Polar Functions in a Daily Desmos Challenge

I was intrigued by Desmos when it was introduced in class, so I was interested in spending a little more time with it in my math work.  Desmos is an online graphing calculator with a lot of neat features.  We took a look at dailydesmos.com in class, which is a website that posts Desmos-generated graphs as challenge problems for other Desmos users to try.  On that particular night of class, Daily Desmos #285 was up on the website, and caught my eye as an interesting graph and a fun challenge.

Daily Desmos #285 (Advanced)

By the look of it, I assumed that this was probably a polar function.  In our precalculus class, we go over some of these – I think the ones we do are called cardioids and limaçon curves, and spirals as well – but I had never seen one that looked quite like this.  I played around with some polar functions on Desmos to try to refresh my memory on how they work.  I triedand then and then , the last of which produces a circle centered at the origin, with radius 1.  I decided this circle equation was the one to begin manipulating, since it does not pass through the origin, or pole.  Having each of those terms squared seemed like it would be a key feature for this rule no matter how it turned out, since 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.

The 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: 

Here’s the result:

My Attempt

The Daily Desmos Challenge

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.