To find the line of reflection that transforms the function \( f(x) = -3x \) into \( f(x) = 3x \), we need to find a line such that when the graph of the first function is reflected over it, it will yield the graph of the second function.
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Identify the slopes: The slope of \( f(x) = -3x \) is -3, and the slope of \( f(x) = 3x \) is 3. The line of reflection must bisect the angle between the two lines.
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Calculate the slope of the line of reflection: The average of the two slopes (-3 and 3) is: \[ m = \frac{-3 + 3}{2} = 0 \] This means the line of reflection is horizontal.
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Find the y-coordinate: The line of reflection will be at the midpoint of the y-values of the two points on the lines for any value of \( x \). Specifically, we can choose \( x = 0 \) (the y-intercept), where:
- For \( f(x) = -3x \), \( f(0) = 0 \).
- For \( f(x) = 3x \), \( f(0) = 0 \). The midpoint of \( 0 \) (y-intercept) is \( 0 \).
Thus, the equation of the line of reflection is:
\[ y = 0 \]
So, the answer is: \[ \text{The line of reflection is } y = 0. \]