https://file.aiquickdraw.com/mathbot/file/1725892482716oh2x2kvi.jpg

This graph represents a sinusoidal (sine or cosine) trigonometric function. Let's analyze the graph to find its equation.

### Key Observations:
1. **Amplitude**: The graph oscillates between \( y = 5 \) and \( y = -5 \). Hence, the amplitude is 5.
2. **Vertical Shift**: The graph oscillates around the line \( y = 0 \), so there is no vertical shift.
3. **Period**: The graph completes one full cycle between \( x = -6\pi \) and \( x = 6\pi \), making the period \( 12\pi \).
   - The general formula for the period of a sine/cosine function is \( \frac{2\pi}{B} \), so we have \( \frac{2\pi}{B} = 12\pi \), which gives \( B = \frac{1}{6} \).
4. **Phase Shift**: The graph appears to be symmetric around the origin, indicating no phase shift.

### Equation of the Graph:
Given that there is no phase shift, and it starts from \( y = 0 \) at \( x = 0 \), this is a sine function. Therefore, the equation of the graph is:

\[
y = 5 \sin\left( \frac{1}{6}x \right)
\]

Would you like any further details or clarifications?

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### 5 Related Questions:
1. How would the equation change if the graph were shifted up by 2 units?
2. What is the equation if the function were cosine-based instead of sine?
3. How do you calculate the amplitude from a trigonometric graph?
4. How do you determine the period of a trigonometric function?
5. What would the equation look like if the period were \( 8\pi \) instead?

### Tip:
To distinguish between sine and cosine graphs, check the starting point: sine starts from 0, while cosine starts from its maximum or minimum.

Write the equation of the trigonometric graph.

Learn how to determine the equation of a sinusoidal graph with an amplitude of 5 and a period of 12π. This problem covers key trigonometric concepts like amplitude, period, and sinusoidal graph analysis. Suitable for high school students in Grades 10-12.

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