Problem: Look at some interesting graphs in polar form.
In this write-up, I will use x instead of theta for convenience, and in all cases x is ranging from 0 to 2pi. The following graph represents the function r = .1x in the polar system.
This provides me a chance to leap into an interesting discussion about functions in the polar plane. Normally, if you had a function like y=.1x in the cartesian plane, you would state that it was a linear function and that you would expect the graph to look like a line. In all cases, you would expect a student to state that the graph represented a function via the vertical line test. However, in the polar plane, this graph is a spiral. Thus, many students would state that this violates the vertical line test and is thus not a function. Teachers must be careful to have their students understand what the vertical line test is actually telling them, and that the test is not really helpful in the polar system. One would need to create more of a "Circle" line test to show there are not multiple values for r for each input x.
Now that I've went off on that tangent, let's look at another graph:
This is the graph of r = 1(sinx-cosx). One might be astounded by this graph at first because it does not hold the properties one would expect a function with trig functions in it to hold in the polar plane. It looks like a "line". Also, it seems to "end" at certain points (that's why I zoomed out). However, I think this is more because of the limitations of the graphing utility and not the function itself, since the output for sine and cosine do get very close at some places.
Let's investigate further with the graph of r = sinx - cosx:
As you can see, this has a more circular look to it, and would be what one would expect out of a trig function in the polar plane. It seems that putting the expressing in the denominator, with 1 as the numerator, changes the function completely, making r tend to large numbers instead of a bounded set.
Now, let's test with even more graphs. The following is a graph of 1/(sin2x-cos2x)
Here, the 2 in front of the x seemed to change everything! Now, we have multiple different parts to the graph. There are 4 lines that appear to make a "tic-tac-toe" type shape, with 4 parabolic looking parts. Rather than just tending to infinity in a "linear" looking manner, r seems to tend to infinity in many different directions.
What happens if we increase the parameter in front of x to 3?
We seem to have even more odd behavior here. Instead of 4 lines, we now have 3 lines that form a triangle around the origin, and there are 3 parabolic looking structures each between two of the lines. Once again r tends to infinity in multiple directions.
Will this trend continue for the parameter being 4x, with r = 1/(sin4x-cos4x)?
Well now there seems to be a pattern arising. Not seen here, I also looked at the case where r=1/(sin5x-cos5x) and it seems like when the parameter is even, the number of asymptote lines is double the parameter, and when the parameter is odd, the number of asymptote lines is the same. This could be due to the periodic nature of trig functions, and the fact that sine is an odd function and cosine is an even function.
Animating for n from 0 to 10, when r = 1/(sin(nx)-cos(nx)), we have:
If nothing else, this is pretty cool for students to look at and explore the effects of parameters on polar functions. As the parameters increase, the number of "parts of the graph" increase as discussed above with even and odd values.
What about some other neat graphs? The following graph is produced when the parameters are made smaller instead of larger. In the below image, we have r = 1/(sin(x/2)-cos(x/2))
Investigating further, we have the graph r = 1/(sin(x/3)-cos(x/3)) below:
It seems that, even though the graph spirals around in a similar fashion in both cases, there reaches a point where the output tends towards infinity. In the case of the second graph, there is asymptote that goes through the point where r is negative 3 on both the horizontal and vertical axis. In the case of the first graph, the line looks "vertical", passing the horizontal axis where r is about -1.4. It seems as though it is very near the square root of 2, which could have something to do with the maximum difference in the denominator, however, more exploration would be necessary to verify this though. When the parameter is 4, this continues. As the parameter gets smaller and smaller, the asymptote line seems to move further and further away, while also making a "circling motion".
What about some other graphs? What is the difference between r = 2*cos(x) -1 and r = 2*cos(x-1)? The two graphs are placed on the same axes below. The first in red and the second in blue.
The blue graph seems like the "standard" look for a trig function in the polar plane. The output of the cosine function where one is subtracted from the input is a circle that seems to have been shifted slightly up and left from the location of r = 2*cos(x). However, the red graph performed a little bit differently than I had expected originally. It appears that subtracting one from the cosine output had an adverse effect on the "circleness" of the graph. At this point, because of the subtraction, and the fact that the graph is periodic and we are taking inputs from 0 to 2*pi, there are multiple values for r, with some input x.
We could also look at graphs like r = n/x, where n varies from zero to ten. See below animation for the results:
In this case, we have a spiral that becomes wider and wider as n gets larger and larger. As n is smaller, you can see that the graph almost becomes a verticle line at the horizontal axis. Indeed, if n was zero, then the radius would be zero and the spiral would not be there. As would be expected, as n gets bigger the value for r gets bigger.
Finally, let's look at a really wild animation to close out. Suppose we had the function r = {n*[(sinx)(cosx)]2 }/[(cosx)5+(sinx)5]where n varies from 0 to 10 (and x from 0 to 2 pi). What would that look like?
As one can see, it is almost like a blooming flower. As n approaches zero, the output is pretty much nothing, as one would expect because n=0 would mean r = 0 and thus there would be only activity at the origin.
I think such graphs (as are all of the ones in this write up) would be interesting for students to see in order to learn how parameters effect the graphs in the polar plane. Many times, students are left with static graphs or worksheets that don't really help them understand what the numbers in the functions actually do. Using a dynamic program like Graphing Calculator helps students see the changes in a new light. Everything can be done in the blink of an eye. Used correctly, technology can create an extremely positive effect in pre-calculus classrooms.