
Michelle E. Chung 

*
EMAT6680 Assignment 11: Polar Equations 

1. Investigate
Note:


First,
let's think about the graph of
when a=b=1 and k
is in [10, 10].
* Since I set
a=b=1, these are the graphs of 'nleaf rose'. 
When k=1 
The
graph has only one leaf when k=1. 

When
k=2 
The
graph has two leaves when k=2. 

When
k=3 
The
graph has three leaves when k=3.
So,
it looks like that the graph has same number of leaves as k. 

When
k=5.3 
Now,
let's think about k is not integer.
The
graph has four complete leaves and twoincomplete leaves when k=5.3.
So,
it looks like that the last leaf isbeing drawn when k is between
two consecutive integers. 

When k=5.3 
Also,
as you see, the graphs are exactly same when k has same value of absolute
vlaues. 

When k is in [10, 10] with 200 steps.
From this movie, you can check the facts we
assume previously are true. 



Second,
let's think about the graph of (Purple)
when a=0 and b=1.
When k=1 
The graph has one leaf that is a circle when k=1. 

When k=2 
The graph has four leaves
when k=2.
So, it looks like that the graph of has two times of k leaves when k is even. 

When k=3 
The graph has threeleaves
when k=3. 

When k=5.3 
Now, let's think about
k is not integer.
The graph has eight complete leaves and one incomplete
leaves when k=5.3.
So, it looks like that the last leaf isbeing drawn
when k is between two consecutive integers. 

When k=5.3 
Also, as you see,
the graphs are exactly same when k has same value of absolute vlaues. 



Third, let's think about
the graph of
comparing to .
When k is
in [10, 10] with 200 steps.
From this movie, you can check the facts we
assume previously are true. 



Fourth,
let's think about the graph of
When k=1

When k=1, the graph of is a circle that has (0,0.5) as its center and 0.5 as its radius. 

When
k=2 
The graph of is RED and the graph of is BLUE.
Both of the graphs have four leaves; however, the graph of is 45 degrees rotated toward the origin. 

When
k=2 
The
graph of is GREEN.
Both graphs have four leaves; however, the graph of is 45 degrees rotated toward the origin. 

When k=3 
When k=3, the graph has three leaves. 

When
k=5.3 
Now,
let's think about k is not integer.
When k=5.3, both graphs have eight complete leaves.



Fifth,
let's think about the graph of when a=1, b=2.
When k=1

The graph of has a different shape of leaf.
When k=1, it has one big leaf and one small leaf, which is inside the big leaf. 

When
k=2 
When k=2, the graph of has four leaves, which are two big leaves and two small leaves. Also, the big leaves pass through (3,0), (0,0), and (3,0), and the small leaves pass through (0,1), (0,0), and (0,1). 

When
k=2 
When k=2, the graph is same as when k=2.


When
k=3 
When k=3, the graph of has three big leaves and three small leaves, which are inside the bigger ones.
So, the graph of has two times of k leaves, which are big and small. The number of big and small leaves are same.
Also, when k is odd, the small leaves are inside the big leaves. 

When k=1.7 
Now,
let's think about k is not integer.
When k=1.7, the graph of has three complete leaves and half of the last one. Since we know that it has four leaves when k=2, we can say that we have complete leaves when k is an integer.
Also, we can see that the small leaves are coming out from the big leaves.


When
k=5.3 
Almost same as above, but because k is closer to odd number, the small leaves are inside the big leaves. 


Sixth,
let's think about the graph of when a=1 and b=2.

Seventh, now let's think about the graph of when a=1 and b=2.
When k=1

The graph of is reflection of the graph of toward yaxis. 


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