select the correct answer. which function has a phase shift of π/2 to the right? a. y = 2 sin(x + π/2) b. y…

Explanation:

Step1: Recall phase - shift formula

For a sinusoidal function $y = A\sin(Bx - C)+D$, the phase - shift is given by $\frac{C}{B}$. A positive phase - shift is to the right and a negative phase - shift is to the left.

Step2: Analyze option A

For $y = 2\sin(x+\frac{\pi}{2})$, we can rewrite it as $y = 2\sin(x - (-\frac{\pi}{2}))$. Here $B = 1$ and $C=-\frac{\pi}{2}$, so the phase - shift is $\frac{-\frac{\pi}{2}}{1}=-\frac{\pi}{2}$, which is a shift to the left.

Step3: Analyze option B

For $y = 2\sin(x-\pi)$, $B = 1$ and $C=\pi$, so the phase - shift is $\frac{\pi}{1}=\pi$, which is a shift of $\pi$ to the right.

Step4: Analyze option C

For $y = 2\sin(2x-\pi)$, $B = 2$ and $C=\pi$, so the phase - shift is $\frac{\pi}{2}$, which is a shift of $\frac{\pi}{2}$ to the right.

Step5: Analyze option D

For $y = 2\sin(\frac{1}{2}x+\pi)=2\sin(\frac{1}{2}x-(-\pi))$, $B=\frac{1}{2}$ and $C = -\pi$, so the phase - shift is $\frac{-\pi}{\frac{1}{2}}=- 2\pi$, which is a shift to the left.

Answer:

C. $y = 2\sin(2x - \pi)$