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# Function symmetry introduction

Learn what even and odd functions are, and how to recognize them in graphs.

#### What you will learn in this lesson

A shape has reflective symmetry if it remains unchanged after a reflection across a line.
For example, the pentagon above has reflective symmetry.
Notice how line $l$ is a line of symmetry, and that the shape is a mirror image of itself across this line.
This idea of reflective symmetry can be applied to the shapes of graphs. Let's take a look.

## Even functions

A function is said to be an even function if its graph is symmetric with respect to the $y$-axis.
For example, the function $f$ graphed below is an even function.
Verify this for yourself by dragging the point on the $x$-axis from right to left. Notice that the graph remains unchanged after a reflection across the $y$-axis!

1) Which of the graphs represent even functions?

### An algebraic definition

Algebraically, a function $f$ is even if $f\left(-x\right)=f\left(x\right)$ for all possible $x$ values.
For example, for the even function below, notice how the $y$-axis symmetry ensures that $f\left(x\right)=f\left(-x\right)$ for all $x$.

## Odd functions

A function is said to be an odd function if its graph is symmetric with respect to the origin.
Visually, this means that you can rotate the figure ${180}^{\circ }$ about the origin, and it remains unchanged.
Another way to visualize origin symmetry is to imagine a reflection about the $x$-axis, followed by a reflection across the $y$-axis. If this leaves the graph of the function unchanged, the graph is symmetric with respect to the origin.
For example, the function $g$ graphed below is an odd function.
Verify this for yourself by dragging the point on the $y$-axis from top to bottom (to reflect the function over the $x$-axis), and the point on the $x$-axis from right to left (to reflect the function over the $y$-axis). Notice that this is the original function!

Which of the graphs represent odd functions?
Algebraically, a function $f$ is odd if $f\left(-x\right)=-f\left(x\right)$ for all possible $x$ values.
For example, for the odd function below, notice how the function's symmetry ensures that $f\left(-x\right)$ is always the opposite of $f\left(x\right)$.