1. **State the problem:** We need to find a sine function of the form $y = -a \sin(bt)$ that models a sound wave with amplitude 2 and period 512. 2. **Recall the formulas:** - Amplitude $a$ is the coefficient in front of the sine function. - Period $T$ is related to $b$ by the formula $$T = \frac{2\pi}{b}$$ 3. **Use the given period to find $b$:** Given $T = 512$, solve for $b$: $$b = \frac{2\pi}{T} = \frac{2\pi}{512} = \frac{\pi}{256}$$ 4. **Check the amplitude:** Amplitude $a = 2$ as given. 5. **Write the function:** The function is $$y = -2 \sin\left(\frac{\pi}{256} t\right)$$ 6. **Compare with options:** - Option 1: $y = 1.024 \sin(2t)$ amplitude 1.024, period $\pi$ (wrong) - Option 2: $y = \sin\left(\frac{1}{512} t\right)$ amplitude 1, period $2\pi \times 512$ (wrong) - Option 3: $y = 2 \sin\left(\frac{1}{512} t\right)$ amplitude 2, period $2\pi \times 512$ (wrong period) - Option 4: $y = -\frac{1}{512} \sin(4t)$ amplitude $\frac{1}{512}$ (wrong) None exactly match $y = -2 \sin\left(\frac{\pi}{256} t\right)$, but since $\frac{\pi}{256} \approx 0.01227$ and $\frac{1}{512} = 0.001953$, the closest period match is option 2 or 3 but they lack the negative sign and have wrong period. **Conclusion:** The correct function is $$y = -2 \sin\left(\frac{\pi}{256} t\right)$$ which is not exactly listed but matches the problem conditions best. **Final answer:** $y = -2 \sin\left(\frac{\pi}{256} t\right)$