Question

find the amplitude, period, and phase shift of the function.
$y = \cos(x - \frac{\pi}{2})$
amplitude
period
phase shift
graph one complete period.

Explanation:

Step1: Recall the general form of cosine function

The general form of a cosine function is \(y = A\cos(B(x - C))+D\). For the function \(y=\cos(x-\frac{\pi}{2})\), we have \(A = 1\), \(B = 1\), \(C=\frac{\pi}{2}\), \(D = 0\).

Step2: Calculate the amplitude

The amplitude of a cosine function \(y = A\cos(B(x - C))+D\) is given by \(|A|\). Since \(A = 1\), the amplitude \(|A|=1\).

Step3: Calculate the period

The period of a cosine function \(y = A\cos(B(x - C))+D\) is given by \(T=\frac{2\pi}{|B|}\). Since \(B = 1\), the period \(T=\frac{2\pi}{|1|}=2\pi\).

Step4: Calculate the phase - shift

The phase - shift of a cosine function \(y = A\cos(B(x - C))+D\) is given by \(C\). Since \(C=\frac{\pi}{2}\), the phase - shift is \(\frac{\pi}{2}\) (to the right).

Answer:

amplitude: \(1\)
period: \(2\pi\)
phase shift: \(\frac{\pi}{2}\) (to the right)