reflection

A reflection flips a graph across an axis as if that axis were a mirror. Every point jumps to the spot directly opposite it, the same distance from the line but on the other side. The shape is preserved, just mirrored.

Two flips matter most. Negating the output, -f(x), reflects the graph across the x-axis: each height becomes its opposite, so the curve flips upside down. Negating the input, f(-x), reflects it across the y-axis: left and right swap places. The first acts on what comes out, the second on what goes in.

Reflections connect directly to even and odd functions. A graph that is its own reflection across the y-axis belongs to an even function, since f(-x) = f(x). A graph unchanged by a half-turn about the origin — which is reflection across both axes at once — belongs to an odd function, since f(-x) = -f(x).