Draw the graphs of the following function
`f(x)=sinabsxxin[-2pi,2pi]`

Answer

To draw the graph of the function \( f(x) = \sin(|x|) \) over the interval \([-2\pi, 2\pi]\), we can follow these steps:

### Step 1: Understand the function
The function is defined as \( f(x) = \sin(|x|) \). The absolute value function \( |x| \) means that:
- For \( x \geq 0 \), \( |x| = x \)
- For \( x < 0 \), \( |x| = -x \)

Thus, we can rewrite the function based on the sign of \( x \):
- For \( x \geq 0 \), \( f(x) = \sin(x) \)
- For \( x < 0 \), \( f(x) = \sin(-x) = -\sin(x) \) (since sine is an odd function)

### Step 2: Determine the key points
Next, we need to identify key points of the sine function:
- \( f(0) = \sin(0) = 0 \)
- \( f(\pi/2) = \sin(\pi/2) = 1 \)
- \( f(\pi) = \sin(\pi) = 0 \)
- \( f(3\pi/2) = \sin(3\pi/2) = -1 \)
- \( f(2\pi) = \sin(2\pi) = 0 \)

For negative values of \( x \):
- \( f(-\pi/2) = -\sin(\pi/2) = -1 \)
- \( f(-\pi) = -\sin(\pi) = 0 \)
- \( f(-3\pi/2) = -\sin(3\pi/2) = 1 \)
- \( f(-2\pi) = -\sin(2\pi) = 0 \)

### Step 3: Plot the points
Now we can plot the points on the graph:
- For \( x \) from \( 0 \) to \( 2\pi \), plot the points from \( (0, 0) \) to \( (2\pi, 0) \) with a peak at \( (\pi/2, 1) \) and a trough at \( (3\pi/2, -1) \).
- For \( x \) from \(-2\pi\) to \( 0 \), plot the points from \( (-2\pi, 0) \) to \( (0, 0) \) with a peak at \( (-3\pi/2, 1) \) and a trough at \( (-\pi/2, -1) \).

### Step 4: Draw the graph
- Draw the curve for \( f(x) = \sin(x) \) for \( x \geq 0 \).
- Draw the curve for \( f(x) = -\sin(x) \) for \( x < 0 \).
- Ensure the graph is continuous at \( x = 0 \) where both parts meet.

### Step 5: Label the axes
- Label the x-axis from \(-2\pi\) to \(2\pi\).
- Label the y-axis from \(-1\) to \(1\).

### Final Graph
The final graph will show a sine wave for \( x \geq 0 \) and an inverted sine wave for \( x < 0 \), both meeting at the origin.