To find which function has a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right, we need to analyze each function and calculate the phase shift.
1. Understanding Phase Shift:
The standard form of a sinusoidal function is:
[tex]\[
y = a \sin(bx + c)
\][/tex]
The phase shift [tex]\(\phi\)[/tex] of this function is given by:
[tex]\[
\phi = -\frac{c}{b}
\][/tex]
A positive value of [tex]\(\phi\)[/tex] indicates a shift to the right, and a negative value indicates a shift to the left.
2. Analyzing Each Function:
Let's break down each option:
A. [tex]\( y = 2 \sin \left(\frac{1}{2} x + \pi\right) \)[/tex]
[tex]\[
\phi = -\frac{\pi}{\frac{1}{2}} = -2\pi
\][/tex]
This corresponds to a phase shift of [tex]\(-2\pi\)[/tex] (or [tex]\(2\pi\)[/tex] to the left), not [tex]\(\frac{\pi}{2}\)[/tex] to the right.
B. [tex]\( y = 2 \sin (2 x + \pi) \)[/tex]
[tex]\[
\phi = -\frac{\pi}{2} = -\frac{\pi}{2}
\][/tex]
This corresponds to a phase shift of [tex]\(-\frac{\pi}{2}\)[/tex] (or [tex]\(\frac{\pi}{2}\)[/tex] to the left), not to the right.
C. [tex]\( y = 2 \sin \left(x + \frac{\pi}{2}\right) \)[/tex]
[tex]\[
\phi = -\frac{\frac{\pi}{2}}{1} = -\frac{\pi}{2}
\][/tex]
This also corresponds to a phase shift of [tex]\(-\frac{\pi}{2}\)[/tex], not [tex]\(\frac{\pi}{2}\)[/tex] to the right.
D. [tex]\( y = 2 \sin (x - \pi) \)[/tex]
[tex]\[
\phi = -\frac{-\pi}{1} = \pi
\][/tex]
This corresponds to a phase shift of [tex]\(\pi\)[/tex] to the right, not [tex]\(\frac{\pi}{2}\)[/tex] to the right.
E. [tex]\( y = 2 \sin (2 x - \pi) \)[/tex]
[tex]\[
\phi = -\frac{-\pi}{2} = \frac{\pi}{2}
\][/tex]
This corresponds to a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right, which is exactly what we are looking for.
Thus, the correct function with a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right is:
[tex]\[
E. \, y = 2 \sin (2 x - \pi)
\][/tex]
So, the function that has a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right corresponds to option E.
