1. **Problem Statement:** Sketch the graph of the function $y = 2 \sin x$ for $0 \leq x \leq 2\pi$. 2. **Formula and Rules:** The sine function $\sin x$ oscillates between $-1$ and $1$ with a period of $2\pi$. Multiplying by 2 changes the amplitude to 2, so the function oscillates between $-2$ and $2$. 3. **Key points to plot:** - At $x=0$, $y=2\sin 0=0$. - At $x=\frac{\pi}{2}$, $y=2\sin \frac{\pi}{2}=2$ (maximum). - At $x=\pi$, $y=2\sin \pi=0$. - At $x=\frac{3\pi}{2}$, $y=2\sin \frac{3\pi}{2}=-2$ (minimum). - At $x=2\pi$, $y=2\sin 2\pi=0$. 4. **Sketching:** Connect these points smoothly to form one full sine wave with amplitude 2 over the interval $0$ to $2\pi$. 5. **Summary:** The graph is a sine wave starting at 0, peaking at 2, returning to 0, dipping to -2, and returning to 0 at $2\pi$. Final answer: The graph of $y=2\sin x$ has amplitude 2 and period $2\pi$ over $0 \leq x \leq 2\pi$.