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The image contains a precalculus review question that appears to be focused on writing the equation for a sine function with given parameters. Here’s the detailed breakdown for this type of problem:

### Problem:
Write the equation of a sine function with:
- Amplitude = 4
- Period = \(3\pi\)
- Phase shift = \(\pi\)
- Midline = \(y = 2\)

### Solution Outline:
A general form for a sine function is:
\[
y = A \sin(B(x - C)) + D
\]
where:
- \(A\) is the amplitude,
- \(\frac{2\pi}{B}\) is the period,
- \(C\) is the phase shift, and
- \(D\) is the vertical shift (midline).

Given values:
1. **Amplitude** \(A = 4\).
2. **Period** \(3\pi\): Using \(\frac{2\pi}{B} = 3\pi\), solve for \(B\):
   \[
   B = \frac{2\pi}{3\pi} = \frac{2}{3}
   \]
3. **Phase Shift** \(\pi\): Since phase shift \(C = \pi\), we keep \(C = \pi\).
4. **Midline** \(y = 2\): This gives the vertical shift \(D = 2\).

### Final Equation:
Substitute all values into the general form:
\[
y = 4 \sin\left(\frac{2}{3}(x - \pi)\right) + 2
\]

Would you like further details or explanations? Here are some related questions to deepen your understanding:

1. How do you calculate the phase shift for a cosine function with similar parameters?
2. What changes if the function were cosine instead of sine?
3. How do you determine the frequency of a sine function from its period?
4. What are some transformations of the sine function on a graph based on these parameters?
5. How can you confirm the midline and amplitude graphically?

**Tip:** For sine and cosine functions, remember that the phase shift moves the graph horizontally, while the vertical shift moves it up or down.

Write the equation of a sine function with amplitude 4, period 3π, phase shift π, and midline y = 2.

This guide covers how to write the equation of a sine function given specific parameters: amplitude of 4, period of 3π, phase shift of π, and a midline at y = 2. It explores key trigonometric concepts such as amplitude, period, phase shift, and midline, and provides a detailed step-by-step solution.

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