Polar
Equations

(Assignment 11)

**by**

Cara Haskins, Matt
Tumlin, and Robin Kirkham

Through the assistance of
Graphing Calculator 3.2, we investigate the different variances when graphing
Polar equations.

Explore the equation **r
= a + b cos (k ****q****) such that 0 ****£**** ****q**** ****£**** 2 ****p**

Since there are three
variables a, b, and k to explore, there are many cases to explore.

1.
When
a and b are equal, and k is an integer, this is referred to

as the Òn-leaf rose.Ó

Let us graph such that :

a = b = 2, k = 1 (red)

a = b = 4, k = 1 (green)

Notice that when a=b , a
and b are scalar factors for the Òn-leaf roseÓ. Also, when k=1, the roots of r = a + b cos (k q) are 0 and a+b.

Next, we observe the
effect of k on the equation.

Below are the graphs of:

a = b = 2, k = 3 (red)

a = b = 4, k = 5 (green)

With these graphs and
further exploration, we observe that k determines the number of leaves in the
Òn-leaf roseÓ figure.

2. LetÕs now look at the
graph **of a=2, b=8, and k=1**.

When a < b and k is an integer, r = a + b cos (k q) have roots at 0, a+b,
and a-b.

When a< b, the function traces a similar path to the
Òn-leaf roseÓ graph, but not on the same scale.

It appears that the results provide a Òk-leaf roseÓ.

One inner ÒleafÓ always is at b-a on the x-axis.

3.
What
happens when a > b and k is an integer?

To see, letÕs investigate
the graph of a = 5, b= 2, and k= 10.

The ÒleavesÓ are merging
towards a circle form.

The leaves come into a
point on the circle centered on the origin with the radius a-b.

The tips of the leaves
work out to a point on the circle centered at the origin with the radius a+b.

The function oscillates
between these two circles k times to produce k ÒleavesÓ.

Once k becomes large enough,
other characteristics can be explored.

Look at a=5, b= 4, and
k=1000

This graph has many
different characteristics that could be explored. Observe that the center is
not filled.

Notice that there is a
five leaf rose in the center as well as the outer leaves (if you will) are also
five.

This seems to be related
to the ÒaÓ value.

When k=2000 is tried,
notice that the internal number of leaves becomes 10.

With some further
investigation it seems that the number of leaves is Ò1000/200 = 5Ó.

4. What about when a = b and k is NOT an
integer?

Let a = b = 5, and k = 3 (green)

and a = b = 5, and k = 3.4 (red)

Observe that the graph is
no longer continuous.

The two graphs are
merging towards each other.

Thus, only when k is an integer
is the graph continuous.

When k is between 3 and 4
then the number of leaves is also between three and four.

5.
When
a < b and k is not an integer, what occurs?

When a < b and k is
not an integer, observe a similar transformation taking place as we witness
when a=b and k is not an integer.

As observed before there
are between 2 and 3 leaves due to k being not an integer.

6.
How
about when a > b and k is not an integer.

When a > b and k is
not an integer, what do you think is observed this time?

Conclusion:

á
As
a, b, and k vary there seems to be many relationship issues that can be
discussed. The number of leaves and the relationship that both a and b have
seem to be related.

á
It
becomes more interesting when the values are no longer integers, that is when
all the changes and predictions change.

This looks like an
investigation that can be shared with high school students allowing them to
draw quite a few different and interesting conclusions as well as what has been
observed.