Let's break down the key points in the problem to determine the correct equation for the graph.

Step 1: Key observations from the graph

• Period: The graph repeats over an interval of $2\pi$. This is a standard period for sine and cosine functions, so this suggests the function is either a sine or cosine wave.
• Amplitude: The graph oscillates between $y = -1$ and $y = 1$, so the amplitude is 1.
• Vertical Shift: The curve oscillates around $y = -0.6$, indicating a vertical shift downward by $0.6$.

Step 2: Identifying the function form

Since the curve is periodic, the possible equations are:

• $y = c + \cos(x + c)$
• $y = c + \sin(x + c)$
• $y = \cos(x - d)$
• $y = \sin(x - d)$

The periodicity and the way the graph behaves suggest it could be a sine or cosine function with a horizontal shift (due to the curve starting at a point not at a maximum or minimum). We also know the curve starts at $(0, -0.6)$, which is not a typical starting point for cosine functions, so it's likely a sine function.

Step 3: Determine the horizontal shift $d$

The sine function usually starts at $(0, 0)$, but our graph starts at $(0, -0.6)$, so it must be a shifted version of the sine curve. From the graph, we can see that the function reaches a maximum at approximately $\frac{7\pi}{10}$. This suggests that the sine function is shifted to the right by $d$, where $d = \frac{\pi}{5}$.

Step 4: Equation of the graph

The general form of a sine function with a horizontal shift is:

$y = a \sin(x - d) + c$

• Amplitude $a = 1$ (since the range is from $-1$ to $1$).
• Vertical shift $c = -0.6$ (since the midline is at $y = -0.6$).
• Horizontal shift $d = \frac{\pi}{5}$.

Thus, the equation of the graph is:

$y = \sin\left(x - \frac{\pi}{5}\right) - 0.6$

Final Answer:

The equation of the graph is:

$y = \sin\left(x - \frac{\pi}{5}\right) - 0.6$

Do you have any questions or need more clarification on this solution?

Here are some follow-up questions you might find helpful:

1. How do the amplitude and vertical shift affect the graph of a sine or cosine function?
2. What happens to the graph if the horizontal shift $d$ is increased?
3. How can you determine the period of a sine or cosine function from its equation?
4. What is the difference between a sine and cosine function with respect to their phase shifts?
5. How can you use the graph to identify the amplitude, period, and vertical shift of a trigonometric function?

Tip: When analyzing periodic functions, always start by identifying key features such as the amplitude, vertical shift, period, and phase shift. This will help you determine the exact form of the function.