1. **State the problem:** We are given a table of $x$ and $y$ values and need to find the equation of the periodic function that fits these points. 2. **Given table:** $$\begin{array}{c|ccccc} x & 0 & \frac{\pi}{2} & \pi & \frac{3\pi}{2} & 2\pi \\ y & 0 & \frac{3}{4} & 0 & -\frac{3}{4} & 0 \end{array}$$ 3. **Analyze the values:** - At $x=0$, $y=0$. - At $x=\frac{\pi}{2}$, $y=\frac{3}{4}$. - At $x=\pi$, $y=0$. - At $x=\frac{3\pi}{2}$, $y=-\frac{3}{4}$. - At $x=2\pi$, $y=0$. 4. **Recall sine and cosine values:** - $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, $\sin \pi = 0$, $\sin \frac{3\pi}{2} = -1$, $\sin 2\pi = 0$. - $\cos 0 = 1$, $\cos \frac{\pi}{2} = 0$, $\cos \pi = -1$, $\cos \frac{3\pi}{2} = 0$, $\cos 2\pi = 1$. 5. **Match the pattern:** The $y$ values match the sine function pattern scaled by $\frac{3}{4}$: $$y = \frac{3}{4} \sin x$$ 6. **Check the options:** - $y = \frac{3}{2} \sin x$ (amplitude too large) - $y = -\frac{3}{4} \sin x$ (wrong sign) - $y = \frac{3}{4} \sin x$ (correct) - $y = \frac{3}{4} \cos x$ (does not match values at $x=0$) **Final answer:** $$y = \frac{3}{4} \sin x$$