Question

The function f(x)= -3x is reflected over a line and results in the equation f(x) =3x. Identify the equation for the line of reflection.
The line of reflection is y =

Answer 1

To find the line of reflection that transforms the function \( f(x) = -3x \) into \( f(x) = 3x \), we can follow a geometric approach.

1. **Identify key points on both graphs**:
- Choose a point on the line \( y = -3x \). For example, if \( x = 1 \), then:
\[
f(1) = -3(1) = -3 \quad \Rightarrow \quad (1, -3)
\]
- Now choose a corresponding point on the line \( y = 3x \). For \( x = 1 \):
\[
f(1) = 3(1) = 3 \quad \Rightarrow \quad (1, 3)
\]

2. **Find the midpoint of the segment connecting these two points**:
- The midpoint \( M \) of the points \( (1, -3) \) and \( (1, 3) \) is:
\[
M = \left( \frac{1 + 1}{2}, \frac{-3 + 3}{2} \right) = \left( 1, 0 \right)
\]

3. **Determine the slope of the line connecting the original point and the reflected point**:
- The slope from \( (1, -3) \) to \( (1, 3) \) is undefined (infinity), which indicates a vertical line.

4. **Determine the equation of the line of reflection**:
- Since the reflection is vertical, the line of reflection must be the horizontal line that runs through the midpoint. The midpoint \( (1, 0) \) gives us the y-coordinate we need:
\[
y = 0
\]

Thus, the line of reflection is given by:
\[
\boxed{y = 0}
\]