https://file.aiquickdraw.com/mathbot/file/1730557760598pgtupa2z.jpg

To determine the equation of this trigonometric graph, let’s analyze the characteristics of the curve:

1. **Amplitude**: The graph oscillates between a maximum of \( y = 5 \) and a minimum of \( y = -5 \), so the amplitude is \( 5 \).

2. **Period**: The graph completes one full cycle between \( x = 0 \) and \( x = 8\pi \), so the period is \( 8\pi \).

3. **Vertical Shift**: There’s no vertical shift, as the graph oscillates symmetrically about the x-axis, centered at \( y = 0 \).

4. **Phase Shift**: The graph starts at \( y = 0 \) and goes up, suggesting it could be a sine function (as sine typically starts from zero).

Given these characteristics, we can construct the general form of a sine function:
\[
y = A \sin(Bx)
\]
where:
- \( A = 5 \) (the amplitude),
- \( B \) is related to the period by the formula \( \text{Period} = \frac{2\pi}{B} \).

Since the period is \( 8\pi \),
\[
8\pi = \frac{2\pi}{B}
\]
Solving for \( B \):
\[
B = \frac{2\pi}{8\pi} = \frac{1}{4}
\]

Therefore, the equation of this graph is:
\[
y = 5 \sin\left(\frac{1}{4}x\right)
\]

### Final Answer
The equation of the trigonometric graph is:
\[
y = 5 \sin\left(\frac{1}{4}x\right)
\]

Would you like additional details or have any questions?

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Here are some related questions that build on this problem:
1. What would the equation be if the amplitude were doubled?
2. How would the equation change if the period were reduced to \( 4\pi \)?
3. What would the equation look like if it were a cosine function instead of a sine?
4. How would a phase shift of \( \pi/2 \) affect the equation?
5. How do you determine the period of a sine or cosine function from its equation?

**Tip:** Remember, the period of a sine or cosine function is determined by the coefficient inside the function (e.g., \( B \) in \( \sin(Bx) \)), with the formula \( \text{Period} = \frac{2\pi}{|B|} \).

Write the equation of the trigonometric graph.

Analyze and determine the equation of a trigonometric graph with given amplitude, period, and other properties. This problem involves understanding the sine function's transformation and calculating the correct period and amplitude. Suitable for students in Grades 10-12.

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