Graphs of Sine and Cosine; Sinusoidal Features

Four numbers describe every sinusoid

Write the function as $y=a\sin\bigl(b(x-c)\bigr)+d$ (or with cosine). Then:

• Midline $y=d$ — the horizontal average $\dfrac{\max+\min}{2}$;
• Amplitude $|a|=\dfrac{\max-\min}{2}$ — distance from midline to an extreme, never max minus min;
• Period $\dfrac{2\pi}{b}$ — so $b$ is not the period, it counts how many cycles fit in $2\pi$;
• Phase shift $c$ — read only after factoring $b$ out: $\sin(3x+\tfrac{\pi}{2})=\sin\bigl(3(x+\tfrac{\pi}{6})\bigr)$, a shift of $\tfrac{\pi}{6}$ left, not $\tfrac{\pi}{2}$.

A negative $a$ reflects across the midline; it does not change the amplitude.

Anchor points

Within one period, a sinusoid visits five equally spaced anchors: midline–max–midline–min–midline for $+\sin$, max–midline–min–midline–max for $+\cos$. Cosine starts at a maximum, sine starts at the midline going up — this is what decides which template fits a given graph or data set.