Write a sine function that has a midline of y, equals 3y = 3, an amplitude of 2, a period of start fraction 2, divided by 7, and an end fraction
7
2
​
and a horizontal shift of start fraction, 1, divided by the end fraction, 14.
14
1
​
to the right.

To write a sine function that satisfies the given conditions, let's break down each component:

1. **Midline**: The midline is \( y = 3 \). This implies that the vertical shift \( D \) is 3.
2. **Amplitude**: The amplitude \( A \) is 2.
3. **Period**: The period \( P \) is \( \frac{7}{2} \). The formula to find the coefficient \( B \) in the sine function is \( B = \frac{2\pi}{\text{Period}} \).
4. **Horizontal Shift**: The horizontal shift is \( \frac{1}{14} \) to the right. This implies \( C = \frac{1}{14} \).

Now, let's use the general form of a sine function:

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

### Step 1: Find the value of \( B \)
\[
B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{\frac{7}{2}} = \frac{2\pi \times 2}{7} = \frac{4\pi}{7}
\]

### Step 2: Substitute the values into the sine function
\[
y = 2 \sin\left(\frac{4\pi}{7}\left(x - \frac{1}{14}\right)\right) + 3
\]

### Final Sine Function
The sine function that meets all the specified conditions is:

\[
y = 2 \sin\left(\frac{4\pi}{7}\left(x - \frac{1}{14}\right)\right) + 3
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the function change if the amplitude were 3 instead of 2?
2. What would be the function if the period were \( \frac{7}{4} \) instead of \( \frac{7}{2} \)?
3. How would the graph of the function look after applying the horizontal shift?
4. Can you rewrite the function in cosine form?
5. What is the effect of the horizontal shift on the phase of the sine function?

**Tip:** To verify the characteristics of a sine function, it's useful to plot the function and visually confirm the amplitude, period, and shifts.

Learn how to write a sine function with a midline y = 3, amplitude of 2, period of 7/2, and horizontal shift of 1/14 to the right. Explore the step-by-step process and formula derivation for constructing this specific sine function.

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