fmap, fold, traverse, join

I guess I'm lacking some background to understand this.

So, I recognize fold, from the above operations ('reduce' in python), what makes it a "functional transformation"?

I'm googling function/functional transformations, and the results refer to translating, flipping or scaling a function, like transforming f(x) to f(x + 2) or f(x) + 2. It doesn't seem like it has anything to do with what you guys are referring to, but correct me if I'm wrong.

function as in functional programmi ng. https://youtu.be/2MXyNS33t2k is an indepth video on how c++ and haskell sequence transformations work and contrast.