This is the write-up for assignment 11.

In this write-up we will be investigating polar equations. Click here for a quick review of the polar coordinate system.

Let's consider the equation and take a look at several graphs while
we let the **b** value remain constant and let the **k **value
vary. We only look at **k** values that are positive integers.

Equations of the form , where **b** is a constant and **k**
is a positive integer, produce graphs called **roses**.

The radius of the rose is determined by the
coefficient **b** in .

We can see that the **k** value effects
the number of petals in the rose. If the **k** value is odd,
there will be exactly **k** petals. If the **k** value is
even, there will be exactly **2k** petals. (In the examples,
the **k** value was always an integer. If **k** is not an
integer, then a complete petal will not be formed.)

Click here
for a dynamic presentation in which the **b **value remains
constant (**b=1) **and the **k **value varies from -10 to
10.

Now, let's take a look at the equation:

Let **k**=1 and vary the **a** and **b**
values ( but we will keep **a**=**b** ) in the equation.

When **a**=**b** in the equation and **k**=1, we
get graphs that are called **cardiods**.

Now, if we let **a** and **b **be different
numbers and keep **k**=1, we get the sample graphs below:

All of the graphs above are called **limacons**.

Some of the graphs above had inner loops. These
are called **limacons** with an **inner loop**.

The limacon has an inner loop. A close look at the graph and the equation reveals that these loops correspond to negative values of r. For , we have r=0 when =/3 or 5/3, and

r<0 for /3 < < 5/3. For these values the curve is drawn on the opposite side of the origin or pole.

These graphs also have symmetry with respect to the x-axis. This symmetry occurs because the cosine function is an even function ( i.e. for any , cos=cos (-) ).

If we were to replace the above equations with the sine function, we would obtain basically the same type graph, except the graph would be symmetric with respect to the y-axis, because the sine function is odd.

Below are some graphs of the above equation letting a=1 and b=1 and varying the k value.

Click here for a dynamic presentation of the graph of where the value of k varies from -10 to 10.

So, when a and b are both the same value in
the equation and
we vary the k value (positive integer values only), we still obtain
a curve that that resembles a rose. The number of petals on the
rose is equal to **k**. This is different when **k **was
an even integer in the equation .

Now, let's take a look at how the **a **and
**b** values effect the equation of . In the previous examples above, we
let **a**=1 and **b**=1. (Notice that in all of the examples,
the petal has a maximum length of 2). Now, if we let the **a
**and the **b** value vary (keeping both the same value)
and let the k value a remain a constant we can see from the examples
below that the length of the petal will depend on the value of
**a** and **b.**

We can see from the examples above that the
lengths of the petals got longer as we increased the values of
**a** and **b **(keeping **a** and **b** the same
value).