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Michelle E. Chung |
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EMAT6680 Assignment 11: Polar Equations |
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1. Investigate
![Eq1](PolarEquations_Word_files/image001.gif)
Note:
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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 'n-leaf rose'. |
When k=1 |
The
graph has only one leaf when k=1. |
![Graph1_1](PolarEquations_Word_files/image014.png) |
When
k=2 |
The
graph has two leaves when k=2. |
![Graph1_2](PolarEquations_Word_files/image017.png) |
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|. |
![Graph1_3](PolarEquations_Word_files/image020.png) |
When
k=5.3 |
Now,
let's think about k is not integer.
The
graph has four complete leaves and two-incomplete leaves when k=5.3.
So,
it looks like that the last leaf isbeing drawn when k is between
two consecutive integers. |
![Graph1_4](PolarEquations_Word_files/image023.png) |
When k=-5.3 |
Also,
as you see, the graphs are exactly same when k has same value of absolute
vlaues. |
![Graph1_5](PolarEquations_Word_files/image026.png) |
When k is in [-10, 10] with 200 steps.
From this movie, you can check the facts we
assume previously are true. |
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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. |
![Image004](PolarEquations_Word_files/image034.png) |
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. |
![image005_N](PolarEquations_Word_files/image037.png) |
When k=-3 |
The graph has three-leaves
when k=-3. |
![](PolarEquations_Word_files/image040.png) |
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. |
![](PolarEquations_Word_files/image043.png) |
When k=-5.3 |
Also, as you see,
the graphs are exactly same when k has same value of absolute vlaues. |
![](PolarEquations_Word_files/image046.png) |
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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. |
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Fourth,
let's think about the graph of ![PolarEq](PolarEquations_Word_files/image089.gif)
When k=1
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When k=1, the graph of is a circle that has (0,0.5) as its center and 0.5 as its radius. |
![image](PolarEquations_Word_files/image094.png) |
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. |
![Image](PolarEquations_Word_files/image100.png)
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When
k=-2 |
The
graph of is GREEN.
Both graphs have four leaves; however, the graph of is 45 degrees rotated toward the origin. |
![Graph1_3](PolarEquations_Word_files/image106.png) |
When k=-3 |
When k=-3, the graph has three leaves. |
![](PolarEquations_Word_files/image109.png) |
When
k=5.3 |
Now,
let's think about k is not integer.
When k=5.3, both graphs have eight complete leaves.
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![image](PolarEquations_Word_files/image115.png) |
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Fifth,
let's think about the graph of when a=1, b=2.
When k=1
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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. |
![](PolarEquations_Word_files/image132.png) |
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). |
![image](PolarEquations_Word_files/image135.png)
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When
k=-2 |
When k=-2, the graph is same as when k=2.
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![image](PolarEquations_Word_files/image138.png) |
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. |
![image](PolarEquations_Word_files/image141.png) |
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.
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![](PolarEquations_Word_files/image153.png) |
When
k=-5.3 |
Almost same as above, but because k is closer to odd number, the small leaves are inside the big leaves. |
![image](PolarEquations_Word_files/image144.png) |
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Sixth,
let's think about the graph of when a=-1 and b=2.
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Seventh, now let's think about the graph of when a=1 and b=-2.
When k=1
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The graph of is reflection of the graph of toward y-axis. |
![](PolarEquations_Word_files/image171.png) |
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