To determine the phase shift of the function [tex]\( y = 4 \cos \left( -3x + \frac{\pi}{2} \right) \)[/tex], we need to rewrite the argument of the cosine function in the standard form [tex]\( A \cos(B(x - C)) \)[/tex].
The given function argument is [tex]\( -3x + \frac{\pi}{2} \)[/tex].
The standard form for the cosine function is [tex]\( A \cos(B(x - C)) \)[/tex], where [tex]\( C \)[/tex] represents the phase shift. To express the given function in this form, we start by factoring out the coefficient of [tex]\( x \)[/tex] inside the cosine:
[tex]\[
-3x + \frac{\pi}{2} = -3 \left( x - \frac{\pi}{6} \right)
\][/tex]
Here’s how we do this:
1. Identify the coefficient of [tex]\( x \)[/tex] in the argument of the cosine function, which is [tex]\( -3 \)[/tex].
2. Rewriting [tex]\( \frac{\pi}{2} \)[/tex] by factoring out [tex]\( -3 \)[/tex]:
[tex]\[
-3 \left( x - \frac{\pi}{6} \right)
\][/tex]
[tex]\( \frac{\pi}{2} = -3 \cdot \left( -\frac{\pi}{6} \right) \text{, since } -3 \times \frac{\pi}{-6} = \frac{\pi}{2} \)[/tex].
So, [tex]\( -3x + \frac{\pi}{2} = -3 \left( x - (-\frac{\pi}{6}) \right) \)[/tex].
Therefore, in the standard form [tex]\( A \cos(B(x - C)) \)[/tex], we have [tex]\( B = -3 \)[/tex] and [tex]\( C = -\frac{\pi}{6} \)[/tex].
The phase shift [tex]\( C \)[/tex] is thus [tex]\( -\frac{\pi}{6} \)[/tex].
A negative phase shift indicates that the graph is shifted to the left. The magnitude of the phase shift is [tex]\( \frac{\pi}{6} \)[/tex].
Thus, the phase shift is [tex]\( \frac{\pi}{6} \)[/tex] units to the left.
Therefore, the correct answer is:
A. [tex]\( \frac{\pi}{6} \)[/tex] units to the left.
