If your basic function is y= f(x), you need to distinguish between "changes to x" (before applying the function f) and "changes to y" (after applying the function f). Obviously, any changes to y affect the graph vertically: y= f(x)+ a moves the graphupby a, y= f(x)- a moves the graphdown. Multiplying, y= af(x) stretches (if a> 1) or squeezes (if a< 1) the graph vertically.

Changes to x (before applying the function f) are horizontal changes. But when you talk about graphing y= f(x), as the notation indicates, you are talking about values ofy, not x. Changes before applying f, that is, changes to x, are more subtle: Suppose . Then will be equal to 0 when , that is, when x= -a, not a. More generally, if , then when or when .