The reflected line will have the same slope (1) but its y-intercept will be negated, resulting in the equation y = -x.
Imagine you have a solid line on a graph, represented by the equation y = x. This line forms a straight diagonal line passing through the origin (0, 0). Now, let's place a two-sided mirror along the x-axis, effectively dividing the graph into two halves.
The dotted line in the image below shows the reflected image of the original line y = x after it bounces off the mirror. We want to find the equation of this reflected line.
Slope: The slope of a line determines its angle of inclination. Since the reflection doesn't alter the angle of the line, the slope remains the same: for y = x, the slope is 1, and the reflected line will also have a slope of 1.
y-Intercept: The y-intercept of a line tells you where the line crosses the y-axis. When we reflect the line across the x-axis, we essentially flip it over. This means the y-intercept changes sign. In this case, the original line y = x has a y-intercept of 0 (it passes through the origin), so the reflected line will have a y-intercept of -0 (essentially 0 mirrored across the x-axis).
Therefore, the equation of the reflected line is:
y = -x
This equation retains the original slope of 1 but negates the y-intercept, reflecting the line across the x-axis.