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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.

Find the formula for the graph of the function f(x) shown in the graph.

Learn how to find the formula for a periodic sine function from its graph. This example involves analyzing amplitude, period, and key characteristics of sine functions. Suitable for high school students in Grades 10-12.

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