**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).**

**Graph**

**x=cos(t)**

**y=sin(t)**

**for
0<t<2pi**

**Graph**

**x=cos(at)**

**y=sin(bt)**

**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?**

**a=4,
b=1**

**a=5,
b=1**

**What
do you think will happen when we change the values of a and b
at the same time?**

**a=2,
b=3**

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