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.