1. The problem is to identify the equation corresponding to a graph that shows a sinusoidal wave with decreasing amplitude as $|x|$ increases, oscillating around $y=0$ with a period of about 100 units. 2. Such behavior suggests a damped sine function, which can be modeled by the formula: $$y = A e^{-kx} \sin\left(\frac{2\pi}{T} x\right)$$ where $A$ is the initial amplitude, $k$ is the damping coefficient (positive), and $T$ is the period of oscillation. 3. From the graph description, the period $T$ is approximately 100 units, so the sine term is: $$\sin\left(\frac{2\pi}{100} x\right) = \sin\left(\frac{\pi}{50} x\right)$$ 4. The amplitude decreases as $|x|$ increases, so the damping factor is an exponential decay $e^{-kx}$ for $x > 0$ and $e^{kx}$ for $x < 0$. To model symmetric damping on both sides, use $e^{-k|x|}$. 5. Therefore, the equation is: $$y = A e^{-k|x|} \sin\left(\frac{\pi}{50} x\right)$$ 6. This equation matches the graph's sinusoidal oscillations with decreasing amplitude as $|x|$ increases and period about 100. Final answer: $$y = A e^{-k|x|} \sin\left(\frac{\pi}{50} x\right)$$ where $A$ and $k$ are positive constants determined by initial amplitude and damping rate.