Let's analyze the function given in the image:

$y = -1 - 2 \cot \left( 3 \left( x - \frac{3\pi}{2} \right) \right)$

Step-by-Step Breakdown:

1. Vertical Shift: The constant $-1$ shifts the entire graph down by 1 unit.

2. Vertical Stretch/Compression and Reflection: The coefficient $-2$ affects the amplitude and reflects the graph across the x-axis. Since it’s a cotangent function ($\cot$), the negative coefficient means the graph will flip vertically.

3. Period and Horizontal Stretch: The argument of the cotangent function is $3(x - \frac{3\pi}{2})$. The $3$ inside the argument affects the period of the cotangent function. The period of $\cot(x)$ is $\pi$, but here it is scaled by $3$, so the period becomes $\frac{\pi}{3}$.

4. Phase Shift: The term $\frac{3\pi}{2}$ represents a horizontal phase shift to the right by $\frac{3\pi}{2}$.

Interpretation:

• The graph is vertically shifted down by 1 unit.
• The graph is vertically stretched by a factor of 2 and reflected.
• The period is $\frac{\pi}{3}$, and the graph is shifted to the right by $\frac{3\pi}{2}$.

Now, let’s analyze the graphs:

• Option A shows a cotangent function with a vertical asymptote near $\pi/6$.
• Option B has asymptotes at $\pi/8$, which is incorrect because the period is based on $\frac{\pi}{3}$.
• Option C has asymptotes at $\pi/6$, matching the correct transformation with the phase shift.

Conclusion:

The correct graph is Option C.

Would you like more details on any part of this explanation?

Here are 5 related questions to explore:

1. What are the key characteristics of the cotangent function?
2. How do we calculate the period of transformed trigonometric functions?
3. How does a phase shift affect the graph of a function?
4. Why does the negative coefficient flip the graph vertically?
5. How do vertical shifts affect the location of asymptotes?

Tip: Always start by identifying the period, phase shift, and amplitude when graphing trigonometric functions.