The task here involves graphing the function $y = -2 - \cos(3\theta)$. Let's break this down step by step:

Step 1: Recognize the base graph

• The base function is $\cos(3\theta)$, which is a cosine graph with a frequency multiplier of 3, meaning the cosine wave completes 3 cycles over $2\pi$. This makes the period $\frac{2\pi}{3}$.
• From the image of the starting graphs, the one that represents the cosine graph should be chosen as the base. The cosine graph typically starts at its maximum value, reaches a minimum, and returns to the maximum, resembling a wave pattern.

Step 2: Apply transformations

Now, we need to apply two transformations to this base cosine function:

1. Vertical Stretch:

   • The factor of $-1$ in front of the cosine function reflects the graph across the horizontal axis. So, instead of starting at a maximum, the graph will start at a minimum.

2. Vertical Shift:

   • The term $-2$ shifts the entire graph downward by 2 units.

Final Function: $y = -2 - \cos(3\theta)$

• This means the graph starts at $y = -3$ (minimum), oscillates between $y = -3$ and $y = -1$, and has a period of $\frac{2\pi}{3}$.

To summarize:

• Choose the cosine graph from the options (likely the second or third from the left).
• Transform it by reflecting it over the x-axis and shifting it downward by 2 units.

Would you like more details on this process? Here are five related questions to deepen understanding:

1. How does the frequency factor $3 \theta$ affect the graph's period?
2. What are the amplitude and range of the transformed graph?
3. What effect does the negative sign in front of the cosine function have?
4. How would the graph change if the transformation included a horizontal shift?
5. How can you determine the key points on the graph of $y = -2 - \cos(3\theta)$?

Tip: Always analyze the effect of both vertical and horizontal transformations separately before combining them.