This covers doing transformations and translations at the same time. In particular we discuss how to determine what order to do the translations/transformations in.

Doing Translations and Transformations at the Same Time.

The general rule of thumb is that the order of the translations/transformations follows the order of operations in the analytic form; with the important rule that everything involving is backward.

As an example, lets suppose we have a function and we define a manipulation of as . Moreover the following is the graph for ;

Let’s consider the manipulation of which we called . There is a lot to unpack in the definition of and we need to make sure that we apply the transformations and translations in the right order, but remember that just means we need to apply them in the order of operations. The easiest way to do this is to pretend to plug in an value and determine what the order is that each computation is applied.

Calculating the Stuff:

If we were to plug in a number for , the first thing we would do is add 1. This appears to be a shift to the right one, but remember everything about is backwards, so it’s really a shift to the left one. Next we would divide the -value by 2, which (again because everything about is backwards) means we stretch the value to twice it’s original width. Thus in total it looks like we would shift left 1 and then make the graph twice as wide as it was originally... but again everything about is backwards, including the order we apply the changes in, so we actually want to make everything twice as wide and then shift everything left one.

Calculating the Stuff:

Now we tackle the values. To do this, we pretend that we have calculated some value for the portion and replace it with a number. In our case then we would pretend to replace with a number, and then see what computations apply from there. Thus the first change (once we have replaced the ‘’ part with a number) would be to multiply the function result by 2, which means stretching the -value to twice it’s height. Finally, adding one to the overall function means we will shift everything up one. Since the -values work the way we would normally expect (unlike the stuff), this means that in total we would make everything twice as tall and then move everything up one.

It may help to do a concrete computation to see that our translation/transformations occur in this order. Below we compute and make note of the translations/transformations as we go.

All this yields the following graph;

Notice that we can use our knowledge of how the transformations/translations work (and what order to do them) to reverse this process. Let’s say we want to take the original and we want to apply the following changes in the following order:

Move it right 3.
Constrict it by a factor of 4 (aka to of it’s normal width).
Move it up 1.
Stretch vertically by a factor of 2.

We can write the analytic form of this description by making sure that the order of operations agrees with the order the instructions were given (reversing the effects on as usual).

To see the effects of changing the various parameters try manipulating the values for , , , and in the following interactive graph. Pay special attention to when these values are positive, negative, and bigger or smaller than 1.

You can also watch a short video giving an in-depth walkthrough example on how to graph the result of translations/transformations of a graph below.

1 : In order to graph multiple translations and transformations at the same time, you should...
Follow order of operations. Always graph -stuff first. Always graph -stuff first. Follow the order of operations in reverse order. Follow the order of operations in reverse order for -stuff (Because everything about is backwards!) and the order of operations in correct order for -stuff.