The function given in the problem is:

$f(x) = 7 \sin \left( \frac{1}{4} \pi x + \pi \right) - 4$

We are tasked with finding the period of this sine function.

Step-by-Step Solution:

1. Standard Form of a Sine Function: The general form of a sine function is: $f(x) = A \sin(Bx + C) + D$ The period of this sine function is given by: $\text{Period} = \frac{2\pi}{|B|}$ where $B$ is the coefficient of $x$.

2. Identify the Coefficient of $x$: In the given function: $f(x) = 7 \sin \left( \frac{1}{4} \pi x + \pi \right) - 4$ The coefficient of $x$ inside the sine function is $\frac{\pi}{4}$, i.e., $B = \frac{\pi}{4}$.

3. Find the Period: The period is calculated as: $\text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 8$

Thus, the period of the function is 8.

Final Answer:

The correct option is $\boxed{8}$.

Would you like more details on any step? Here are some related questions:

1. What role does the coefficient $B$ play in determining the period of trigonometric functions?
2. How do phase shifts (like $\pi$) affect the sine function?
3. Why does the amplitude or vertical shift not affect the period of the sine function?
4. How would the period change if the coefficient of $x$ was $\frac{1}{2} \pi$ instead of $\frac{1}{4} \pi$?
5. How do you determine the phase shift from a sine function's equation?

Tip: For any sine or cosine function, the coefficient of $x$ inside the trigonometric function directly influences the period. Always divide $2\pi$ by that coefficient to find the period.