https://file.aiquickdraw.com/mathbot/file/1726870288918a4ynsaz6.jpg

To find a function of the form \( f(x) = a \sin(bx - c) \) that matches the graph in the image, let's analyze the given sinusoidal graph step by step.

### Key Observations:
1. **Amplitude**: The graph oscillates between 6 and -6, so the amplitude \(a\) is \(6\). Therefore, \( a = 6 \).

2. **Period**: The graph completes one full oscillation (from peak to peak) between \(x = -3\) and \(x = 3\), indicating that the period is 6. The period \(T\) of a sine function is given by \( T = \frac{2\pi}{b} \). In this case:
   \[
   6 = \frac{2\pi}{b} \implies b = \frac{2\pi}{6} = \frac{\pi}{3}
   \]
   So, \(b = \frac{\pi}{3}\).

3. **Phase Shift**: The graph appears to be shifted horizontally. The sine function typically starts at 0 when \(x = 0\), but this graph's sine wave appears shifted. The maximum occurs at \(x = -3\), meaning the graph has been shifted left. The phase shift \(c\) corresponds to this horizontal shift. Since the maximum occurs at \(x = -3\), and we know that for a sine function \( \sin(bx - c)\), the maximum occurs when \( bx - c = \frac{\pi}{2} \), we solve for \(c\):
   \[
   b(-3) - c = \frac{\pi}{2}
   \]
   \[
   -\frac{\pi}{3}(-3) - c = \frac{\pi}{2} \implies c = 0
   \]
   So, there's no phase shift.

### Conclusion:
The equation of the function is:
\[
f(x) = 6 \sin\left(\frac{\pi}{3}x\right)
\]

Would you like further explanation or details on any part of this solution?

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Here are 5 related questions to expand your understanding:
1. How does the period of a sine function change when the coefficient \(b\) is altered?
2. What happens to the graph when the phase shift \(c\) is changed in the equation \(a \sin(bx - c)\)?
3. How would the function change if the amplitude were reduced to 4 instead of 6?
4. What would the graph look like if a cosine function \(a \cos(bx - c)\) were used instead of sine?
5. How can you identify the vertical shift of a sinusoidal graph?

**Tip**: The horizontal shift (phase shift) of sinusoidal functions can be determined by looking at the starting point of the wave (e.g., where it reaches its maximum or minimum).

Find a function of the form f(x) = a sin(bx - c) such that the graph of f matches the figure.

Learn how to determine the sine function f(x) = a sin(bx - c) that matches a given graph. This example involves identifying the amplitude, period, and phase shift of the sine function, suitable for students in grades 10-12.

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