Amplitude
Amplitude is the maximum displacement of a periodic function from its midline, determining the height of peaks and depth of troughs.
Formula
\text{amplitude} = |A| \text{ in } y = A\sin(Bx + C) + D
Definition
Amplitude is how tall a wave is from its midline to its highest (or lowest) point; for the basic sine wave, the amplitude is $1$. For $y = A\sin(Bx + C) + D$ or $y = A\cos(Bx + C) + D$, the amplitude is $|A|$, the distance from the midline $y = D$ to either the maximum or minimum, so the graph oscillates between $D - |A|$ and $D + |A|$, and equivalently $|A| = (\max f - \min f)/2$. In signal processing, amplitude corresponds to the magnitude of a Fourier coefficient: for $f(x) = \sum A_n \sin(nx + \phi_n)$, each $|A_n|$ is the amplitude of the $n$th harmonic, and total energy is proportional to $\sum A_n^2$ by Parseval's theorem.
Example
$y = 3\sin(x)$ has amplitude $3$, going from $-3$ to $+3$, while $y = -4\cos(2x) + 1$ has amplitude $4$, midline $y = 1$, max $5$, and min $-3$. A sound wave at $440$ Hz (A4) with amplitude $0.8$ Pa is $p(t) = 0.8\sin(2\pi \cdot 440t)$; doubling the amplitude to $1.6$ increases sound intensity by $4\times$, since intensity scales with amplitude squared.
Key Insight
Think of amplitude as the volume of a sound wave or the size of an ocean wave: bigger amplitude means bigger swings up and down, and amplitude is always positive (it is the absolute value of $A$), a negative $A$ simply flips the graph upside down without changing its size. In quantum mechanics, the amplitude of a wave function determines the probability density, $|\psi(x)|^2$, via the Born rule, making amplitude central to quantum measurement theory.