### Video Transcript

If a function is even, then its curve is symmetric about what.

Remember, a function π of π₯ is said to be even if π of negative π₯ is equal to π of π₯ for every π₯ in that functionβs domain. So thinking about its curve, letβs consider the graph of some function π¦ equals π of π₯. Suppose we know that the domain of this function definitely includes values of π₯ in the closed interval negative two to two. We could look to plot this on a graph by substituting each value of π₯ into the expression for the function. So when π₯ is equal to negative two, π¦ is equal to π of negative two. When π₯ is negative one, π¦ is π of negative one. When π₯ is zero, π¦ is π of zero, and so on.

But remember, this is an even function. So for all values of π₯ in the functionβs domain, π of negative π₯ is equal to π of π₯. So when π₯ is equal to negative two, π¦ is equal to π of two. And when π₯ is equal to negative one, π¦ is equal to π of one. Notice this means that when π₯ is equal to negative two and π₯ is equal to two, the value of the function is exactly the same. Similarly, when π₯ is equal to negative one and π₯ is equal to one, we get π of one. This means, in fact, that our function is entirely symmetrical about the line π₯ equals zero. But of course the line π₯ equals zero can also be called the π¦-axis. So we say that if a function is even, its curve is symmetric about the π¦-axis.