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?

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).