https://file.aiquickdraw.com/mathbot/file/1729609231989pitg4sdi.jpg

The graph you provided shows a sinusoidal function, most likely a variation of either a sine or cosine function due to its periodic behavior and shape. Let's analyze it step by step:

### Key Features:
1. **Amplitude**: The graph reaches a maximum value of 5 and a minimum value of 0, so the amplitude (the distance from the middle to the peak) is 2.5. However, since the graph starts at a higher value (not zero), it seems to have a vertical shift.
   
2. **Period**: The distance from one peak to the next is \( 2\pi \), which is typical for sine and cosine functions.

3. **Vertical Shift**: The graph oscillates between 0 and 5. The middle of this range is \( 2.5 \), suggesting that the function is vertically shifted by 2.5 units upwards.

4. **Type of Function**: This graph resembles a **cosine function**, because it reaches a maximum at \( x = 0 \). A typical cosine function \( \cos(x) \) starts at a maximum when \( x = 0 \).

### General Equation:
The form of the cosine function is given by:
\[
y = A \cos(Bx - C) + D
\]
where:
- \( A \) is the amplitude,
- \( B \) affects the period (\( \text{Period} = \frac{2\pi}{B} \)),
- \( C \) is the phase shift (horizontal shift),
- \( D \) is the vertical shift.

From the graph:
- The amplitude \( A = 2.5 \),
- The period is \( 2\pi \), so \( B = 1 \),
- The vertical shift \( D = 2.5 \),
- No horizontal shift is observed, so \( C = 0 \).

Thus, the equation of the graph is:
\[
y = 2.5 \cos(x) + 2.5
\]

This represents a cosine function with an amplitude of 2.5, a vertical shift of 2.5 units upwards, and a period of \( 2\pi \).

Would you like more details or have any further questions?

### Related Questions:
1. How would the graph look if the amplitude were doubled?
2. What happens to the graph if the period is halved?
3. How does a sine graph compare to this cosine graph?
4. What effect does changing the vertical shift to 0 have on the graph?
5. How would the graph change if there were a phase shift?

### Tip:
When analyzing trigonometric graphs, always check for amplitude, period, phase shift, and vertical shift to write the correct equation!

What is the equation of the sinusoidal graph shown?

Analyze a sinusoidal graph and determine its equation based on amplitude, period, and vertical shift. This example explores the graph of a cosine function and helps students understand key concepts in trigonometry.

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