An Exploration of Polar Equations

by

Michelle Jones and Vicki Tarleton

Investigate the general form of a polar equation:

r = a + b cos (kQ)

We will begin by examining

r = b cos (kQ)

where a = 0 and b = 1.

Effects of a, b, and k on the graphs:

a determines how the leaf unfolds.

By keeping b =1 and k=4, and varying a we see what happens as a is increased.

b determines where the leaves intersect the x and y axis.

Keeping k constant at k=2, we can study the effects of changing the value of b.

When b = 1 or -1, the leaves intersect at 1 and -1 on both axes.

When b = 2 or -2, the leaves intersect at 2 and -2 on both axes.

When b = 3 or -3, the leaves intersect at 3 and -3 on both axes.

When b = 4 or -4, the leaves intersect at 4 and -4 on both axes.

It does not matter if b > 0 or b < 0, unless b is odd, the place of the intersection is still the same.

k determines the number of leaves

When k=1, there is a one leaf rose

When k=2, there is a four-leaf rose

When k=3, there is a three-leaf rose

When k=4, there is an eight-leaf rose

We obtain the following conclusion:

When k is odd, the number of leaves is k.

When k is even, the number of leaves is 2k.

When a and b are equal, keeping k = 1, the rose

a = b = 1

a = b = 2

a = b = 3

a = b = 4

When a and b are equal, keeping k = 2, the rose

When a and b are equal, keeping k = 3, the rose

Finally, when a and b are equal, keeping k = 4, the rose