The Department of Mathematics and Science Education

tiffani c. knight

Parametrics…hmmmm. Well, first I graphed the following three equations for 0≤ t ≤2pi:

And for all three, I got a circle…the unit circle. Of course my a and b for the last two were 1, which is probably why they all look the same.

Let’s explore the last two graphs in two ways: first, when a ranges from 2 to 5 while b stays at 1, and when b ranges from 2 to 5 while a stays at 1.

When a = 2

When a = 3

When a = 4

When a = 5

When b = 2

When b = 3

When b = 4

When b = 5

Okay, these proved to be interesting. For , when b stayed constant at b = 1, I saw a pattern for a and it was based on whether a or b was an even or odd number. Seems like a dictates the number of times the curve will cross the y-axis. When a was an even number, the curve crossed a times. Look up, when a was 2, the curve crossed the y axis 2 times and when a was 4, the curve crossed 4 times.

Now, when a was odd, the number of times the curve crossed the y axis is a + 1. So, when a was 3, the curve crossed the y axis 4 times. And when a was 5, it crossed 6 times.

When a was kept constant at 1 and b was varied, a pattern emerged that b dictated how many times the curve crossed the x-axis. And it didn’t matter whether b was even or odd, the number of times the curve crossed was b+1. So, when b was 2, 3, 4, and 5, the curves crossed the x axis 3, 4, 5, and 6 times, respectively.

And obviously, the graphs were no longer of circles.

When a = 2

When a = 3

When a = 4

When a = 5

When b = 2

When b = 3

When b = 4

When b = 5

Well, for , I continued to get circles, not the unit circle of course, but circles nonetheless. I observed a few things, the first being that when a increased, the circle got narrower horizontally. And when b was increased, the circle got narrower vertically. It appears that a determines where the circle will cross the x-axis. And those two places are +a and –a. And be determines where the circle will cross the y axis, and those two places are +b and –b.

I wondered if this had anything to do with me keeping a and b constant at 1, so I graphed a parametric when a=3 and b=5. I was expecting to get a circle that touched the x axis at +3 and -3, and touched the y axis at +7 and -7. Did I get this?

Yep! Sure did.