To determine the phase shift and vertical shift of the function [tex]\( f(\theta) = -\frac{2}{3} \cos \left( \frac{1}{2} \theta + 20^{\circ} \right) - \frac{3}{2} \)[/tex], let's break it down step by step.
### Step 1: Determine the Phase Shift
The general form of a cosine function is:
[tex]\[ f(\theta) = A \cos(B(\theta - C)) + D \][/tex]
In the given function, [tex]\( f(\theta) = -\frac{2}{3} \cos \left( \frac{1}{2} \theta + 20^{\circ} \right) - \frac{3}{2} \)[/tex], see how it compares to the general form:
- The coefficient [tex]\( A \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex].
- The coefficient [tex]\( B \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
- The phase shift is determined by solving the inside of the cosine function: [tex]\( \frac{1}{2} \theta + 20^\circ \)[/tex].
Set [tex]\( \frac{1}{2} \theta + 20^\circ = 0 \)[/tex] to find the horizontal displacement (phase shift):
[tex]\[ \frac{1}{2} \theta + 20^\circ = 0 \][/tex]
[tex]\[ \frac{1}{2} \theta = -20^\circ \][/tex]
[tex]\[ \theta = -40^\circ \][/tex]
Hence, the phase shift is [tex]\( -40^\circ \)[/tex], which means the function is shifted [tex]\( 40^\circ \)[/tex] to the left.
### Step 2: Determine the Vertical Shift
The vertical shift is given directly by the constant term outside the cosine function. In this function, the constant term is [tex]\( -\frac{3}{2} \)[/tex].
Therefore, the function is shifted vertically downward by [tex]\( \frac{3}{2} \)[/tex] units.
### Conclusion
Combining both results, the phase shift is [tex]\( 40^\circ \)[/tex] to the left and the vertical shift is [tex]\( \frac{3}{2} \)[/tex] units downward. Thus, the correct option is:
[tex]\[ 40^\circ \text{ to the left and } \frac{3}{2} \text{ units down} \][/tex]
