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Translations of a graph

The graph of a real function f, the set of points (x, f (x)) {\displaystyle (x,f(x))}, is often pictured in the real coordinate plane with x as the horizontal coordinate and y = f (x) {\displaystyle y=f(x)} as the vertical coordinate.

Starting from the graph of f, a horizontal translation means composing f with a function x ↦ x − a {\displaystyle x\mapsto x-a}, for some constant number a, resulting in a graph consisting of points (x, f (x − a)) {\displaystyle (x,f(x-a))}. Each point (x, y) {\displaystyle (x,y)} of the original graph corresponds to the point (x + a, y) {\displaystyle (x+a,y)} in the new graph, which pictorially results in a horizontal shift.

A vertical translation means composing the function y ↦ y + b {\displaystyle y\mapsto y+b} with f, for some constant b, resulting in a graph consisting of the points (x, f (x) + b) {\displaystyle {\bigl (}x,f(x)+b{\bigr)}}. Each point (x, y) {\displaystyle (x,y)} of the original graph corresponds to the point (x, y + b) {\displaystyle (x,y+b)} in the new graph, which pictorially results in a vertical shift.

For example, taking the quadratic function y = x 2 {\displaystyle y=x^{2}}, whose graph is a parabola with vertex at (0, 0) {\displaystyle (0,0)}, a horizontal translation 5 units to the right would be the new function y = (x − 5) 2 = x 2 − 10 x + 25 {\displaystyle y=(x-5)^{2}=x^{2}-10x+25} whose vertex has coordinates (5, 0) {\displaystyle (5,0)}. A vertical translation 3 units upward would be the new function y = x 2 + 3 {\displaystyle y=x^{2}+3} whose vertex has coordinates (0, 3) {\displaystyle (0,3)}.

The antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other.