This section describes the geometric interpretation of what makes a transformation
Here is a video!
In essence, a (function) transformation in this context is the process of stretching, contracting, or flipping the graph. It is important to notice that, unlike the rigid translation case, transformations involve changing the shape of the graph, although it might be more accurate to say it involves exaggerating the shape of the graph.
It can sometimes be difficult to tell whether a function has been compressed in one direction or stretched in another however. Consider the following two graphs;
Is the second graph a vertical stretch, or a horizontal compression? It turns out, it could be either one in this case.
This is why the graphical representation can be difficult to determine exactly using just the geometric perspective. Nonetheless, it is clear that a compression or stretch has occurred by looking at the graph, thus understanding what happened is easier to see in the graph, even if the how is a bit harder to determine.