Parametric Equations

By: Carly Cantrell

Before we consider graphing in the polar plane, letÕs understand the relationship between Cartesian and polar planes. Remember, in the Cartesian plane y is a function of t, where t goes from 0 ˆ 2. And in polar form, x is a function of t, where t goes from 0 ˆ 2 and x is the radial length, which is the horizontal distance from the origin.

Graph: y = cos(t) in the Cartesian plane (left)

Graph: x = cos(t) in the polar plane (right)

Notice, in both situations:

 Input Output 0 1 0 -1 3/2 0 2 1

The points are mapped in the same manner; they are just represented different based on the plane.

Parametric equations are graphed in the polar plane.

Similarly,

Graph: y = sin(t) in the Cartesian plane (left)

Graph: y = sin(t) in the polar plane (right)

Now, in polar form y represents the vertical distance away from the origin and the following table represents both graphs:

 Input Output 0 0 1 0 3/2 -1 2 0

LetÕs investigate the results when we plug in various a and b in the forms:

x = cos(at)

y = sin(bt)

for 0

When a and b are both 2:

x = cos(2t)

y = sin(2t)

When a and b are both 3:

x = cos(3t)

y = sin(3t)

When a and b are both 4:

x = cos(4t)

y = sin(4t)

When and b are both  :

Question: How come for even integers there are twice as many petals as the integer, but for odd integers there are the same number of petals as the integer?

Answer: When there are even integers, from 0 ˆ 2, there is no overlap. Whereas, when mapping with an odd integer, there is overlap. This means from 0 ˆ  and from  ˆ 2 the same points are being swept by those angle measures.

Conjecture:  When the integer is less than one, the graphs are identical! Investigate for yourself!

LetÕs investigate the results when we plug in various a and b in the forms:

x = acos(t)

y = bsin(t)

for 0

This results in a radial scaling away from the origin.

x = 2cos(t)

y = 2sin(t)