Video Transcript
If a function is even, then its curve is symmetric about what.
Remember, a function 𝑓 of 𝑥 is said to be even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥 for every 𝑥 in that function’s domain. So thinking about its curve, let’s consider the graph of some function 𝑦 equals 𝑓 of 𝑥. Suppose we know that the domain of this function definitely includes values of 𝑥 in the closed interval negative two to two. We could look to plot this on a graph by substituting each value of 𝑥 into the expression for the function. So when 𝑥 is equal to negative two, 𝑦 is equal to 𝑓 of negative two. When 𝑥 is negative one, 𝑦 is 𝑓 of negative one. When 𝑥 is zero, 𝑦 is 𝑓 of zero, and so on.
But remember, this is an even function. So for all values of 𝑥 in the function’s domain, 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. So when 𝑥 is equal to negative two, 𝑦 is equal to 𝑓 of two. And when 𝑥 is equal to negative one, 𝑦 is equal to 𝑓 of one. Notice this means that when 𝑥 is equal to negative two and 𝑥 is equal to two, the value of the function is exactly the same. Similarly, when 𝑥 is equal to negative one and 𝑥 is equal to one, we get 𝑓 of one. This means, in fact, that our function is entirely symmetrical about the line 𝑥 equals zero. But of course the line 𝑥 equals zero can also be called the 𝑦-axis. So we say that if a function is even, its curve is symmetric about the 𝑦-axis.
