#### Exploring Parametric Equations

by
#### Julia Neal

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** and your work with parametric equations should
pay close attention to 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.

Let's look at the graph of

for

This graphs the Unit Circle.

How can we change the graph?

What would constants in front of **t** do
to the graph?

Let's take a look at the family of graphs

We have already investigated the case where
a and b both are 1.

Let's look at the case where **a** = 1,
and **b** = 2.

This graph is called the bow tie graph,
for obvious reasons.
Now, let's consider the case where **a**
= 1, and **b** = 3.

Notice that this graph has 3 "loops."
There appears to be a pattern.

**b** = 1, 1 "loop"

**b** = 2, 2 "loops"

**b** = 3, 3 "loops"

Will the pattern hold for **b** = 4?

The pattern continues to hold.
Now, we need to consider the impact that the
constant **a** has on the graph.

Will the constant **a** affect the graph
in the same way that **b** did, except along the vertical axis?

Let's consider the case where **a** = 2,
and **b** = 1.

This is not the expected graph.
Perhaps we can gain some insight by examining
the case where **a** = 3, and **b** = 1.

This is something more along the lines
that we were expecting, initially.
Now let's examine the case where **a** =
4, and **b** = 1.

It appears that the "loop" theory
needs to be abondoned. Notice, though that when

**a** = 2, the graph
crossed the vertical axis twice.

**a** = 4, the graph
crossed 4 times.

It appears that for the constant **a**,
even numbers and odd numbers affect the graph in different ways.

Odd numbers affect the graph in a pattern similar
to the way the constant **b** did, except in respect to the
vertical axis

Even numbers seem to indicate how many times
the graph will cross the vertical axis in our interval.

The pattern appears to hold for the even values
of **a**.

The pattern also holds for the odd values of
**b**.

Negative values for **a** and **b** result
in graphs that are identical to the graphs with positive values
of **a** and **b**. This may be a result of reflection across
the lines of symmetry of the graph.

This exploration focused only on integer values
of a and b. It would be interesting to investigate values of **a**
and **b** between 0 and 1.

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