Help find the amplitude, A, period, P, and the Midline of the function:

f(x)=2(cos(x-(5pi)/6)+3)

To find the amplitude \( A \), period \( P \), and midline of the function:

\[
f(x) = 2 \left( \cos \left( x - \frac{5\pi}{6} \right) + 3 \right)
\]

Let's break this down step by step:

### 1. **Amplitude, \( A \)**

The amplitude of a cosine function in the form \( y = a \cos(bx - c) + d \) is given by the absolute value of the coefficient of the cosine function, \( |a| \). In this case, the coefficient in front of the cosine is \( 2 \). So, the amplitude is:

\[
A = 2
\]

### 2. **Period, \( P \)**

The period of a cosine function is determined by the formula \( P = \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \) inside the cosine function. In this case, \( b = 1 \), so the period is:

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

### 3. **Midline**

The midline of the function is the vertical shift, which is given by the constant added to the cosine function. In this case, the constant is \( +3 \). Therefore, the midline is:

\[
y = 3
\]

### Final Answer:

- Amplitude \( A = 2 \)
- Period \( P = 2\pi \)
- Midline \( y = 3 \)

Let me know if you need more details or have any questions.

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Here are 5 related questions to explore:

1. How does the phase shift affect the graph of the function?
2. What is the maximum value of \( f(x) \)?
3. What is the minimum value of \( f(x) \)?
4. How would the period change if the coefficient of \( x \) was 2 instead of 1?
5. How can you rewrite the function in sine form?

**Tip**: The midline is always the average of the maximum and minimum values of the function.

Help find the amplitude, A, period, P, and the Midline of the function: f(x)=2(cos(x-(5pi)/6)+3)

Discover how to find the amplitude, period, and midline of the trigonometric function f(x) = 2(cos(x - 5π/6) + 3) with a step-by-step solution. This problem explores core trigonometric concepts, including the cosine function properties. Suitable for high school students in Grades 10-12.

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