Question

find the amplitude, period, and phase shift of the functi
$y = 4\cos(3x - \frac{\pi}{4})$
amplitude
4
period
phase shift

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \(y = A\cos(Bx - C)+D\). For the given function \(y = 4\cos(3x-\frac{\pi}{4})\), we have \(A = 4\), \(B = 3\), \(C=\frac{\pi}{4}\), \(D = 0\).

Step2: Calculate the period

The formula for the period of \(y = A\cos(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B = 3\) into the formula, we get \(T=\frac{2\pi}{3}\).

Step3: Calculate the phase - shift

The formula for the phase - shift of \(y = A\cos(Bx - C)+D\) is \(\text{Phase - shift}=\frac{C}{B}\). Substituting \(B = 3\) and \(C=\frac{\pi}{4}\) into the formula, we get \(\text{Phase - shift}=\frac{\frac{\pi}{4}}{3}=\frac{\pi}{12}\).

Answer:

The period is \(\frac{2\pi}{3}\) and the phase - shift is \(\frac{\pi}{12}\).