https://file.aiquickdraw.com/mathbot/file/172680533559129g1r94n.jpg

The problem asks for the formula of a trigonometric function that matches the graph shown, with \( f(x) = 0.5 \cos\left(\frac{\pi x}{5}\right) \) as the suggested form.

### Steps to Analyze the Graph and Formula:
1. **Amplitude**: The graph's maximum and minimum values range from 0.5 to -0.5, which suggests the amplitude of the function is 0.5. The given equation, \( 0.5 \cos(\cdots) \), matches this.

2. **Period**: The period of a cosine function \( \cos(Bx) \) is \( \frac{2\pi}{B} \). 
   - The graph shows a full cycle from \( -5 \) to \( 5 \), giving a period of 10.
   - In the formula \( \cos\left(\frac{\pi x}{5}\right) \), the coefficient of \( x \) inside the cosine is \( \frac{\pi}{5} \), so the period is \( \frac{2\pi}{\frac{\pi}{5}} = 10 \), which matches the graph.

3. **Phase Shift**: There is no horizontal shift in the graph, and the function starts at its maximum at \( x = 0 \), which is characteristic of a cosine function.

Thus, the proposed formula:
\[
f(x) = 0.5 \cos\left(\frac{\pi x}{5}\right)
\]
is a valid representation of the graph.

### Would you like more details or clarifications?

Here are 5 related questions to consider:
1. How can we adjust the formula if the graph had a phase shift?
2. What is the effect of changing the amplitude on the graph?
3. How would a sine function look compared to this cosine function?
4. How does the period change if the coefficient of \( x \) inside the cosine changes?
5. What would happen if the graph was reflected across the x-axis?

**Tip**: When analyzing trigonometric functions, always check amplitude, period, and phase shift to determine the correct formula.

Find a possible formula for the trigonometric function graphed below. Use x as the independent variable in your formula.

This problem involves finding the trigonometric function formula from a graph. The graph represents a cosine function with an amplitude of 0.5 and a period of 10, leading to the formula f(x) = 0.5 cos((π x)/5). This example is suited for high school students learning trigonometric functions.

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