Reflecting Across the Line y = x

The line y = x is the set of points where the x-coordinate and y-coordinate are equal, like (3,3) or (-5,-5). It acts as a perfect mirror because it's tilted at 45 degrees through the origin, equally close to both axes. Fold the plane along that line and the horizontal axis lands exactly on the vertical axis. So whatever distance a point sits from the x-axis becomes its distance from the y-axis after folding, and vice versa. That swap of distances is exactly a swap of coordinates.

Mechanically, take a point (a, b). The segment connecting it to its mirror image (b, a) is always perpendicular to y = x, and the midpoint of that segment always lands exactly on y = x. You can check this with slope: the segment from (a,b) to (b,a) has slope -1, which is perpendicular to the line y = x's slope of 1. No rotation, no shifting — just a clean swap of the two coordinate values.

This idea connects to other reflections in the same unit. Reflecting across y = -x also swaps coordinates, but then negates both: (a, b) maps to (-b, -a). Mixing these two up is the single biggest trap in this topic. Reflecting a whole graph, like a line or curve, works the same way: swap x and y in the equation and solve for y again. That's how you find the inverse of a function graphically, which is why this reflection shows up again once inverse functions enter the picture.