https://file.aiquickdraw.com/mathbot/file/1729874705669sr0lhmsz.jpg

The graph you provided is of a sinusoidal function. Let's analyze the graph to determine the equation.

### Key Observations:
1. **Amplitude**: The graph oscillates between a maximum of 5 and a minimum of -5, so the amplitude is \( A = 5 \).
2. **Period**: The graph completes one full cycle between \( x = -3\pi \) and \( x = 3\pi \), so the period is \( 6\pi \). The formula for the period of a sine or cosine function is \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \). Therefore, \( \frac{2\pi}{B} = 6\pi \), giving us \( B = \frac{1}{3} \).
3. **Vertical Shift**: The midline of the graph is at \( y = 0 \), so there is no vertical shift.
4. **Phase Shift**: The graph of the sine or cosine function appears to have no horizontal shift.

### Equation Form:
A sine or cosine function can be written as:
\[
y = A \cdot \sin(Bx + C) + D
\]
or
\[
y = A \cdot \cos(Bx + C) + D
\]

In this case, since the graph starts at a maximum at \( x = -3\pi \), it resembles a cosine function. So the equation should be a cosine function of the form:
\[
y = 5 \cdot \cos\left(\frac{1}{3}x\right)
\]

This is the equation of the given trigonometric graph.

Would you like more details or have any questions?

### Related Questions:
1. What is the general form of a sinusoidal function?
2. How do you find the period of a trigonometric function?
3. What is the difference between sine and cosine graphs?
4. How would the graph change if we added a vertical shift?
5. How do you calculate the phase shift in a trigonometric graph?

### Tip:
To find the period of any trigonometric function, use the formula \( \text{Period} = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) in the function.

Write the equation of the trigonometric graph.

Discover how to find the equation of a cosine function with an amplitude of 5 and a period of 6π, given the graph of the sinusoidal function. This problem covers key trigonometric concepts, including graph analysis and equation formation. Suitable for students in Grades 10-12.

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