|Daily Desmos #285 (Advanced)|
From teaching precalc, I knew that a function of the form would produce a spiral, so I knew I would need a term like this as well. Looking at the successive passes of the curve on one axis at a time gave me the idea that this term should be increasing r values by a little bit less than 5 for each full rotation. I decided to try 4.6 as my first estimate. To figure out what k should be for this term, I solvedfor k to get .
With these guesses and estimates, I put together my first attempt at really matching the graph:
I was pretty satisfied that I had at least discovered the right form for my polar function. All it would take to get things to match more accurately would be to adjust my constants a little bit.
In retrospect, I didn’t take advantage of Desmos as an exploratory tool very well. If you type in a function in Desmos with letters in place of numbers, it can automatically create sliders for the would-be constants. It might have made more sense to set up a rule with sliders for the constants, once I had decided on a form for the rule. That is, I would have entered into Desmos. That would have made it easier to investigate different values for each constant. At the same time, it was good to do some analysis of the points and make educated guesses.
This was a fun way for me to review some polar functions concepts, but it would also be a great activity for my precalculus students in the future. Providing students with laptops, or going to a computer lab, and then presenting the with graphs like this would be a great way for them to test and expand their understanding at the close of a polar functions unit. Something similar could certainly be done in other classes and units as well, with other types of functions. If computers aren't readily available, this could be done with graphing calculators as well. Taking the time to do this activity deepened and reinforced my understanding of polar functions and how they work, and got me thinking about the features of several types of polar graphs. I think it could do the same for my students.