Trigonometric Functions and Graphs

$y = a\sin(b(x - c)) + d$

• $|a|$ = amplitude.
• Period = $\frac{2\pi}{|b|}$ (or $\frac{360°}{|b|}$).
• $c$ = phase shift (horizontal).
• $d$ = vertical shift.

• $\sin x$: starts at 0, period $2\pi$, range $[-1, 1]$.
• $\cos x$: starts at 1, period $2\pi$, range $[-1, 1]$.
• $\tan x$: period $\pi$, asymptotes at $x = \frac{\pi}{2} + n\pi$.

$y = 3\sin(2x - \frac{\pi}{3}) + 1$.

Amplitude: 3. Period: $\pi$. Phase shift: $\frac{\pi}{6}$ right. Vertical shift: 1 up. Range: $[-2, 4]$.

Find all solutions in $[0, 2\pi]$: $\sin x = \frac{1}{2}$ → $x = \frac{\pi}{6}, \frac{5\pi}{6}$.