All functions transformations
This is not a post just about quadratics or parabolas, but I am placing it in this context, as I want to explain my thinking on this topic of teaching.
In talking to teachers, I have been told a variety of techniques to teach the transformations of different function families. The one thing that I realized is that there is no one way that teachers use to teach this idea. There is, however, and efficiency to be found in the way I approach this topic.
On day 1 of my classroom in Algebra 2, this is what the learners saw on the board, and we used this every day in class.
Sometime in around day 3 or 4 of the school year we would do a “what do you notice” and “what do you wonder” exercise with this, and we would build the ideas of similarity of a and (h,k) actions, and inverses of functions.
I leaned heavily into the similarity of how a, h, k, (and b, but that is typically outside of the Alg 2 curriculum and into the Trig/Precalc curriculum). I had a rule for myself in my teaching, which was: “Teach something once, use it every day.”
So for translations, I would teach it once. I used Desmos to assist with this, and we used a type of this calculator file. It has ALL the things. I would start it quite simply with this:
and work up to all the functions.
This is messy, ugly, and not clear from this image, but if you use the sliders, you will see that all the functions will move together. Rational functions, transcendental functions (sine, cosine, tangent), exponential functions, logarithms, as well as all the polynomial functions all work the same way.
Teach it once, use it every day.
This is essential to my teaching philosophy and approach, but it also saves so much time. We can talk about “what do you remember?” and challenge learners to dig into their notes and memory to reconnect knowledge. “Make a prediction as to what happens when I do this, based upon what you learned earlier,” and learners will be able to use what they learned, instead of learning something new.
The translation form of all the functions isn’t always useful, but it serves the purpose of tying all the different functions we talk about together, so that it is clear that as we are learning one idea or topic, we are not forgetting the other ideas we learned.
Everything is connected. We just have to spend a little bit of time thinking about how the topics are connected, and build the connections with the learners.