Symmetry of a Graph — Cheat sheet

CHEAT SHEET · FREE

Key method

$-f(x)$ reflects across the horizontal axis. $f(-x)$ reflects across the vertical axis. $-f(-x)$ reflects through the origin. Reflection across $y=x$ exchanges coordinates and describes an inverse relation.

Example

If $(2,5)$ lies on $f$ , then $(2,-5)$ lies on $-f$ , $(-2,5)$ lies on $f(-x)$ , and $(-2,-5)$ lies on $-f(-x)$ . For the even function $x^2$ , horizontal input reflection leaves the graph unchanged, but output reflection gives a downward-opening parabola.

Avoid this mistake

The graph $y=-x^2$ is not a right-to-left reflection of $y=x^2$ ; it is an up-to-down reflection.