**by**

**Behnaz
Rouhani**

In this assignment we are going to investigate the following polar equation cases.

**Case I:**

for various b and k.

**Case II:**

for various a, b, and k.

**Case III: **Special investigation

We will explore the above polar equation for various b and k values.

b = 1 : red

b = 2: blue

b = 3: green

b = 6: magenta

Notice that multiplying the above two graphs by a factor of b causes the graphs to become larger by a factor of b. In these graphs the sine polar curve is symmetrical about the vertical line, whereas the cosine polar curve is symmetrical about the polar axis (horizontal axis). When k = 2, we have a 4-leaved rose.

Let's see what happens when k = 4.

In the above sine and cosine graphs when k = 4, we have 8 petals. Based on the above graphs we could conclude that:

- The sine polar curve remains symmetrical about the vertical axis, and the cosine polar curve remains symmetrical about the polar axis.
- Varying b will expand the graph by a factor of b.
- When k is even the number of rose petals will be 2k.

b = 1 : red

b = 2: blue

b = 3: green

b = 6: magenta

From the above graphs we could conclude the following:

- Varying b will expand the graph by a factor of b
- When k is odd the number of petals remain as k

Different values of a and b are represented in the following graph.

a = b = 1: red

a = b = 3: blue

a = b = 6: magenta

** Consider k=1** ----this is graph of

This curve is called Cardioid because
it is shaped like a heart. This polar curve is symmetrical about the polar
axis (horizontal axis).

__Consider k = 2__

This polar curve is symmetrical about
the vertical axis.

__Consider k = 3__

From the above we can make the following observations:

- The different values of a = b will cause the graph to be larger
- The number of petals is same as k

Different values of a and b are considered in the following graphs.

a = b = 1 : red

a = b = 3: blue

a = b = 6: magenta

** Let k =1** --- case of a

This polar curve is symmetrical about
the vertical axis.

__Let k = 2__

This polar curve is symmetrical about
the pole.

__Let k = 4__

As we vary a = b, the graph just gets larger by that factor. For instance when a = b = 3, then the graph gets larger by a factor of 3. In this case the number of petals is same as k.

Following are graphs of a **Limacon**.

In all the above cases whether k is odd or even we
get 2k petals half of which are small and half are large. The only difference
is that when k is odd the small petals are inside the large ones, but when
k is even the small petals are outside the large ones.

This following is the graph of a **spiral**.