Assignment 10:

Parametric Curves

A parametric curve in the plane is a pair of functions

where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t.  In many applications, we think of x and y as "varying with time t " or the angle of rotation that some line makes from an initial location.
For this assignment, I will be using Graphing Calculator 3.2.

Let's Start Our Investigation by Looking at the Following Parametric Equations:

x = cos(t)

y = sin(t)

where t ranges from 0 to 2pi

What do you think the graph will look like?

Now Let's Change the Equation a Little!

For various a and b, investigate:

x = cos(at)

y = sin(bt)

where t ranges from 0 to 2pi

What do you think the graph will look like when a = b?

What do you think the graph will look like when a > b?

What do you think the graph will look like when a<b?

Do you see a pattern forming from our observations?

I have another idea ... maybe my observation only works when 'a' or 'b' is 1 and 'a' is an odd number!