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.