Amplitude and Period of a Graph

For a periodic graph, the amplitude is the maximum distance of the curve from the $x$-axis — equivalently, the height from the axis up to a peak (or down to a trough). The period is the horizontal length required for the graph to complete exactly one full repeating cycle.

Two structural facts used throughout the chapter:

1. The graph of a periodic function is nothing but the SAME portion — one period's worth — repeated end to end indefinitely in both directions. So it always suffices to work out the shape on ONE interval of length equal to the period, then copy-paste it.
2. The graph of an odd function is symmetric about the origin (a $180°$ rotation maps the curve to itself); the graph of an even function is symmetric about the $y$-axis (a mirror reflection in the $y$-axis maps the curve to itself).

For the general sinusoids $y=A\sin(\alpha x)$ and $y=B\cos(\beta x)$: both satisfy $-A\le A\sin(\alpha x)\le A$ and $-B\le B\cos(\beta x)\le B$ respectively, so their amplitudes are $A$ and $B$, and their periods are $\dfrac{2\pi}{\alpha}$ and $\dfrac{2\pi}{\beta}$ respectively. Graphing either is simply a matter of extending the portion drawn on $\left[0,\tfrac{2\pi}{\alpha}\right]$ (resp. $\left[0,\tfrac{2\pi}{\beta}\right]$). …