The problem in the image asks:

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

To form the equation of a sine function, we use the general formula: $y = A \sin(B(x - C)) + D$ where:

• $A$ is the amplitude,
• $B$ is related to the period by $\text{Period} = \frac{2\pi}{B}$,
• $C$ is the phase shift,
• $D$ is the vertical shift, which also defines the midline.

Given values:

1. Amplitude $A = 4$,
2. Period $3\pi$, so $B = \frac{2\pi}{3\pi} = \frac{2}{3}$,
3. Phase shift $\pi$, so $C = \pi$,
4. Midline $y = 2$, so $D = 2$.

Plugging these values in:

$y = 4 \sin\left(\frac{2}{3}(x - \pi)\right) + 2$

Final Answer:

The equation of the sine function is: $y = 4 \sin\left(\frac{2}{3}(x - \pi)\right) + 2$

Would you like further clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. What would happen to the equation if the phase shift was $\frac{\pi}{2}$ instead of $\pi$?
2. How would the equation change if the amplitude was 5?
3. Can you rewrite this equation in terms of cosine instead of sine?
4. What would the equation look like if the period was $4\pi$?
5. How does the midline affect the range of the function?

Tip: The phase shift $C$ in the equation $y = A \sin(B(x - C)) + D$ shifts the graph horizontally.