Write a sine function that has a midline of y, equals, 3*y*=3, an amplitude of 4, a period of 2, pi2*π*, and a horizontal shift of start fraction, 2, pi, divided by, 3, end fraction32*π*​ to the right.

To write a sine function that meets the given criteria, let's define each part:

1. **Midline**: The midline is \( y = 3 \), so the vertical shift \( D = 3 \).
2. **Amplitude**: The amplitude \( A = 4 \).
3. **Period**: The period \( P = 2\pi \), so the coefficient \( B \) is calculated as \( B = \frac{2\pi}{P} \).
4. **Horizontal Shift**: The horizontal shift is \( \frac{2\pi}{3} \) to the right, so \( C = \frac{2\pi}{3} \).

The general form of a sine function is:

\[
y = A \sin(B(x - C)) + D
\]

### Step 1: Find the value of \( B \)
Since the period \( P = 2\pi \),

\[
B = \frac{2\pi}{2\pi} = 1
\]

### Step 2: Substitute the values into the sine function
\[
y = 4 \sin\left(1 \cdot \left(x - \frac{2\pi}{3}\right)\right) + 3
\]

### Simplified Sine Function
\[
y = 4 \sin\left(x - \frac{2\pi}{3}\right) + 3
\]

This is the sine function that satisfies all the given conditions.

Would you like further details or have any questions?

### Related Questions:
1. How would the function change if the amplitude were 2 instead of 4?
2. What would be the effect on the function if the horizontal shift were to the left instead of the right?
3. Can you find the phase shift in degrees instead of radians?
4. What happens to the function if the midline changes to \( y = 5 \)?
5. How does the graph of this function compare to the standard sine function \( y = \sin(x) \)?

**Tip:** Always check that your function reflects the correct amplitude and shifts by comparing it to a standard sine graph.

Learn how to write a sine function with a midline of y = 3, an amplitude of 4, a period of 2π, and a horizontal shift of π/3 to the right. Detailed step-by-step solution provided. Suitable for Grades 10-12.

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