An even function is symmetric about the y-axis: fold the coordinate plane along that axis and the two halves of the graph land exactly on top of each other.

The Fold Test

Fold the graph of

along the y-axis (the dashed line) and the right half lands exactly on the left half. Once you know a function is even, you only need to draw one side — then reflect it to complete the picture.

More Examples

and

sits below the x-axis instead of above it, but that doesn’t matter for evenness — both sides still match when folded along the y-axis.

The Algebra Test

A function is even if for every in its domain — opposite inputs give the same output.

and

Opposite values, same value — exactly what means.

A Shortcut Halve the work in integration

To find the area between

and the x-axis from

, you don’t need the whole interval: find the area from

and multiply by

. The matching left half comes for free.

Summary

An even function is symmetric about the y-axis — fold along it and both sides match.
, , and are all even, whether they sit above or below the x-axis.
Algebraically, a function is even when for every in its domain.
Symmetry gives a shortcut in integration: compute half the area and double it.

Function Symmetry