Question

What is the difference between f(x) and y
(Quadratic Functions)

Answer 1

f(x) is not necessarily for graphs as it can be used with composites eg fg(x) but can be shown graphically but y= is purely for a graph

Answer 2

There is no difference. y and f(x) is the exact same thing. They are both the result after you apply a function to x. That is what f(x) stands for.