To solve this problem, we need to identify the equation of the given trigonometric function $f(x)$ by analyzing its amplitude, frequency, phase shift, and vertical shift.

Step-by-step Analysis:

1. Amplitude: The amplitude is the distance from the midline of the graph to its maximum or minimum. In this case, the maximum value is $3$ and the minimum value is $-5$. The amplitude is half the range: $\text{Amplitude} = \frac{\text{max} - \text{min}}{2} = \frac{3 - (-5)}{2} = 4$ However, the vertical shift affects the observed amplitude.

2. Vertical Shift: The vertical shift is the midline of the graph. The midline is calculated as: $\text{Midline} = \frac{\text{max} + \text{min}}{2} = \frac{3 + (-5)}{2} = -1$ Thus, the graph is shifted down by 1 unit.

3. Period: The period of the function is determined by the distance it takes for the wave to repeat. From the graph, one full cycle occurs from $x = 0$ to $x = \pi$. Thus, the period is: $\text{Period} = \pi$ The formula for the period of a sine or cosine function is: $\text{Period} = \frac{2\pi}{B}$ Solving for $B$: $B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{\pi} = 2$

4. Phase Shift: The graph appears to be shifted horizontally. To determine the phase shift, note that the graph reaches its maximum at $x = \frac{\pi}{4}$. This corresponds to the phase shift: $\text{Phase Shift} = \frac{\pi}{4}$

5. Choice of Function: From the shape of the graph, it resembles a cosine wave. The cosine function has its maximum at $x = 0$ before shifting. Considering the phase shift, vertical shift, and amplitude, the function takes the form: $f(x) = 3\cos\left(2\left(x - \frac{\pi}{4}\right)\right) - 1$

Thus, the correct answer is: $\boxed{(A) \ 3\cos\left(2\left(x - \frac{\pi}{4}\right)\right) - 1}$

Do you want further clarification or details about any step?

Related Questions:

1. How do we determine if the function is sine or cosine-based?
2. Why does the vertical shift affect the midline of the graph?
3. Can the phase shift be calculated differently if using a sine function?
4. How do we generalize the formula for sinusoidal functions?
5. Why does the amplitude remain positive, regardless of the graph's orientation?

Tip:

When analyzing trigonometric graphs, always check amplitude, vertical shift, period, and phase shift systematically to avoid errors.