Summary: Symmetry of a Function

Key Concepts

The point that is symmetric to [latex](a, b)[/latex] about the [latex]x[/latex]-axis is [latex](a, -b)[/latex].
The point that is symmetric to [latex](a, b)[/latex] about the [latex]y[/latex]-axis is [latex](-a, b)[/latex].
The point that is symmetric to [latex](a, b)[/latex] about the origin is [latex](-a, -b)[/latex].
A function is symmetric about the y-axis if [latex](-x, y)[/latex] is on the graph of the function whenever [latex](x, y)[/latex] is on the graph.
A function is symmetric about the origin if [latex](-x, -y)[/latex] is on the graph of the function whenever [latex](x, y)[/latex] is on the graph.
If [latex]f(-x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an even function. An even function is symmetric about the [latex]y[/latex]-axis.
If [latex]f(-x)=-f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] is an odd function. An odd function is symmetric about the origin.
To test whether an equation with two variables is symmetric about the [latex]x[/latex]-axis, substitute [latex]-y[/latex] for [latex]y[/latex].
To test whether an equation with two variables is symmetric about the [latex]y[/latex]-axis, substitute [latex]-x[/latex] for [latex]x[/latex].
To test whether an equation with two variables is symmetric about the origin, substitute [latex]-x[/latex] for [latex]x[/latex] and [latex]-y[/latex] for [latex]y[/latex].

even function: a function whose graph is unchanged by horizontal reflection, [latex]f\left(x\right)=f\left(-x\right)[/latex], and is symmetric about the [latex]y\text{-}[/latex] axis

odd function: a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\left(x\right)=-f\left(-x\right)[/latex], and is symmetric about the origin