Parametric Curves

A parametric curve in the plane is a pair of functions where the two functions are the ordered pairs (x,y).

x = f(t)

y = g(t)

First we will graph the functions

x = cos (t)

y = sin (t)

from 0 to 2 pi

Now we will graph the functions:

x = cos (at)

y = sin (bt)

where a = 1 and b = 1 and still from 0 to 2 pi

Now we will change the values of the a and b. In this instance let's say a = 1 and b = 2.

You see the graph crosses dividing it into two distinct parts.

Now we will try once again leaving a = 1 and changing b = 3.

I am beginning to see a pattern here. It breaks this graph into three parts. We will try one more and then we will look at the graph when a is varied.

Let's try a = 1 and b = 4.

Now let's change the values of a in the same ways as we did for b.

Let a = 2 and b = 1.

Hmm! Who would have expected it to look like that. Let's try a = 3 and b = 1. Any guesses as to what might happen? Let's see.

Well isn't that interesting! Now what?

Let's try and see.

a = 4 and b = 1

Not quite the same pattern as for b, but there seems to be a pattern.

Just for fun let's change the values for both a and b and see what happens.

Let's say a = 2 and b = 3

Hey! Now that's kind of cool.