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Determine the correct equation of the following graph.

The graph shows a periodic function, and it appears to be a sine or cosine function. To determine the correct equation, we need to analyze the graph's characteristics: amplitude, period, phase shift, and vertical shift.

1. **Amplitude**: The maximum value is approximately 4, and the minimum value is approximately -4. Therefore, the amplitude is 4.

2. **Period**: The distance between two consecutive peaks (or troughs) determines the period. From the graph, it appears that the period is 60 units (from -30 to 30 or from 0 to 60).

3. **Phase Shift**: The graph appears to cross the y-axis at the origin (0,0), which suggests there is no horizontal shift for the cosine function. 

4. **Vertical Shift**: The graph oscillates around the x-axis (y=0), indicating there is no vertical shift.

Given these characteristics, the equation could be of the form \( y = A \cos(Bx + C) + D \).

Plugging in the values:
- \( A = 4 \) (amplitude)
- \( B = \frac{2\pi}{\text{period}} = \frac{2\pi}{60} = \frac{\pi}{30} \) (since the period is 60)
- \( C = 0 \) (no phase shift)
- \( D = 0 \) (no vertical shift)

Thus, the equation of the graph is:

\[ y = 4 \cos\left(\frac{\pi}{30} x\right) \]

Would you like further details or have any questions?

Here are five more questions you might want to ask:
1. How do you determine the amplitude of a graph?
2. What is the significance of the period in trigonometric functions?
3. How do you find the phase shift of a function from its graph?
4. What are the steps to convert a trigonometric function into its general form?
5. Can you explain how vertical shifts affect the graph of a trigonometric function?

**Tip**: When identifying the equation of a trigonometric function from its graph, carefully note the amplitude, period, phase shift, and vertical shift, as these parameters define the function's behavior.

Learn how to determine the correct equation of a graph showing a periodic function resembling a sine or cosine function. Understand the amplitude, period, phase shift, and vertical shift to formulate the equation y = A cos(Bx + C). Suitable for high school students studying trigonometry.

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