Symmetry

Symmetry is simply when one half of a graph is a mirror image of the other, the way your left hand mirrors your right. The formulas up top name the two kinds you'll see most: $f(-x) = f(x)$ (going the same distance left or right gives the same height) and $f(-x) = -f(x)$ (going left gives the opposite height). The names "even" and "odd" come from powers of $x$: even powers like $x^2$ and $x^4$ have the first kind, odd powers like $x^3$ have the second. It's useful because it lets you draw half a graph and get the other half free.

Why the tests work. Reflecting across the y-axis sends each point $(x, y)$ to $(-x, y)$. If the graph looks the same afterwards, every $(-x, y)$ must also satisfy the equation, and that's exactly what substituting $-x$ checks. The other tests work the same way with their own reflections.

Three tests:

• y-axis: replacing $x$ with $-x$ gives the same equation. Functions with this symmetry are called even, like $x^2$, $x^4 - 3x^2$ and $|x|$.
• Origin: replacing both $x$ with $-x$ and $y$ with $-y$ gives the same equation. Functions with this symmetry are odd, like $x^3$, $\frac1x$ and $x^3 - 3x$. Symmetry about the origin means a half turn leaves the graph unchanged.
• x-axis: replacing $y$ with $-y$ gives the same equation, as for $x = y^2$. Such graphs (other than $y = 0$) fail the vertical line test, so they aren't functions: each $x$ has a $y$ and a $-y$.

Symmetry halves the work of graphing: plot $x \ge 0$, then reflect.

A common mistake. It's tempting to think every function is either even or odd. Most are neither. $x^2 + x$ mixes an even power with an odd one, and $f(-x) = x^2 - x$ matches neither $f(x)$ nor $-f(x)$.

Physicists use symmetry to cut calculations in half, and engineers balance rotating parts by making them symmetric. Fourier series of even signals contain only cosines.