# 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 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.

#### Various graphing technology,
such as the TI-81, TI-82, TI-83, TI-85, TI-86, TI-89, TI-92. Ohio
State Grapher, xFunction, Theorist, Graphing Calculator 3.2, and
Derive, can be readily used with parametric equations. Try Graphing
Calculator 3.2 or xFunction for what is probably the friendliest
software.

1. Graph

As you can see, the solution appears to be
a circle with center at the origin and a radius of 1.

In order to further investigate, I will observe
the parametric equations

for various a's and b's.

Let's observe some examples:

1) a=b

It is obvious that when a=b, then the solution
will always be a circle about the origin with a radius of 1.

2)a=2

As you can see, when a=2, then the solution
is a series of curves that look like loops.

The number of loops depends on b. You can see
that the number of loops = 1/2(b).

3) b=2

These graphs look very different, so I will
look at the ones where a is odd first. When a is odd, you can
see that there are always 2 maxs and mins for y, and there are
a maxs and mins for x.

When a is even, there appears to only be one
max and min for y and 1/2(a) maxs and mins for x. When a=4, this
notion is not true. This makes me wonder if this is because a=2b.
So, now I will look at some of these examples.

4)a=2b

So, it is true that when a=2b the graph is
always in this shape.

5) Some other fun examples

As you can see from these examples, the number
of maxs and mins for x is always a, and the number of maxs and
mins for y is always b. Another observation is that the graph
is always in the boundaries -1<x<1 and -1<y<1.

Another way to help me further my investigation
is to look at the parametric equations

for various a's and b's.

Let's observe some examples.

1)a=b

It is obvious that when a=b, the solution is
a circle with center at the origin and radius equal to a and b
(r=a=b).

2)Some other fun examples

As you can see, the solution is always an ellipse
with the center at the origin and the x max=a, x min=-a, y max=b,
and y min= -b.