You've provided an image of a graph with the task of finding the formula for the graph of the function $f(x)$ shown in the graph. Let's analyze the characteristics of the graph to determine the formula.

1. Shape and Nature of the Graph:

   • The graph shows a periodic wave, which suggests that the function is a trigonometric function.

2. Periodicity:

   • The graph completes one full cycle between $x = 0$ and $x = 2\pi$, indicating that the basic function might be $\sin(x)$ or $\cos(x)$ since both have a natural period of $2\pi$.

3. Amplitude:

   • The graph ranges from about $-4$ to $4$, suggesting an amplitude of $4$.

4. Phase Shift and Vertical Shift:

   • There doesn’t seem to be any phase shift or vertical shift as the graph starts at zero and oscillates symmetrically around the x-axis.

Based on this analysis, the likely formula for $f(x)$ could be: $f(x) = 4\sin(x) \quad \text{or} \quad f(x) = 4\cos(x)$ Considering the standard starting points of sine and cosine functions, the function closely matches $4\sin(x)$ because it starts at $x = 0, y = 0$ and reaches the first peak at $x = \frac{\pi}{2}, y = 4$.

Would you like further details on this analysis or have any other questions? Here are some related queries you might consider:

1. How would the formula change if there was a phase shift in the graph?
2. What would the graph look like if the amplitude was doubled?
3. How does changing the period affect the graph of a sine function?
4. Could the function be a cosine function instead, and why might it appear so?
5. What are some real-world applications of sine and cosine functions?

Tip: When analyzing periodic functions from a graph, always note the maximum and minimum points along with the x-intercepts to determine amplitude, period, and phase shifts effectively.