What's the Effect of a Vertical or Horizontal Translation on a Graph?

A vertical translation moves every point on a graph up or down, while a horizontal translation moves every point right or left. The graph’s shape and orientation remain unchanged.

This is part of IB Mathematics: Analysis and Approaches, SL 2.11: Transformation of functions.

How Translations Work

Starting with $y=f(x)$, a vertical translation changes the function’s output, whereas a horizontal translation changes its input.

Equivalently, if an original point ((p,q)) lies on $y=f(x)$, use the point mappings below rather than substituting several trial values. A translation is a rigid transformation, so distances between corresponding points are preserved.

The horizontal sign appears reversed because replacing $x$ with (x-a) delays each original output until the input is $a$ units larger. For example, the vertex of $y=x^2$ is $(0,0)$. In , the graph moves right by and up by , so the vertex becomes .

For another example, if $f(4)=1$, then the graph of $g(x)=f(x+3)-5$ contains $$. The inside shifts the point left by , and the outside shifts it down by . This point-mapping method works even when the formula for is not given.

A common misconception is that $f(x-2)$ moves the graph left because of the minus sign. This is incorrect: changes inside the function have the opposite horizontal direction, while changes outside have the expected vertical direction.

IB Exam Technique

For a sketch, preserve the graph’s shape and translate all key features, including intercepts, vertices, endpoints and asymptotes. On Paper 1, show the coordinate mapping or identify the translation explicitly; on Paper 2, technology may confirm the graph, but you should still state the transformation using correct notation. If asked to “describe” the transformation, give both its direction and magnitude; “translate right” alone is incomplete.