Let's analyze each function to find which one has a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right.
Recall that a sine function of the form [tex]\(y = a \sin(bx - c)\)[/tex] has a phase shift given by [tex]\(\frac{c}{b}\)[/tex] to the right.
Let's apply this to each option:
### Option A: [tex]\(y = 2 \sin \left(x + \frac{\pi}{2}\right)\)[/tex]
Here, the function is of the form [tex]\(y = 2 \sin \left( x - \left(-\frac{\pi}{2}\right) \right)\)[/tex]. The phase shift is:
[tex]\[ \text{Phase shift} = \frac{-\frac{\pi}{2}}{1} = -\frac{\pi}{2} \][/tex]
This corresponds to a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the left.
### Option B: [tex]\(y = 2 \sin (2x - \pi)\)[/tex]
Here, the function is of the form [tex]\(y = 2 \sin \left( 2x - \pi \right)\)[/tex]. The phase shift is:
[tex]\[ \text{Phase shift} = \frac{\pi}{2} = \frac{\pi}{2} \][/tex]
This corresponds to a phase shift of [tex]\(\frac{\pi}{2}\)[/tex] to the right.
### Option C: [tex]\(y = 2 \sin \left(\frac{1}{2}x + \pi\right)\)[/tex]
Here, the function is of the form [tex]\(y = 2 \sin \left( \frac{1}{2}x - (-\pi) \right)\)[/tex]. The phase shift is:
[tex]\[ \text{Phase shift} = \frac{-\pi}{\frac{1}{2}} = -2\pi \][/tex]
This corresponds to a phase shift of [tex]\(2\pi\)[/tex] to the left.
### Option D: [tex]\(y = 2 \sin (x - \pi)\)[/tex]
Here, the function is of the form [tex]\(y = 2 \sin \left( x - \pi \right)\)[/tex]. The phase shift is:
[tex]\[ \text{Phase shift} = \frac{\pi}{1} = \pi \][/tex]
This corresponds to a phase shift of [tex]\(\pi\)[/tex] to the right.
Based on the analysis above, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
