Let's analyze the given problem step by step:

Graph Characteristics:

The equation is of the form:

$y = A \cos(\omega x) + B$

where:

• $A < 0$ (amplitude),
• $B > 0$ (vertical shift), and
• $0 < \omega < 2\pi$ (angular frequency).

The graph has a maximum value of 3 and a minimum value of -3, which gives the following information:

• The vertical shift $B$ is the average of the maximum and minimum values, calculated as:

$B = \frac{3 + (-3)}{2} = 0$

Thus, $B = 0$, and there is no vertical shift. The graph oscillates around the x-axis.

The amplitude $A$ is the distance from the midline (in this case, the x-axis) to the maximum or minimum values, which gives $|A| = 3$. Since $A < 0$, we conclude $A = -3$.

Period:

To determine $\omega$, we need the period of the function. The period is the length of one complete cycle, which is from $-\frac{7}{8}$ to $\frac{7}{4}$.

• The period is the difference between these two values:

$\text{Period} = \frac{7}{4} - \left(-\frac{7}{8}\right) = \frac{7}{4} + \frac{7}{8}$

To add these, let's find a common denominator:

$\frac{7}{4} + \frac{7}{8} = \frac{14}{8} + \frac{7}{8} = \frac{21}{8}$

Thus, the period of the cosine function is $\frac{21}{8}$.

The period of a cosine function is given by $\frac{2\pi}{\omega}$, so we can solve for $\omega$:

$\frac{2\pi}{\omega} = \frac{21}{8}$

Solving for $\omega$, we get:

$\omega = \frac{2\pi \cdot 8}{21} = \frac{16\pi}{21}$

Final Equation:

The equation of the graph is:

$y = -3 \cos\left(\frac{16\pi}{21}x\right)$

Would you like more details or explanations? Here are some related questions:

1. How do you calculate the amplitude from a graph?
2. What does the period of a cosine function represent?
3. How can you determine the vertical shift of a trigonometric function?
4. What is the general form of the cosine function, and how do different parameters affect it?
5. How do you solve for $\omega$ using the period of a cosine function?

Tip: When solving trigonometric equations, carefully observe the amplitude, period, and vertical shift to correctly determine each parameter.