__Parametric
Explorations__

By: Brandie Thrasher

Wolframmathworld.com
describes parametric equations as Òa set of equations that express a set of quantities as explicit functions of a number of independent
variables, known as ÔparametersÕ.Ó An example of this type of equation set can
yield the graph of a circle.

The parameter (or
quantity that defines the system and its characteristics) is usually expressed
with the variable *t*. We will be using t where 0 __<__ t __<__ 2p.

When we introduce *a*
and *b*
into our equation, our graph begins to take shape, as long as a and b are not
equal.

Here, a = 2 and b =
1

Here, a = 1 and b =
2

If we change our
definitions (swapping) x and y, our graphs shift.

here,
a = 1 and b=2 then the second graph has a = 2 and b = 1

It appears that our
line of symmetry is dependent on what parameter is set to x. In our first set,
x was equal to the sin function, yielding a graph with a line of symmetry at q = ^{p}/_{2} while
x that is set equal to the cos function yields a line of symmetry at q = 2p.

With our first set,
if we leave b =1, our graphs shows the amount of loops, represented by the
value of a. here is a = 3, 4, and 5

If we change *a* from the
second set to the same values, we would actually receive the same graphs for
the odd values, but a = 4 would display