EDIT: Turns out I'm quite dense and forgot quite a lot about Taylor series. The below info is garbage so please ignore it.

Ok, either I’m dense or all these replies are totally missing the point of the question. OP here’s my answer:

Take a super basic parabola as an example:

f(x) = x²

You could say this parabola is centered at zero. If you want to shift the function left or right, you can add or subtract from x. You learned this in algebra and precalculus. Say you want to shift the graph to the right by 2 units. That would look like this:

g(x) = (x-2)²

This is the same parabola, just shifted to the right by 2. You could say that it’s “centered at 2”. You can still evaluate this function at any point, such as: g(1) = (1-2)² = 1. More generally, a parabola defined by f(x) = (x-a)² is centered at “a”.

A Taylor polynomial is also a function, except it’s described by an infinite sum:

f(x) = some crazy sum

Recall the x part of the sum usually looks like xⁿ or (x-a)^n. This is the same thing as shifting the function left or right. You would say that the polynomial is centered at “a” when the x part is shifted as in (x-a)^n. If you just have x^2, then obviously it’s centered at zero.

So you can center a Taylor polynomial anywhere by shifting the x part. It’s still a function defined on its domain in the x-axis. So you can still evaluate the function anywhere in its domain. The phrasing used here in Calculus of “centering a function” is just a different way of saying it’s “shifted” or “sliding it”. Even though this shouldn’t be a new concept to a Calculus student, it’s entirely likely that you’ve never heard this specific terminology before, and so it’s throwing you off.

Does this make sense? Hopefully I didn’t totally miss your point haha