f(x) = sin(2x) for x ∈ [−2π, 2π] explain each step needed in order to solve a question like this asking to sketch the graph

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

f(x) = sin(2x) for x ∈ [−2π, 2π] explain each step needed in order to solve a question like this asking to sketch the graph.

Learn how to sketch the graph of f(x) = sin(2x) over the interval [-2π, 2π]. This step-by-step solution explains the period, key points, and behavior of the sine function with a horizontal compression due to the factor 2. Suitable for students in Grades 10-12.

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