The graph of [tex]$y=\sin \left(x-\frac{3 \pi}{2}\right)$[/tex] is the graph of [tex]$y=\sin (x)$[/tex] shifted:

A. [tex][tex]$\frac{3 \pi}{2}$[/tex][/tex] units to the left
B. [tex]$\frac{3 \pi}{2}$[/tex] units to the right
C. [tex]$\frac{3 \pi}{2}$[/tex] units up



Answer :

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]

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