Question

find the period, amplitude, and phase shift of the function.

$y = \frac{1}{2}\sin(\pi x - \frac{\pi}{2}) - 2$

give the exact values, not decimal approximations.

period:

amplitude:

phase shift:

Explanation:

Step1: Recall the general form of a sine function

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

Step2: Calculate the amplitude

The amplitude of \(y = A\sin(Bx - C)+D\) is given by \(|A|\). Substituting \(A=\frac{1}{2}\), we get \(|A|=\frac{1}{2}\).

Step3: Calculate the period

The period of \(y = A\sin(Bx - C)+D\) is \(T=\frac{2\pi}{|B|}\). Substituting \(B = \pi\), we have \(T=\frac{2\pi}{\pi}=2\).

Step4: Calculate the phase - shift

The phase - shift of \(y = A\sin(Bx - C)+D\) is \(\frac{C}{B}\). Substituting \(C=\frac{\pi}{2}\) and \(B=\pi\), we get \(\frac{C}{B}=\frac{\frac{\pi}{2}}{\pi}=\frac{1}{2}\).

Answer:

Period: \(2\)
Amplitude: \(\frac{1}{2}\)
Phase shift: \(\frac{1}{2}\)