*Daily Desmos #285*was up on the website, and caught my eye as an interesting graph and a fun challenge.

Daily Desmos #285 (Advanced) |

*r*looks to stay positive throughout the graph. If

*r*

*did*ever change from positive to negative, the graph would have pass through the pole.

*Daily Desmos*graph begins at about. Since it goes out about that far, and just a little farther (maybe 2 units?), when , I figured I would not add a constant to my rule. I thought that would make the graph lop-sided, and while it's not actually symmetric, it's close (can you be

*close*to symmetric?). Instead, I decided to try a multiplier of 17 on the term of my rule. Made a similar decision for the term of my rule. Since the term would drop out to 0 on the

*y-*axis whenever is a multiple of , the would be the main player at those points. The first of these points on

*Daily Desmos #285*is about , but I decided that the multiplier on should be just a little bit lower than 4, due to the spiral effect going on in this graph.

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.