__Assignment 11: Polar Equations__
by
Melissa Silverman

I
will begin my investigation with a simple polar equation, where
a, b and k are equal to 1.

As the absolute value of a increases,
the shape becomes more circular. If -1<a<1, then the shape
becomes smaller and an inner loop is formed.

What happens if the value of b is changed?
If b is positive and increasing, then the graph increases in size
and forms an inner loop. The graph is oriented towards the positive
x-axis.

If b is negative and increasing, then
the graph increases in size and forms an inner loop. The graph
is oriented towards the negative x-axis.

If 0<b<1, then the graph becomes
more circular and is oriented towards the positive x-axis.

If -1<b<0, then the graph becomes
more circular and is oriented towards the negative x-axis.

The last variable to test is k. As
the absolute value of k increases, the graph is flower with k
number of petals, all centered around the origin.

If -1<b<1, then the graph is
a spiral.

What if the cosine function is replaced
by the sine function? How do a, b, and k affect the graph? We
can begin by graphing a simple polar equation, where a, b, and
k equal 1.

As the absolute value of a increases,
the shape of the graph becomes more circular and the size of the
enclosed figure increases.

If -1<a<1, then the graph forms
an inner loop. The orientation of the graph does not change whether
b is positive or negative.

If b is positive and increasing, the
graph increases in size, forms an inner loop, and is oriented
about the positive y-axis. Inversely, if b is negative and increasing,
the graph increases in size, forms an inner loop, and in oriented
about the negative y-axis.

The value of k determines how many
petals form the flower graph. If k is positive, the petals are
centered at (0,0) and are symmetric about the line y=x, without
the line y=x bisecting the petal(s) in the first quadrant. If
k is negative, the petals are centered at (0,0) and are symmetric
about the line y=x, with the line y=x bisecting a petal in the
first quadrant. The sign of k does affect the orientation of the
flower.

It
seems that the key to this polar equation is the k value. The
k value determines the number of petals that the flower has. The
a and b terms change the look and size of the flower, but cannot
determine the flower. It is also interesting that both sine and
cosine produce similar flowers, something which may not have easily
been predicted without using a graphing program.
One frustration with this type of investigation
is the myriad of possible scenarios. One can use sine or cosine.
One can change the variables a, b, and k for different discoveries.
Different combinations of a, b, and k values will also affect
the graph and the flower shape. Playing with positives and negatives
changes the graph's shape and/or its orientation. A graphing program
makes this easier to accomplish because the results are instant.
But there certainly seem to be infinite possibilities to this
investigation!

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