We’re in the “World of Functions” unit in IMP4. This may be my favorite unit in the whole curriculum.
Instead of a separate sections on linear, quadratic, radical, rational, exponential, …, we’re studying functions. Delving deeper into the relationship between the table, the graph, and the equation. Working on translating from one form to another to another. We’re doing application problems too. Lots of them.
Of course, they’ve used to working from multiple representations. They’ve been doing it for almost four years now.
This week we began algebra of functions.
I gave my students the following graph and asked them to draw 
I didn’t tell them what to do. They didn’t need me to. They figured it out. They helped each other. They debated. They used numerical examples. They figured it out.2
Did they all do it perfectly on the first try? No. After working on it in their groups? Yep, most of them. Did they all try? Yep. Not one person sat there waiting for me to tell them what to do. No one said “I can’t.” A few are still working on solidifying their understanding of this. I’m okay with that. Not everyone gets it at the same time.
I’m blown away on a daily basis by my seniors in IMP4. Both by the mathematical understanding they’re demonstrating and they willingness with which they tackle new problems. I’m not sure which impresses me more.
1While not exactly a “What Can You Do with This” image, I still like it.
2We also discussed when f/g would be undefined and why f*g needs a scale. Any other ideas as to where we could have taken this?
Continuities 
You have attached a single, compelling question (“find (f + g)(x)”) to a clear, unaffected image, one which clears a trail to more rigorous questioning (“what about (f – g)(x)”, “where are the intercepts of (f + g),” etc.
I guess what I’m trying to promote with WCYDWT? is an aesthetic which can be applied to heady conceptual stuff like projectile motion and skill-driven procedure equally.
Fun stuff, thanks.
I think it would be worthwhile to look at some familiar functions that combine in intuitively unexpected ways. A simple example, such as f(x) = x^2 and g(x) = 5-x^2, would lead to some interesting results. You could also try a square root function and a quadratic, a cos function and a linear function, etc.
I wish all students had the opportunity to explore concepts this deeply. The students I see at the center have only learned how to combine functions algebraically. I’m not sure they would know what to do with a problem like this one. There’s only one way to find out, though. Thanks for the inspiration.
Dan Don’t get me wrong, I love your WCYDWT series. And I wholeheartedly agree with your statements that most textbooks take away any thought required on the part of the reader.
I do think though, that working from this image requires conceptual understanding too, no?
Colleen We’ve been combining other functions too – both from tables and from equations. I think the multiple representations are important. Some students “get it” more from the graph or table. They can then transfer this knowledge too the other forms. If you do end up doing this (or something like it), I’d love to hear how it goes.
Right. Conceptual knowledge isn’t the same thing as application problems. I dig Colleen’s approach also.
Have you covered the composition of functions? It might be interesting (if you whack a scale on that graph) to use the graph to determine f(g(x)), g(f(x)) and maybe investigate what it means when the two are equal.
Since I have just started functions and I love your visual approach, I’ve tried to create a similar looking visual. My results are nowhere near as slick as yours. I’m curious to know what you used to create it?
Yep, we just started composite functionsClint. We’re starting with situations (word problems) and then going to tables/graphs/equations. Which is leading us to f(g(x))=g(f(x)) and inverse functions. 🙂
I created this with Graph Sketcher.
For classes with many advanced students, you may want to try some modifications. Depending on where you place the tick marks for 1 and -1 on the y-axis, you can get some interesting results for f(x)*g(x).
Better yet is to use the same graph twice with difference placements for the tick marks for 1 and -1 and have them sketch f(x)*g(x). It’s good practice for students to think about fractions.
This skill comes in handy when working with damped trigonometric graphs.
For the function addition, the skill they use here also ties in with slant asymptotes. Try f(x)=1/x and g(x)=x. If g(x) is too steep, try g(x)=(1/2)x or g(x)=(1/4)x. Since it doesn’t seem that you’re giving them the function equations, the exact slope won’t matter much as long as they’re somewhat shallow so the resulting graph fits in the “viewport” of the graph.
I had fun playing with Gelfand’s Functions and Graphs. It might inform your questions.
There are ways to play with f without g that are challenging (not simple transformations), ex |f(x)| or 1/f(x).
Jonathan