**Assignment
#11**
**By
Nikki Masson**
**Polar
Equations**

**Review:**

Recall that we define
a point (x,y) on the plane as x units to the right of the origin
and y units to the left of the origin. Below is a graph of the
point (1,2).

This works great for lines
and parabolas, but now we are going to graph more complex figures
using polar coordinates.

**Definition
of Polar Coordinates:**
Now a point will be given as such: (r, theta). This r is the distance
from the origin to the point and the second coordinate, theta,
is the angle that the vector from the origin to the point makes
with the positive x-axis. To convert cartesian equations to polar
equations use the following rules:

__Investigation
into Polar Equations__

We are
going to be exploring the above polar equation for different values
of *a, b* and k and see how the values affect the graph.

__Part
I : __*A=B* and k is an integer that will vary:

What
seems to be the pattern here? As k increases, the number of "leaves
in the flower increases"? If we increase k=4, we should get
four "leaves."
Let's
look at k=10 and see if we the pattern still works.
In
all the above graphs, *A=B*=1 and k varied. What if *A=B*=5,
would k still be the numbe of leaves?
If
you compare the above equation to the equation earlier equation
with four leaves, you will see that as you increase *a=b*,
the graph becomes "leaves" become larger.
Conclusion:
When *A=B*, then k= the number of "leaves in the flower."
__Part
II : A>B and k is an integer that will vary:__

This
graph looks very similar to the above when k=1 and* a=b*,
let's keep investigating and see if *a>b* has an impact
on the graph.
Now,
the leaves are not meeting in the middle, let's keeping investigating:
As,
we can see, when *a>b*, the leaves do not meet at the
origin as they did when *a*=*b*. In all the above polar
equations for part II, a=3 and b=2. Let's see what happens when
a=4 and b=2.
Conclusion:
In comparing the two above equations we see that as the difference
from *a* to *b* gets larger, the leaves meet further
and further away from the origin.
__Part
III : __*A*<*B a*nd k is an integer that will vary:

To help
us better compare how *a *and *b* are affecting the
graphs, I will draw three graphs on each picture, the new graphs
when a<b will be in blue.** **

The
blue curve has one loop at the origin when k=1. When we increase
k=2, let's see what happens:
Now,
we see two extra loops when k=2. Let's compare the above graph
with a couple more that we have already seen:
What
do we expect for k=10, when *a<b*? From the above patterns,
the graph will look just like the graph when *a=b*, but there
will be 10 extra loops inside the larger loops or "leaves".
__Part
IV : Investigate__

__Part
A:__
b=1 and k varies

__Part
A:__
b varies and k=1

**You
can continue investigating polar equations in Graphing Calculator
3.2 or TI-83 and similar calculators.**
**Suggestions
would be to change cos to sin and investigate as we have just
done. **

**Return**