Tiffany N. KeysŐ Assignment 10:

Parametric
Curves

A
parametric curve in the plane is a pair of functions where the two functions
are the ordered pairs (x,y).

x
= f(t)

y
= g(t)

EXPLORATION: For
various **a **and
**b**,
investigate x = cos (at)

y
= sin (bt) from 0 __<__ t __<__
2P

LetŐs
graph the functions where a = 1 and b = 1: x
= cos (at)

y
= sin (bt) from
0 __<__ t __<__ 2P

LetŐs
see what happens when different values for b
are substituted in the equation.

When
a = 1 and b = 2, the following graph is obtained:

When
a = 1 and b = 3, the following graph is obtained:

When
a = 1 and b = 4, the following graph is obtained:

OBSERVATIONS:

á
It
appears that as the value of b is increased the number of curves increased that
appear along the x-axis.

Now,
letŐs see what happens when different values for a
are substituted in the equation.

When
a = 2 and b = 1, the following graph is obtained:

When
a = 3 and b = 1, the following graph is obtained:

When
a = 4 and b = 1, the following graph is obtained:

OBSERVATIONS:

á
It
appears that as the value of a is increased the number of curves donŐt
necessarily increase.

á
When
a has an even value, the curves appear to not be complete or as if the graph is
incomplete.

á
However,
when a has an odd value, the number of curves is equal to the value of a and
they extend along the y-axis instead of the x-axis.

What
will happen if both the value for a and b were changed?

When
a = 2 and b = 3, the following graph is obtained:

When
a = 3 and b = 4, the following graph is obtained:

OBSERVATIONS:

á
It
appears that the curves are created along the x- and the y-axis.