To determine how the graph of [tex]\( y = \sin\left(x - \frac{3\pi}{2}\right) \)[/tex] is shifted relative to the graph of [tex]\( y = \sin(x) \)[/tex], we need to look at the transformation properties of the sine function.
1. Horizontal Shifts: The sine function has a general form:
[tex]\[
y = \sin(x - b)
\][/tex]
where [tex]\( b \)[/tex] is a constant. This represents a horizontal shift of the graph of [tex]\( y = \sin(x) \)[/tex].
2. Determining the Direction:
- If [tex]\( b \)[/tex] is positive, the graph shifts to the right.
- If [tex]\( b \)[/tex] is negative, the graph shifts to the left.
In the given function, [tex]\( y = \sin\left(x - \frac{3\pi}{2}\right) \)[/tex]:
- We can see that [tex]\( b = \frac{3\pi}{2} \)[/tex], which is a positive value.
3. Since [tex]\( \frac{3\pi}{2} \)[/tex] is positive, this shifts the graph to the right by [tex]\( \frac{3\pi}{2} \)[/tex] units.
Therefore, the graph of [tex]\( y = \sin\left(x - \frac{3\pi}{2}\right) \)[/tex] is the graph of [tex]\( y = \sin(x) \)[/tex] shifted [tex]\( \frac{3\pi}{2} \)[/tex] units to the right.
So the correct choice is:
[tex]\[ \frac{3 \pi}{2} \text{ units to the right} \][/tex]
