x) stretches/compresses vertically. y = f(bx) stretches/compresses horizontally.
Multiplying the output by a scales the graph vertically (|a|>1 stretches, 0<|a|<1 compresses); multiplying the input by b scales it horizontally, but in the opposite sense (|b|>1 compresses, 0<|b|<1 stretches).
- a
- — the vertical scale factor
- b
- — the horizontal scale factor (acts inversely — a larger b compresses)
When to use it: Whenever a graph needs to be made taller/shorter or narrower/wider.
Worked Example
Apply a horizontal compression
The function y = f(x) has a key point at (4, 5). Find the corresponding point on y = f(2x).
Why Does This Work?
y=f(bx) evaluates f at bx instead of x — to reach the same output that f(x) gave at some original input x₀, the new function needs bx = x₀, i.e. x = x₀/b — so every key x-coordinate shrinks by a factor of b (for b>1), which is a compression, not a stretch, despite b appearing to "multiply."
Real-Life Example
Adjusting a sound wave's amplitude and frequency
An audio engineer increases a sound wave's volume (a vertical stretch) and separately increases its pitch/frequency (a horizontal compression).
These are literally the two transformations covered here — amplitude scaling is vertical (y=a·f(x)), frequency scaling is horizontal (y=f(bx)) and behaves inversely.
Practice
Which transformation makes the graph of y=f(x) narrower (compressed horizontally)?
HardCommon mistake
Assuming y=f(bx) with b>1 stretches the graph horizontally (matching the intuition from the vertical case, y=a·f(x)) — horizontal scaling is inverted: b>1 compresses, 0<b<1 stretches.
Quick Review
- y = a·f(x): vertical scale by a. |a|>1 stretches; 0<|a|<1 compresses.
- y = f(bx): horizontal scale, but inverted. |b|>1 compresses; 0<|b|<1 stretches.
- Horizontal scaling is the one genuinely counter-intuitive transformation — always double-check it.