https://file.aiquickdraw.com/mathbot/file/1727276390594kwwzt08n.jpg

Let's analyze the given graph and find the equation of the sinusoidal function:

### Characteristics from the graph:
1. **Amplitude**: The maximum height from the centerline (y = 2) is 2 units, so the amplitude is 2.
2. **Vertical Shift**: The centerline is at \( y = 2 \), meaning the graph is shifted 2 units up.
3. **Period**: The graph completes one cycle from \( 0 \) to \( 2\pi \), but since there are two full cycles between \( 0 \) and \( 2\pi \), the period is \( \pi \). This indicates a horizontal compression, so the angular frequency \( b = 2 \) (as the period \( P = \frac{2\pi}{b} \)).
4. **Function Type**: The graph starts at a peak at \( x = 0 \), which indicates it is a **cosine** function, not sine.

### Equation Form:
The general equation of a cosine function is:
\[
y = A \cos(Bx) + D
\]
where:
- \( A = 2 \) (amplitude),
- \( B = 2 \) (from the period),
- \( D = 2 \) (vertical shift).

Thus, the equation is:
\[
y = 2 \cos(2x) + 2
\]

### Answer:
The correct choice is **B) \( y = 2 \cos(2x) \)**.

Would you like a detailed explanation of any part of this process? Here are 5 related questions:

1. How do you determine the period of a sinusoidal function from a graph?
2. What are the key differences between sine and cosine graphs?
3. How does the vertical shift affect the graph of a sinusoidal function?
4. Can you explain how to derive the amplitude from a graph?
5. How does the phase shift affect sinusoidal functions?

**Tip**: The angular frequency \( B \) in sinusoidal functions determines how many cycles fit into \( 2\pi \). The larger \( B \), the more compressed the graph is.

Write the equation of the sinusoidal function shown in the graph.

Learn how to write the equation of a sinusoidal function from a graph. This example covers trigonometric concepts like amplitude, period, and vertical shift using a cosine function. Suitable for students in Grades 10-12.

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