Understanding Symmetry in Graphing Functions

Symmetry is a helpful idea when drawing graphs of functions. It’s especially useful when we talk about even and odd functions. Let’s break these concepts down!

1. Even Functions
Even functions are the ones that look the same on both sides of the y-axis. This means if you pick any point on the graph, like $(x, y)$, you can also find a point $(-x, y)$ on the graph.

A common example of an even function is $f(x) = x^2$.

To show this, if you calculate $f(-x)$, you get $(-x)^2$ which equals $x^2$. This proves it’s an even function!

Try This:
If you plot the points for $x = -2, -1, 0, 1, 2$, you’ll see the points are like a mirror image on either side of the y-axis.

2. Odd Functions
Odd functions are different. They have a kind of twist. For every point $(x, y)$ on the graph, you can also find a point $(-x, -y)$.

A great example of an odd function is $f(x) = x^3$.

If you calculate $f(-x)$, you get $(-x)^3$, which simplifies to $-x^3$. This shows that it’s an odd function!

Try This:
If you plot points for this function, you’ll find that going from one corner of the graph to the opposite corner gives you a matching point.

Using symmetry makes graphing functions easier! It helps you figure out other points on the graph without having to plot every single one.