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 = 4sin(pi x + 1)-4 ) 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.) 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 a sine function \(y = A\sin(Bx - C)+D\), the amplitude is \(|A|\).
In the function \(y = 4\sin(\pi x + 1)-4\), \(A = 4\). So the amplitude is \(|4|=4\).

Step2: Find the period

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

Step3: Find the phase shift

The formula for the phase shift of a sine function \(y = A\sin(Bx - C)+D\) is \(\text{Phase Shift}=\frac{C}{B}\).
Rewrite \(y = 4\sin(\pi x + 1)-4\) as \(y = 4\sin(\pi(x+\frac{1}{\pi}))-4\), so \(C=- 1\) (in the form \(y = A\sin(Bx - C)+D\)), \(B = \pi\). Then the phase shift is \(\frac{-1}{\pi}=-\frac{1}{\pi}\).

Answer:

The amplitude is \(4\).
The period is \(2\).
The phase shift is \(-\frac{1}{\pi}\).