GIVE ME PERIOD AND MIDLINE FOR EQUAR\TION 3(COS(X-3PI/2)+1)

Let's break down the given equation \( 3 \left(\cos\left(x - \frac{3\pi}{2}\right)\right) + 1 \) to find the period and midline.

### General form of the cosine function:
The general form of a cosine function is:
\[
f(x) = A \cos(Bx - C) + D
\]
where:
- \( A \) is the **amplitude**,
- \( \frac{2\pi}{B} \) is the **period**,
- \( D \) is the **midline** (vertical shift),
- \( C \) affects the **phase shift** but is not needed for amplitude or period calculation.

### For the equation:
\[
f(x) = 3 \left(\cos\left(x - \frac{3\pi}{2}\right)\right) + 1
\]

1. **Amplitude**: The coefficient in front of the cosine function is \( 3 \), but we aren't asked to find the amplitude.

2. **Period**: The period is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \). In this case, \( B = 1 \) (since the equation is \( \cos(x - \frac{3\pi}{2}) \), the coefficient of \( x \) is 1). Therefore, the period is:
   \[
   \text{Period} = \frac{2\pi}{1} = 2\pi
   \]

3. **Midline**: The midline is the constant added outside the cosine function, which in this case is \( D = 1 \).

### Summary:
- **Period**: \( 2\pi \)
- **Midline**: \( 1 \)

Would you like further details or explanations? Here are some related questions to consider:

1. How do phase shifts affect the cosine function graph?
2. How does the amplitude influence the range of a cosine function?
3. How would a negative amplitude affect the cosine graph?
4. How can you determine the phase shift for a cosine function?
5. What are the similarities and differences between sine and cosine functions?

**Tip:** When identifying the period of trigonometric functions, always check the coefficient of \( x \) inside the trigonometric expression to calculate it properly.

Learn how to find the period and midline of the trigonometric function f(x) = 3(cos(x - 3π/2)) + 1. Understand the concepts of amplitude, period, and midline in trigonometry with detailed explanations. Suitable for advanced high school students.

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