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A parametric curve in the plane is a pair
of functions

(**x = f(t)** and **y = g(t)**),
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 **and your work with parametric
equations should pay close attention the range of **t**. In many applications, we think of
**x **and **y **"varying with time **t **" or the angle of
rotation that some line makes from an initial location.

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Let’s look at the
following parametric equations and vary **a** and **b**:

**for ** 0 __<__
t __<__ 2pi

What happens when
we set **a** and **b** equal?

Let’s look at **a = 2 ** and **b = 2**

** **

You will
notice that we get a circle with radius one.

Further investigations revealed to me that this will always
be the case when **a** = **b**.

Let’s see what
happens when **a** and **b** aren’t equal.

We’ll let **a**
= 2 and vary the values of **b**.

What happens when **b**
= 4?

What happens when **b**
= 6?

And when **b**
= 10 :

Notice in these examples that the number of circular shapes
created is equal to the value of **b** divided by the value of **a**.

Now let’s see what
happens when we hold **b** constant and vary **a**.

Let’s make **b**
= 2.

Let **a** = 4

These
values of b seem to produce the same curves for a, except that they are rotated
90 degrees.

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Now let’s see an
example for large values of **a** and **b**.

Here **a** = 19
and **b** = 29