https://r2.erweima.ai/mathbot/image/17242492875876v04vomm.jpg

To solve the problem presented in the image, we need to determine the equation of the cosine function that has been graphed. 

Here’s the step-by-step process:

1. **Identify the amplitude**:
   - The amplitude of a cosine function is the distance from the middle of the wave to the peak (or trough). 
   - From the graph, the maximum value is 2, and the minimum value is -2, so the amplitude \( A \) is \( 2 \).

2. **Determine the period**:
   - The period is the length of one complete cycle of the wave.
   - In this graph, the function completes one full cycle from \( x = -\pi \) to \( x = \pi \). Therefore, the period \( T \) is \( 2\pi \).
   - The standard period of the cosine function is \( 2\pi \). Since this graph also has a period of \( 2\pi \), the frequency \( B \) is 1, meaning the equation has the form \( \cos(x) \).

3. **Check for horizontal shifts (phase shift)**:
   - The cosine function normally starts at its maximum value when \( x = 0 \). In this graph, the function also starts at \( x = 0 \) without any shift, so there is no horizontal phase shift, meaning \( C = 0 \).

4. **Check for vertical shifts**:
   - The midline of the graph is \( y = 0 \), meaning there is no vertical shift, so \( D = 0 \).

5. **Form the equation**:
   - With all the identified parameters, the equation of the cosine function is:
   \[
   y = 2 \cos(x)
   \]

The equation that models the graph provided is \( y = 2 \cos(x) \).

Would you like more details or have any questions about this solution?

Here are 5 related questions you might explore:
1. How would the equation change if the graph was shifted horizontally?
2. What is the effect of changing the amplitude in a cosine function?
3. How can you determine the period of a cosine function from its equation?
4. What happens to the graph if we add a vertical shift?
5. How would the equation change if the function were a sine function instead?

**Tip**: To identify the amplitude of a trigonometric function quickly, look at the distance between the maximum (or minimum) value and the midline.

Explore how to determine the equation of a cosine function from a graph. This detailed guide covers identifying amplitude, period, and phase shifts, with step-by-step examples. Suitable for advanced high school students and those studying trigonometry.

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