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### Course: Algebra 2 > Unit 9

Lesson 3: Symmetry of functions- Function symmetry introduction
- Function symmetry introduction
- Even and odd functions: Graphs
- Even and odd functions: Tables
- Even and odd functions: Graphs and tables
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even & odd functions: Equations
- Symmetry of polynomials

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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 $y$ -axis.

**even function**if its graph is symmetric with respect to theFor 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!

#### Check your understanding

### An algebraic definition

Algebraically, a function $f$ is even if $f(-x)=f(x)$ for all possible $x$ values.

For example, for the even function below, notice how the $y$ -axis symmetry ensures that $f(x)=f(-x)$ 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!

#### Check your understanding

### An algebraic definition

Algebraically, a function $f$ is odd if $f(-x)=-f(x)$ for all possible $x$ values.

For example, for the odd function below, notice how the function's symmetry ensures that $f(-x)$ is always the $f(x)$ .

*opposite*of## Reflection question

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