Stretch and Shrink
Functions & Advanced AlgebraStretching or shrinking a function scales its graph vertically or horizontally, making it taller, shorter, wider, or narrower.
Formula
g(x) = a \cdot f(b \cdot x)
Definition
Stretching makes a graph taller/wider; shrinking makes it shorter/narrower. Multiplying the output by a number stretches or shrinks the graph vertically.
Example
$f(x) = x^2$ is the parent parabola. $g(x) = 3x^2$ is stretched vertically (taller and narrower-looking). $h(x) = 0.5x^2$ is shrunk vertically (flatter).
Key Insight
A vertical stretch with factor $a > 1$ makes all $y$-values $a$ times larger. A factor between $0$ and $1$ compresses. Think of it as zooming in or out vertically on the graph.
Definition
Vertical stretch/shrink: $g(x) = a \cdot f(x)$. If $|a| > 1$, vertical stretch; if $0 < |a| < 1$, vertical shrink (compression). Horizontal stretch/shrink: $g(x) = f(b \cdot x)$. If $|b| > 1$, horizontal shrink; if $0 < |b| < 1$, horizontal stretch. (Horizontal effects are reciprocal.)
Example
$f(x) = \sin(x)$: $g(x) = 2\sin(x)$ doubles the amplitude (vertical stretch). $h(x) = \sin(2x)$ doubles the frequency/halves the period (horizontal shrink). These are different transformations.
Key Insight
In the form $g(x) = a \cdot f(b(x - h)) + k$, the factor $a$ controls vertical scaling and $b$ controls horizontal scaling. Note: horizontal scaling by $b$ compresses by factor $1/b$ (counterintuitive but consistent with the substitution rule).
Definition
Scalings are linear transformations. Vertical scaling by $a$ is postcomposition with multiplication: $T_a(y) = ay$. Horizontal scaling by $b$ is precomposition with dilation: $f(bx)$. In Fourier analysis, if $F(f(t)) = F(w)$, then $F(f(bt)) = (1/|b|) \cdot F(w/b)$, the scaling theorem.
Example
The scaling theorem in Fourier analysis shows a fundamental trade-off: compressing a signal in time ($b > 1$) spreads its frequency content. This is the time-frequency uncertainty principle, formalizing why you cannot have arbitrarily sharp time and frequency resolution simultaneously.
Key Insight
This uncertainty underlies the Heisenberg uncertainty principle in quantum mechanics: position and momentum are Fourier transform pairs, and scaling one domain compresses the other.