https://file.aiquickdraw.com/mathbot/file/17301360571452qf7970j.jpg

The sinusoidal graph shown resembles the form of a sine or cosine function. To write its equation, let's identify key components of the sinusoid, specifically its amplitude, period, and vertical shift:

1. **Amplitude**: The maximum value of the wave is \(5\), and the minimum value is \(-5\), giving an amplitude of:
   \[
   A = 5
   \]

2. **Period**: The wave completes one full cycle between \(-2\pi\) and \(2\pi\), so the period \(T\) is \(4\pi\).

   The formula for the period of a sine or cosine function is \(T = \frac{2\pi}{B}\), where \(B\) is the horizontal stretch factor. Solving for \(B\):
   \[
   4\pi = \frac{2\pi}{B} \Rightarrow B = \frac{1}{2}
   \]

3. **Vertical Shift**: Since the sinusoid oscillates symmetrically around the x-axis (no vertical shift), \(D = 0\).

Since the wave appears to start at the origin, this function aligns with the sine function format. Thus, the equation of the sinusoid is:
\[
y = 5 \sin\left(\frac{1}{2}x\right)
\]

Let me know if you would like further details or have any questions!

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**Related Questions:**

1. How do you determine the equation of a cosine function from a graph?
2. What changes in the equation if the sinusoid has a phase shift?
3. How would a vertical shift affect the sinusoidal equation?
4. Can the amplitude ever be negative, and what would that indicate?
5. How does the period of a sinusoidal function affect its graph?

**Tip:** Always check the starting point of the wave to decide if it's a sine or cosine function.

Learn how to write the equation of a sinusoidal graph by identifying amplitude, period, and vertical shifts. This example covers essential trigonometric concepts and guides students through the steps to derive the equation.

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