1. The problem is to understand and graph the function $y = \sin(2x)$.\n\n2. The function $\sin(2x)$ is a sine function with its input multiplied by 2, which affects the frequency of the wave.\n\n3. The general sine function is $y = \sin(x)$, which oscillates between -1 and 1 with a period of $2\pi$.\n\n4. When the input is multiplied by 2, the function becomes $y = \sin(2x)$, which means the period is shortened. The period $T$ of $\sin(bx)$ is given by $T = \frac{2\pi}{b}$. Here, $b=2$, so $T = \frac{2\pi}{2} = \pi$.\n\n5. This means the sine wave completes one full cycle every $\pi$ units along the x-axis, making it oscillate twice as fast as $\sin(x)$.\n\n6. The amplitude remains 1, so the wave oscillates between -1 and 1.\n\n7. The graph starts at the origin $(0,0)$, goes up to 1 at $x=\frac{\pi}{4}$, back to 0 at $x=\frac{\pi}{2}$, down to -1 at $x=\frac{3\pi}{4}$, and returns to 0 at $x=\pi$, completing one full cycle.\n\nFinal answer: The function $y = \sin(2x)$ is a sine wave with amplitude 1 and period $\pi$, oscillating twice as fast as $y = \sin(x)$.