A parametric curve in the plane is a pair of functions
x = f(t)
y = g(t)
where the two continuous functions define ordered pairs (x,y).
where a=1, b=1 and 0<t<2pi
Let's see what happens when we change the values of a and b. What happens when a=1, but b=2?
Notice that it divided the shape into two parts. What about when a=1 and b=3?
Notice that the new value for b divided the graph into three parts. Can you guess what might happen when we change the value of b to 4?
That's right. It divided the shape into four parts. Now let's see what happens when we change the value of a. Here is what happens when a=2 and b=1.
Before we make any assumptions let's try another value for a. Here is what happens when a=3 and b=1.
Notice for a=2 we got an open graph, but for a=3 we got a closed graph. What do you think will happen for a=4? a=5?
What do you think will happen when we change the values of a and b at the same time?
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