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.