When we draw both even and odd functions on a graph, we can see some cool patterns that show how they work differently.

Even Functions:

These functions are balanced on both sides of the y-axis (the vertical line in the middle of the graph).

For example, take the function $f(x) = x^2$. This is an even function because if you plug in a negative number, it gives you the same answer as plugging in a positive number.

So, $f(-x) = f(x)$ for every value of $x$.

When you graph it, it looks like this:

$$\text{Graph of } y = x^2$$

Think of it like a U-shape that looks the same on both sides of the y-axis.

Odd Functions:

Now, odd functions are different. They are balanced around the origin, which is the point where the x-axis and y-axis cross.

A good example of an odd function is $g(x) = x^3$.

Here, when you use a negative number, it gives you the opposite answer compared to a positive number.

So, $g(-x) = -g(x)$.

The graph looks like this:

$$\text{Graph of } y = x^3$$

It's like an arrow that points away from the center of the graph.

Combining the Graphs:

When you put both $y = x^2$ and $y = x^3$ on the same graph, you'll see some interesting things:

• The even function ($y = x^2$) stays perfectly symmetric around the y-axis.

• The odd function ($y = x^3$) has a different kind of balance that goes through the origin.

This mixing of even and odd functions shows us how beautiful symmetry can be in math!