Sketching Graphs - Transformations

Stage: 5 Challenge Level:
One useful and important technique in graph sketching is to consider transformation of functions. If you know the basic shape of a function, you can use that to work out what translations, reflections or stretches of that function will look like.

Watch the video below for an explanation of how the graph $y=x^2$ can be transformed by simple translations.

The video shows what happens when we add something to a function or variable. But what if we multiply?
Can you think of a clear way to explain how the graphs $y=f(2x)$ and $y=2f(x)$ are related to the graph $y=f(x)$?

Now, using some graphing software such as GeoGebra, or Desmos, have a go at some of the following challenges:

Parabolic Patterns
Parabolas Again
Cubics
Tangled Trig Graphs