Introduction

Let’s consider the two quadratic functions and , defined for all real values of . We know what their graphs look like:

The graphs are very similar, down to the horizontal “width”. In fact, drawing them together, we may see that they only differ by a horizontal translation:

Algebraically, one can see that this happens because This hints at the following general fact: doing horizontal shifts to the graph of a function amounts to replacing with “” inside . In this unit, we’ll understand in more detail how to work with this, and also how to deal with vertical shifts, as opposed to horizontal shifts. Since vertical shifts are much easier to understand, that’s where we’ll begin.

Shifting a function vertically

Let’s consider a very simple situation, where we have two functions and . Graphing them, in order, we have that

Clearly, and are directly related via , and seeing their graph together, we have that:

In other words, the graph of was obtained from the graph of by shifting it up exactly by units. This is a very general phenomenon, that happens for any functions who differ by a constant.

In the above setting, it is useful to call the parent function.

Shifting a function horizontally

Consider again the example given in the introduction, where we have and . The first thing we would like to address is a source of frequent confusion when first learning this topic. Namely, we have replaced with in the formula for , but the graph of the modified function ended up shifted to the right, even though one might expect the shift to have happened to the left, due to the negative sign in the factor!

Here is one safe way to think about it: imagine that you are standing on the -axis and, say, at the origin of the cartesian plane, but that the graph of is already drawn. Replacing with does move the -axis to the left. But you, the observer, standing on the -axis, sees the graph move to the right!

Alternatively, compare this with what happened with vertical shifts, but switching the roles of the -axis and -axis. More precisely, start with the graph of , then rotate it by clockwise (this switches the axes). Replacing with now brings the graph down by unit. Finally, rotate everything back by counterclockwise (this undoes the switching of the axes). The resulting graph is obtained from the original one by shifting it to the right, not left.

As before we’ll continue to call the parent function, whose graph will be drawn in blue, while the graphs of the “child” functions will be indicated in red.