Polar Equations
By: Kimberly Young and Mindy Swain
Investigate varying a,b, c, and k.
The affects of k on the function 
.
Let a=1 and k vary. The below animation
captures the affect that k has on the function 
.
Conclusions about k-values:
When k is even, there are are a total of 2k
petals with k petals above the x-axis and k petals below the x-axis.
The graph is the same for all positive and negative even k values.
Consider the following function 
.
When k is odd, there are total of k petals.
When k is an odd positive value, there one
more petal above the x-axis than below. We could further conclude
that the number of petals below the x-axis is [k/2], (where [
] is the greatest integer function). And the number of petals
above the x-axis is [k/2] + 1. Consider the following function
.
When k is an odd negative value the opposite
is true. The number of petals above the x-axis is[k/2]. And the
number of petals below the x-axis is [k/2] + 1. Consider the
following function 
.
When k = 0, the function is a single point,
(0,0).
The affects of a on the function 
.
Let k=1 and a vary. The below animation
captures the affect that a has on the function .
Conclusion about a values:
a determines
the radius of the circle formed by the function.
When a is positive the circle is above
the x-axis. Consider 
.
When a is negative the circle is below
the x-axis. Consider 
.
The affects of b on the function
.
Let k=1 and a = 1 and b vary. The below
animation captures the affect that b has on the function 
.
When b=0, the graph of the function is a
circle centered at (0,1) with radius of 1.
When -1<b<1, the graph of the function
is a curve that loops inside itself.
When b<-2 or b>2, the graph of the
function is a cardioid that approaches a circle as |b| increases.
The affects of a and k on 
Recall that a affected the radius of
a circle and k affected the number of petals. Consider the following
graphs.
Again, k has affected the number of petals
and a determines the amplitude, or size of the petal. The
observations that we made earlier about k still hold. The amplitude
of the petal can be determined by the a value and is equal
to 2a.
The negative values for both a and k will
cancel each other out. As seen in the following graph.
The affects of a, k and b on 
Now lets consider changes in a, k, and b.
Consider the following animation, where n = b.
Again, as |b| increases, the graph of the
function approaches a circle.
When b=0, the obvious is true.
When -2a<b<2a, the graph of the function
loops inside of itself.
A New Function: 
We can now look at how these graphs compare
to 
. We began by looking
at how the a values compared. We noticed that a still determines
the radius of a circle or the amplitude of the petals. However,
for the cosine functions, when holding k constant, the graphs
of the functions were rotated 90 degrees. We also noticed that
for the k values, there was another shift. For even values of
k, the graph of the sine function can be rotated 45 degrees to
equal the graph of the cosine function. For odd values of k, the
same is true with a 90 degree rotation. Similarly, the b value
of the cosine function has the same effect as on the sine function
except there is a 90 degree rotation. When a, k and b vary, the
cosine graphs are equivalent to rotated sine graphs. The rotation
is determined by the descriptions above. Consider the following
graph.
Another Function: 
When exploring this function, we found interesting
characteristic in the graph. We found that the k value affects
the number of lines that form asymptotes. For values of k that
are even there are 2k asymptotes and for odd values of k there
are k asymptotes. As |k| gets larger, the graph rotates and a
k-sided regular polygon approaches the origin as well as shrinks
in size. The a and b values determine the distance from the origin
to the vertices of the hyperbola and cause rotations, a in a positive
direction and b in the negative direction. For |c|>1, the k-sided
polygon formed by the asymptotes has a larger area and as a result
the graph approaches the origin at a slower pace. For |c|<1,
the k-sided polygon formed by the asymptotes has a smaller area
and as a result the graph approaches the origin at a faster pace.
