Phase Shifts and Periodic Models

For y=asin(b(x-h))+d, with a,b 0, amplitude is |a|, period is T= 2pi|b|, and the midline is y=d. The shift is h to the right. A negative a reflects the curve in its midline. Use the same parameters for cosine. For tangent, T= |b| and there is no amplitude. Factor the argument before reading the shift: 2x-pi=2(x- 2). Example A water level varies from 2 m to 8 m over a 12-hour cycle, with a maximum at t=1. One model is H(t)=5+3cos(6(t-1)). Check a maximum, a minimum and one period against the context. A model is an approximation: restrict its time domain to the period over which its assumptions are reasonable. For degree arguments use 360 in the period formula.