One of my favorite topics to teach is transformation of functions. The reason I like it is the same reason a 3-year-old likes pushing buttons and turning knobs: when you make a change you can see what happens–immediately.
I have labored to make the development of these ideas seamless in my classes. I think I found a really good way to teach it.
I often do a discovery exercise using all different types of functions (square-root, quadratic, cubic, etc.) to see what each parameter in an equation does to the graph of the equation. However, this time I opted for a piecewise function so that I could really make use of the function notation.
The point that I really wanted to emphasize is to answer the question: Why does a horizontal transformation behave “opposite” how I would expect? Why does f(x+2) shift the graph of f(x) to the left and not the right?
I am attaching a copy of the investigation worksheet. Also, here’s the link to the desmos graph that I created for my students to tinker with.
Here’s the questions that I had them answer as they explored:
Word Document: desmos-exploration-transforming-a-piecewise-function
Moser Math Blog 