I know the period of sin(x) is 2π but I keep messing up which number changes it.
Accepted answer
Only the coefficient **on x** changes the period. In `y = a·sin(b(x − h)) + k`:
**period = 2π / |b|**
Here `b = 2`, so the period is `2π/2 = π`. The graph completes a full cycle twice as fast.
The others don't touch it: `a = 3` is the amplitude (vertical stretch), `h = π/4` is the phase shift (horizontal slide), `k = 1` is the midline (vertical slide). A common slip is reading `b` off an unfactored form — `sin(2x − π/2)` must be factored to `sin(2(x − π/4))` before `h` is readable, though `b` (and so the period) is the same either way.
For tangent the base period is π, so it's `π/|b|`.