Question

find the amplitude (if one exists), period, and phase shift of the function. graph the function. be sure to label key points. show at least two periods. ( y = 3sin(pi x + 4)-3 ) o a. the amplitude is (simplify your answer. type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.) o b. the function does not have an amplitude. what is the period? (simplify your answer. type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.) what is the phase shift? (simplify your answer. type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.) use the graphing tool to graph the function.

Explanation:

Step1: Find the amplitude

For the function \(y = A\sin(Bx - C)+D\), the amplitude is \(|A|\).
In the function \(y = 3\sin(\pi x + 4)-3\), \(A = 3\). So the amplitude is \(|3|=3\).

Step2: Find the period

The period of the function \(y = A\sin(Bx - C)+D\) is given by \(T=\frac{2\pi}{|B|}\).
Here \(B=\pi\), so \(T=\frac{2\pi}{\pi}=2\).

Step3: Find the phase - shift

The phase - shift of the function \(y = A\sin(Bx - C)+D\) is given by \(\frac{C}{B}\).
Rewrite \(y = 3\sin(\pi x + 4)-3\) as \(y = 3\sin(\pi(x+\frac{4}{\pi}))-3\), so \(C=- 4\) (in the form \(y = A\sin(Bx - C)+D\)), and \(B = \pi\). The phase - shift is \(\frac{-4}{\pi}\) (or \(x =-\frac{4}{\pi}\)).

Answer:

A. The amplitude is \(3\).
The period is \(2\).
The phase - shift is \(-\frac{4}{\pi}\).