https://file.aiquickdraw.com/mathbot/file/17296060642602oq282ex.jpg

The graph you've provided shows a sinusoidal wave. To determine the equation of the wave, we need to analyze key aspects of the graph, such as:

1. **Amplitude**: The vertical distance from the midline to a peak or trough. The graph reaches a maximum of \(-1\) and a minimum of \(-3\), so the amplitude is half the distance between \(-1\) and \(-3\), which is \(1\).

2. **Vertical Shift**: The midline of the wave is located at \(y = -2\) (midway between \(-1\) and \(-3\)).

3. **Period**: The horizontal length for one complete cycle of the wave. The wave completes one full cycle from \(0\) to \(2\pi\), so the period is \(2\pi\).

4. **Phase Shift**: The graph starts at \(x = 0\) and appears to follow a cosine-like wave, though reflected vertically. 

Based on these observations, we can write the equation of the sinusoidal wave as:

\[
y = -\cos(x) - 2
\]

Where:
- The negative sign reflects the wave vertically.
- The \( \cos(x) \) suggests it follows a cosine function.
- The \(-2\) represents the vertical shift.

Let me know if you'd like more details or clarifications on this.

### Related Questions:
1. What is the general form of a sinusoidal function?
2. How do you determine the phase shift of a trigonometric graph?
3. What does the period of a wave tell you in trigonometric functions?
4. How does changing the amplitude affect the graph of a trigonometric function?
5. What is the difference between sine and cosine waves?

**Tip**: To graph any trigonometric function, it's important to first identify the amplitude, period, phase shift, and vertical shift.

Write equations based on the provided graph of a sinusoidal wave.

Learn how to write the equation of a sinusoidal wave based on its graph. This problem covers key trigonometric concepts such as amplitude, vertical shift, and period, and is suitable for students in Grades 10-12.

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