Investigating Polar Equations
By Sharon K. OŐKelley
Let us begin with an investigation of a basic polar equationÉ.
Below are several graphs of this polar equation when b and k are varied. LetŐs see if we can find some patterns.
1. When b = 1 and k =1, the result is a circle.
2. When b = 1 and k =2, the result is a figure with four petals.
Could it be that when k is even, the number of petals is twice the value of k? If that is the case, then when k=4, the figure will have 8 petals. LetŐs check it outÉ.
(Notice that the tips of the horizontal and vertical petals are at 1 and -1.)
3. What would happen if we changed b to a value other than 1? LetŐs experiment with the last equation and make the value of b equal to 3.
Notice that the horizontal and vertical tips are now at -3 and 3 which means the original figure has been stretched by a factor of 3.
Does this mean that if b is less than 1, it will shrink? LetŐs seeÉ.
4. What would happen if k were odd? LetŐs try oneÉ.
In this case, it appears that the number petals corresponds to the value of k so that should mean that if k =7, then the figure should have 7 petals.
Based on the previous work, we can guess that if b is equal to any number other than 1 the figure will stretch or shrink. A case in pointÉ.
LetŐs make it interesting and investigate the polar equation...
1. LetŐs make a = 1, b = 1, and k = 1.
2. Now, letŐs make a = 1, b = 1, and k = 4.
Notice that the value of k, although even, now corresponds to the number of petals? What if k were odd in this same situation?
The result appears to hold.
3. What happens if we hold k and b constant and vary the values of a? ConsiderÉ.
When a is greater than 1, the figure appears to stretch.
Notice that if a is less than one an interesting situation develops.
To see an animation of the values of a changing, go here. (Be patient!)
4. What happens if we hold k and a constant and vary the values of b? ConsiderÉ.
Note that the petal number is the same and appears to be dictated by the value of k and again there appears to be a flower within a flower. If the value of b increases, so does the size of the figure.
5. What if the value of a is greater than the value of b?
The inside figure disappears and the figure expands.
Would any of these patterns hold for sine? LetŐs look at oneÉ. ConsiderÉ.
So if k =2, there should be four petals based on our previous work.
It worked! So if k = 5, there should be five petalsÉ.
It appears then that some of the basic patterns are the same for sine and cosine in polar equations!