The function given is:

$y = -2 \cos\left(\frac{3}{4}x\right) - 2$

This is a transformed cosine function, and we can analyze its key features:

1. Amplitude:

The amplitude is the absolute value of the coefficient in front of the cosine term, which is $|-2| = 2$. The negative sign indicates that the graph is reflected over the x-axis.

2. Period:

The period of a cosine function is given by:

$\text{Period} = \frac{2\pi}{|B|}$ where $B$ is the coefficient of $x$. In this case, $B = \frac{3}{4}$, so the period is:

$\text{Period} = \frac{2\pi}{\frac{3}{4}} = \frac{2\pi \times 4}{3} = \frac{8\pi}{3}$

3. Phase Shift:

Since there is no horizontal shift term (no $(x - c)$ inside the cosine), the phase shift is zero.

4. Vertical Shift:

The entire graph is shifted down by 2 units due to the "-2" at the end of the function.

5. Reflection:

The negative sign in front of the amplitude reflects the graph over the x-axis, meaning the peaks and troughs are flipped.

Summary:

• Amplitude: 2 (reflected)
• Period: $\frac{8\pi}{3}$
• Vertical Shift: Down 2
• No phase shift
• Reflected over x-axis

Would you like me to plot the graph or explain anything in more detail?

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Here are 5 related questions to explore further:

1. How would the graph change if the period were doubled?
2. What if the vertical shift was positive? How would that affect the graph?
3. How does the amplitude affect the height of the peaks and troughs?
4. How would adding a phase shift affect the starting point of the graph?
5. What happens if you change the sign of the cosine function inside?

Tip: When analyzing trigonometric functions, always check the amplitude, period, phase shift, and vertical shift to understand how the graph behaves.