How are transformations and functions related

In general, transformations are used to turn one function into another one. For example if I have a line f (x) = 2x + 3 and I want to turn it into a line that passes through (0,6) and has the same slope, I can write a new function which moves (translates) f (x) vertically 3 units: g (x) = f (x) + 3. Substituting for f (x) will show you what happened in the transformation - g (x) = (2x + 3) + 3 = 2x + 6.