Translating a Graph

A translation moves every point of a graph the same distance in the same direction, so the shape, size, and orientation never change — only the location does. There are two separate moves to track. Outside the function, f(x) + k shifts the graph up when k is positive and down when k is negative, because every output value just gets k added to it. That one behaves exactly the way it looks.

Inside the function, f(x + h) shifts the graph horizontally, but the direction is the opposite of what the sign suggests. Think about why: g(x) = f(x + h) reaches the same output that f reaches, just h units earlier, at a smaller x-value. To get the output f(5) out of g, you need x + h = 5, so x = 5 - h — a smaller input than 5. That means the whole graph slides left when h is positive, and right when h is negative. The rule of thumb: horizontal shifts go opposite the sign you see written in the argument.

This topic is the algebra behind every family of functions you'll graph from here on. Once you can read (x - p) as a shift right by p and +q as a shift up by q, you can sketch transformed parabolas, absolute value graphs, square roots, and later exponential and trig graphs without plotting a single extra point — you just move the parent graph's key features.