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.

---

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.