To sketch the graph of $f(x) = \sin(2x)$ for $x \in [-2\pi, 2\pi]$, follow these steps carefully, as each one builds on understanding the function's behavior over the given interval:

Step 1: Understand the Function

• The function is a sine function, but it has been modified by the factor of 2 inside the argument. The general sine function is $\sin(x)$, which oscillates between -1 and 1 with a period of $2\pi$.
• The factor of 2 inside $\sin(2x)$ indicates a horizontal compression, meaning the sine wave will oscillate faster (its period will be shorter).

Step 2: Identify the Period

• The period of the sine function $f(x) = \sin(kx)$ is given by $\frac{2\pi}{|k|}$. Here, $k = 2$, so the period is: $\text{Period} = \frac{2\pi}{2} = \pi$
• This means the function completes one full cycle (from 0 to 1 to 0 to -1 and back to 0) every $\pi$ units, rather than $2\pi$ units as in $\sin(x)$.

Step 3: Determine the Key Points

• For $f(x) = \sin(2x)$, the key points are based on the behavior of the sine function. Over one period, the sine function starts at 0, rises to 1, returns to 0, falls to -1, and returns to 0. The key points for one cycle of $\sin(2x)$ are:
  • $x = 0$ : $f(0) = \sin(0) = 0$
  • $x = \frac{\pi}{2}$ : $f\left(\frac{\pi}{2}\right) = \sin(\pi) = 0$
  • $x = \frac{\pi}{4}$ : ( f\left(\frac{\파 tracing