Assignment 11: Polar Equations
By Dorothy Evans
Let’s investigate the polar
equations below
Purple 

Red 



Here is the first case
where a = 1, b = 1, and k = 1. As you
can see it’s ok, but let’s see when we change k. 
Here a = 1, b = 1, and k = 2. As you can see a flower like shape emerges. Notice the length of the red petals are 1 and the length of the purple are 2. Also notice the number of petals. At this point there are 4 red petals and 2 purple. Let’s see what happens next. 


Now a = 1, b = 1, and k =
3. Notice the shape now has 3 petals
that are still 1 long in the red and 2 long in the purple. So what do you think k changes? Let’s try another. 
Here a = 1, b = 1, and k =
4. It would appear when k is even we
get 2k petals in the red graph and k petals in the purple graph. Now let’s see what happens as we change a. 


In this case a = 1, b = 2,
and k = 4. So, it would appear we now
have 2k petals in both the red and purple, but in the purple graph the petals
are different sizes. In the red graph
the petals are b long and in the purple graph the petals are ka and a in
length. 
In this case a = 1, b = 3,
and k = 4. The flower got bigger. Notice the larger purple petals are 4. So maybe the length of the larger purple leaves
is actually a+b instead of ka. Also
it appears the red are still b in length and the smaller purple petals are
not a, but ba in length. Let’s try
another and see if our equations hold true. 


Now a = 1, b = 4, and k =
4. From our previous investigation we
hypothesized: # of petals: 2k Size of red petals = b =
4 true Size of larger purple
petals = a+b = 1+4 = 5 true Size of smaller purple
petals = b – a = 41 = 3 true 
Looks like we got the
pattern figured out. Now for those of
you wondering how I verified the length of the smaller purple petals because
none of them are on an axis. It’s
simple, the good old Pythagorean Theorem.
I took an estimate of my (x,y) coordinates and calculated the distance
to the origin. Now in this case I used
an approximation. How would you prove
it? For that matter how would you
prove any of the distances we hypothesized? 
To explore these graphs further click here for the Graphing Calculator file.