Answer:
A transformation refers to a way in which a function is modified such that the resulting curve differs in some geometric way. For example, scaling and translating are both transformations. Mathematically, a transformation can be described as two functions [tex]d(x)[/tex] and [tex]r(x)[/tex] such that for some function [tex]f(x)[/tex], a transformed version of [tex]f[/tex] is any function [tex]g(x) = r(f(d(x)))[/tex]
A translation specifically refers to some real numbers h and k such that given a linear function [tex]f(x)[/tex], a translation of [tex]f[/tex] is any function [tex]g(x) = f(x - h) + k[/tex]. More generally, a translation is a type of transformation where [tex]d(x) = x - h[/tex] and [tex]r(x) = x + k[/tex].
Consider the function [tex]f(x) = x^2[/tex]. A translation is any function of the form [tex]g(x) = f(x - h) + k = (x - h)^2 + k[/tex]. Now consider how a transformation differs from a translation in terms of scaling [tex]f(x)[/tex], where [tex]d(x) = 2x[/tex] and [tex]r(x) = x[/tex] such that a transformed version of [tex]f[/tex] is some function [tex]h(x) = r(f(d(x))) = 4x^2[/tex]. For all values of h and k, [tex]g(x) \neq h(x)[/tex]
