4.7 Transformations of functions
4.7.1 Y-axis transformations
These are the easier case (at least in my view) to think about. When transforming a function in the y-axis, there are two key transformations to be aware of - stretching and translating.
To translate a function in the y-axis we can just add something to it, e.g. to shift the graph of three units up, define a variable, e.g. - the 2D graph of this function will then be shifted three units above. This is illustrated on the graph below:
4.7.2 X-axis transformations
To transform a function in the X-axis, we just evaluate , where is a function which maps values from the - plane (i.e. the usual set of axis we plot things on) to one in the - plane (i.e. like the usual set of axis we plot things on, except that wherever we had (where stands for any number) we now want ).
This deserves a bit of explanation. Let’s imagine that . If we plot against , we might get something like this (for this specific )
We don’t want a graph of against , though! We want one of against . To do this, we need to work out how to write in terms of , and then work out where every point on the -axis should be on the -axis.
As if we add two to each side, we obtain that . This means that if we shift every point on the -axis two to the right then we would have the X-axis!
Thus, the graph of and looks like
We can transform the X-axis in many ways, another one is stretching the graph. For example, if we set , then to work out where every point on the -axis should be on the X-axis, we first rearrange , obtaining that
and thus we stretch (not, as commonly misconceived, squish) the graph. I try to visualise it as the graph stretching as the infinite number of points on the axis are doubled (moved twice as far away as they once were).