Draw the graph of the function `f(x)=-x|x|`.

Answer

To draw the graph of the function \( f(x) = -x|x| \), we will first rewrite the function in a piecewise manner based on the definition of the absolute value function.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = -x|x| \) involves the absolute value of \( x \). We need to consider two cases based on the sign of \( x \):
- When \( x \geq 0 \)
- When \( x < 0 \)

2. **Case 1: \( x \geq 0 \)**:
- For \( x \geq 0 \), \( |x| = x \).
- Thus, \( f(x) = -x|x| = -x \cdot x = -x^2 \).
- The graph of \( f(x) = -x^2 \) is a downward-opening parabola starting from the origin (0,0) and going downwards.

3. **Case 2: \( x < 0 \)**:
- For \( x < 0 \), \( |x| = -x \).
- Thus, \( f(x) = -x|x| = -x \cdot (-x) = x^2 \).
- The graph of \( f(x) = x^2 \) is an upward-opening parabola starting from the origin (0,0) and going upwards.

4. **Combining the Cases**:
- For \( x \geq 0 \), the graph is \( f(x) = -x^2 \).
- For \( x < 0 \), the graph is \( f(x) = x^2 \).
- Both parts meet at the origin (0,0).

5. **Graphing**:
- Plot the downward-opening parabola \( f(x) = -x^2 \) for \( x \geq 0 \).
- Plot the upward-opening parabola \( f(x) = x^2 \) for \( x < 0 \).
- The final graph will show a downward parabola for non-negative \( x \) and an upward parabola for negative \( x \), both meeting at the origin.

### Final Graph:
- The graph will look like this:
- For \( x < 0 \): It will rise upwards like \( x^2 \).
- For \( x \geq 0 \): It will fall downwards like \( -x^2 \).