The graph provided appears to be a cosine function, and the multiple-choice options represent variations of transformed cosine equations. To analyze which equation best represents the graph, let's break down the key features of the graph and match it with the options:

Key Features of the Graph:

1. Amplitude: The graph oscillates between $y = 5$ and $y = -5$, giving an amplitude of $5$.
2. Vertical Shift: There does not appear to be any vertical shift, as the graph is centered around the x-axis, so no constant is added to shift the graph up or down.
3. Period: To find the period, we observe the distance between two peaks. From the graph, one cycle (from peak to peak) appears to take $\frac{8\pi}{5}$. The general form of the cosine function is: $y = A \cos(B(x - C)) + D$ where:
   • $A$ is the amplitude.
   • $B$ affects the period, where the period is $\frac{2\pi}{B}$. Based on the graph, $B$ can be calculated to give a period of $\frac{8\pi}{5}$, so $B = \frac{4}{5}$.
   • $C$ represents the horizontal shift.
4. Horizontal Shift: The graph seems shifted to the right by $\frac{5\pi}{8}$.

Matching with the Equation Options:

Now, the equation must have:

• Amplitude = 2.
• A horizontal shift = $\frac{5\pi}{8}$ to the right.
• Period is governed by $\frac{4}{5}$.

Among the options, the correct equation that matches the graph is: $y = 2 \cos \left( \frac{4}{5}(x - \frac{5\pi}{8}) \right)$

This equation correctly accounts for the amplitude, period, and phase shift. Therefore, the correct answer is the first option: $y = 2 \cos \left( \frac{4}{5} (x - \frac{5\pi}{8}) \right)$

Would you like further details or have any other questions?

Here are 5 related questions:

1. How do we determine the amplitude of a cosine function from its graph?
2. How is the period of a trigonometric function calculated?
3. What is the effect of a horizontal shift in the cosine function's graph?
4. How can you tell the difference between a cosine and sine graph?
5. How would the equation change if the graph were vertically shifted up?

Tip: To find the period of a cosine graph, calculate the horizontal distance between two peaks (or troughs) and use that to adjust the $B$-value in the equation $y = A \cos(B(x - C))$.