Introduction

In this section, we will review our earlier discussion about function transformations. In addition, we’ll explore what happens when multiple function transformations occur.

Review of Single Transformations

The following table gives the formulas and descriptions of all the function transformations we learned about. If is the parent function, then the formula in the left column gives the function that corresponds to the transformation of the graph given in the middle column. The right column gives the new location of the point on the graph of .

Putting it Together

Transformations may be performed one after another. If the transformations include stretches, shrinks, or reflections, the order in which the transformations are performed may make a difference. In those cases, be sure to pay particular attention to the order.

The previous example shows how to construct the formula of a function given a sequence of transformations. One might wonder how to find the transformations applied to a parent function’s graph given a complicated formula. That is, if we know a formula , can we reconstruct the sequence of transformations? There are two parts to this: finding the transformations and finding the order of those transformations. The following example gives some of the reasoning behind the process.

Let’s see how this fits into a more complicated example.

Given a function , note that we can apply the same reasoning to to find the order of the transformations applied to the graph of :

(a)
horizontal shift,
(b)
horizontal stretch or compression,
(c)
vertical stretch or compression, and
(d)
vertical shift.

One might wonder where reflections fit into all this. Let’s see with another example.

We can summarize all this information as follows. Given a function , the graph of can be found using the following transformations in order:

(a)
horizontal shifts given by ,
(b)
horizontal stretches or compressions given by ,
(c)
vertical stretches or compressions given by ,
(d)
reflections given by the sign of and , and
(e)
vertical shifts given by .

Note that this is not the only order you can use to graph transformations of functions. This is just one order that works every time, provided you start with a function of the form .

Below is a link to a Desmos graph containing for a function where you can adjust the values of , , , and to see how they affect the graph. The function in the link below is the formula for a half-circle, but don’t worry too much about where it came from.