Symmetry

In math, shape symmetry refers to a shape that can still look the same after undergoing some sort of transformation.

In functions there are even and odd functions

Even Function

A function is even if it satisfies the condition \(f(-x) = f(x)\). All even functions have line symmetry about the y-axis which is to say it looks the same when reflected along the y-axis.

Odd Function

A function is odd if it satisfies the condition \(f(-x) = -f(x)\) which is to say \(f(x) + f(-x) = 0\). All odd functions have point symmetry meaning it relfects about the y = x line so that \((a, b) \to (b, a)\). As point symmetry is a special case of rotational symmetry, odd functions also have rotational symmetry about the origin after 180°.

Identification

If the exponent on all terms of a polynomial function is even, the function is even. If all exponents on all terms of a polynomial function is odd, the function is odd. When a function satisfies neither conditions, the function has no symmetry.

Point and Line Symmetry

It is possible for a function to have line symmetry without being an even function or to have point symmetry without being an odd function. These cases would be translated functions, eg. \(x^2\) (red) is even with line symmetry but \((x - 2)^2\) (blue) is not even yet still has line symmetry (reflects on the line x = 2).