Phase Shifts and Periodic Models

For \(y=a\sin(b(x-h))+d\), with \(a,b\ne0\), amplitude is \(|a|\), period is \(T=\dfrac{2\pi}{|b|}\), 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, \(T=\dfrac\pi{|b|}\) and there is no amplitude. Factor the argument before reading the shift: \(2x-\pi=2\left(x-\dfrac\pi2\right)\).

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 \(\mathrm{H}(t)=5+3\cos\left(\dfrac\pi6(t-1)\right)\).

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, the period is \(360^\circ/|b|\) for sine and cosine, and \(180^\circ/|b|\) for tangent.