Stretch and Shrink
Stretching 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 or wider; shrinking makes it shorter or narrower. Vertical stretch/shrink is $g(x) = a \cdot f(x)$: if $|a| > 1$ it's a vertical stretch, if $0 < |a| < 1$ a vertical shrink; horizontal stretch/shrink is $g(x) = f(b \cdot x)$, where $|b| > 1$ shrinks horizontally and $0 < |b| < 1$ stretches horizontally (the horizontal effect is reciprocal). These scalings are linear transformations: vertical scaling by $a$ is postcomposition with multiplication, $T_a(y) = ay$, and 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
$f(x) = x^2$ is the parent parabola; $g(x) = 3x^2$ is stretched vertically (taller, narrower-looking) while $h(x) = 0.5x^2$ is shrunk vertically (flatter). For $f(x) = \sin(x)$: $g(x) = 2\sin(x)$ doubles the amplitude (vertical stretch), while $h(x) = \sin(2x)$ doubles the frequency and halves the period (horizontal shrink), two genuinely different transformations. The scaling theorem in Fourier analysis reveals a fundamental trade-off: compressing a signal in time ($b > 1$) spreads its frequency content, the time-frequency uncertainty principle, formalizing why you cannot have arbitrarily sharp time and frequency resolution simultaneously.
Key Insight
A vertical stretch with factor $a > 1$ makes all $y$-values $a$ times larger, like zooming in vertically on the graph, while a factor between $0$ and $1$ compresses it. In the general form $g(x) = a \cdot f(b(x - h)) + k$, the factor $a$ controls vertical scaling and $b$ controls horizontal scaling, and horizontal scaling by $b$ compresses by factor $1/b$ (counterintuitive but consistent with the substitution rule). This time-frequency uncertainty underlies the Heisenberg uncertainty principle in quantum mechanics: position and momentum are Fourier transform pairs, and scaling one domain compresses the other.