Symmetry occurs often in nature and in art. For example, when viewed from the front, the bodies of most animals are at least approximately symmetric. This means that each eye is the same distance from the center of the bridge of the nose, each shoulder is the same distance from the center of the chest, and so on. Architects have used symmetry for thousands of years to enhance the beauty of buildings.

A knowledge of symmetry in mathematics helps us graph and analyze equations and functions.

Consider the points (4, 2) and (4, −2) that appear on the graph of $x=y^2$, as shown below. Points like these have the same x-value but opposite y-values and are reflections of each other across the x-axis. If, for any point (x, y) on a graph, the point (x, −y) is also on the graph, then the graph is said to be symmetric with respect to the x-axis. If we fold the graph on the x-axis, the parts above and below the x-axis will coincide.

Consider the points (3, 4) and (−3, 4) that appear on the graph of $y=x^2-5$, as shown below. Points like these have the same y-value but opposite x-values and are reflections of each other across the y-axis. If, for any point (x, y) on a graph, the point (−x, y) is also on the graph, then the graph is said to be symmetric with respect to the y-axis. If we fold the graph on the y-axis, the parts to the left and right of the y-axis will coincide.

Consider the points $\left(-3,\sqrt{7}\right)$ and $\left(3,-\sqrt{7}\right)$ that appear on the graph of $x^2=y^2+2$, as shown below. Note that if we take the opposites of the coordinates of one pair, we get the other pair. If, for any point (x, y) on a graph, the point (−x, −y) is also on the graph, then the graph is said to be symmetric with respect to the origin. Visually, if we rotate the graph $180°$ about the origin, the resulting figure coincides with the original.

Now we relate symmetry to graphs of functions.

An algebraic procedure for determining even functions and odd functions is shown at left. Below we show an even function and an odd function. Many functions are neither even nor odd.