Write a sine function that has a midline of y equals 4 an amplitude of 3 and a period of 5pi/3
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To write the equation of a sine function with the given characteristics:

- **Midline**: \( y = 4 \)
- **Amplitude**: \( 3 \)
- **Period**: \( \frac{5\pi}{3} \)

The general form of a sine function is:

\[
y = A \sin(Bx + C) + D
\]

Where:
- \( A \) is the amplitude.
- \( B \) is related to the period by the formula \( \text{Period} = \frac{2\pi}{B} \).
- \( D \) is the vertical shift (midline).

Given the information:
- \( A = 3 \)
- The period is \( \frac{5\pi}{3} \), so \( B \) can be found using \( \frac{2\pi}{B} = \frac{5\pi}{3} \). Solving this gives \( B = \frac{6}{5} \).
- \( D = 4 \)

Since no horizontal shift is mentioned, we assume \( C = 0 \).

So the equation of the sine function is:

\[
y = 3 \sin\left(\frac{6}{5}x\right) + 4
\]

Would you like to go into more details or have any questions?

Here are 5 related questions for you to explore further:

1. How would the function change if the period were doubled?
2. What is the phase shift if \( C \) were \( \frac{\pi}{4} \)?
3. How would you write the equation for a cosine function with the same properties?
4. What are the maximum and minimum values of this function?
5. How do you determine the phase shift from a given function?

**Tip:** When writing trigonometric functions, always check if the equation needs any horizontal or vertical shifts!

Learn how to write a sine function with a midline of y = 4, an amplitude of 3, and a period of 5π/3. This detailed explanation covers the sine function formula, amplitude, period calculation, and midline adjustment. Suitable for high school students and anyone interested in trigonometry concepts.

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