To find the equation of the sinusoidal function $f(x) = A \sin(B(x - D))$, let's follow these steps:

1. Determine the period:

The sine function has zeros at $x = -4$ and $x = 2$, meaning these are consecutive zeros of the sine wave. The distance between them is $2 - (-4) = 6$. This distance represents half of a full period of the sine wave, so the full period is: $\text{Period} = 2 \times 6 = 12$

2. Find $B$:

The period of a sine function is related to $B$ by the formula: $\text{Period} = \frac{2\pi}{B}$ Substituting the known period: $12 = \frac{2\pi}{B}$ Solving for $B$: $B = \frac{2\pi}{12} = \frac{\pi}{6}$

3. Determine $D$ (Phase Shift):

Since the sine function typically starts at $x = 0$, but here the sine wave has a zero at $x = -4$, the function has been shifted horizontally. This shift is $D$, and it corresponds to the first zero point at $x = -4$. Therefore, $D = -4$.

4. Determine $A$ (Amplitude):

From the problem, we know the sine function passes through $(x = -4, y = 0)$ and $(x = 2, y = 0)$, but we don't have explicit information about the maximum or minimum values of the sine function. However, assuming the standard sine function shape, we can set the amplitude $A = 1$, unless other information is provided to suggest otherwise.

Final Equation:

The equation of the sinusoidal function is: $f(x) = \sin\left(\frac{\pi}{6}(x + 4)\right)$

Would you like more details on any part of this process?

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Here are 5 questions to extend this topic:

1. How does changing the value of $A$ affect the graph of the sinusoidal function?
2. What happens to the graph if you change the value of $B$?
3. How can we modify the equation if the sine function had a maximum at $x = -4$?
4. What would the equation be if the curve was a cosine function instead of a sine function?
5. How can you determine the vertical shift of a sinusoidal function?

Tip: The period of any sinusoidal function can be easily calculated using the formula $\frac{2\pi}{|B|}$, where $B$ is the coefficient of $x$ inside the sine or cosine function.