Let's break down the function $f(x) = 2 \left( \cos \left( x - \frac{\pi}{4} \right) + 2 \right)$ to determine the amplitude, period, and midline.

1. Amplitude (A):
   The amplitude is the coefficient of the cosine function. In this case, the coefficient of $\cos \left( x - \frac{\pi}{4} \right)$ is 2.
   Thus, 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, the function is $\cos \left( x - \frac{\pi}{4} \right)$, which means $B = 1$.
   Therefore, the period is: $P = \frac{2\pi}{1} = 2\pi$

3. Midline (y):
   The midline is the vertical shift of the function. In this case, the entire cosine function is shifted up by 2 units.
   Hence, the midline is: $y = 2$

To summarize:

• Amplitude: 2
• Period: $2\pi$
• Midline: 2

Let me know if you would like more details or have any questions!

Here are five related questions for further understanding:

1. How would the function change if the amplitude were 3 instead of 2?
2. What happens to the graph if the period is halved?
3. Can you explain how to find the phase shift of the function?
4. How does the midline affect the overall shape of the graph?
5. What would happen to the period if the equation inside the cosine were $2x$?

Tip: The period of a trigonometric function is always influenced by the coefficient of $x$. If you increase that coefficient, the period decreases, and vice versa.