Summary: Symmetry of a Function

Key Concepts

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

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

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