Question Video: Understanding Parity of Functions Mathematics

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

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Video Transcript

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

Remember, a function 𝑓 of π‘₯ is said to be even if 𝑓 of negative π‘₯ is equal to 𝑓 of π‘₯ for every π‘₯ in that function’s domain. So thinking about its curve, let’s consider the graph of some function 𝑦 equals 𝑓 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, 𝑦 is equal to 𝑓 of negative two. When π‘₯ is negative one, 𝑦 is 𝑓 of negative one. When π‘₯ is zero, 𝑦 is 𝑓 of zero, and so on.

But remember, this is an even function. So for all values of π‘₯ in the function’s domain, 𝑓 of negative π‘₯ is equal to 𝑓 of π‘₯. So when π‘₯ is equal to negative two, 𝑦 is equal to 𝑓 of two. And when π‘₯ is equal to negative one, 𝑦 is equal to 𝑓 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 𝑓 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 𝑦-axis. So we say that if a function is even, its curve is symmetric about the 𝑦-axis.

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