* *

*Investigating Polar Equations*

*By Sharon K. OŐKelley*

* *

** **

**Part One**

** **

**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.**

** **

** **

** **

**Part Two**

** **

**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.**

** **

** **

**Part Three**

** **

**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!**

** **