In this exploration, we will investigate the
polar equation *r = a + b cos (k*** q)** where

First, let's look at several graphs of *r
= a + b cos (k*** q)** where

Notice that in each of these graphs, we let
**a** = **b** = 1. When **k** = 1, the graph was a 1-leaf
rose. When **k** = 2, the graph was a 2-leaf rose. As **k**
changes, the number of leaves changes to **k**. So, if **k**
= 8. we would get a 8-leaf rose. Click HERE to change **k** and observe the graphs. What happens
when we change **a** and **b** to something other than 1
but keep them equal? Let's look at several graphs of *r =
a + b cos (k*** q)** where

Notice that these graphs are similar to the
above graphs, but these are just larger. These graphs are just
dilations of the first set of graphs by a scale factor of **a**
(or **b** since **a** = **b**). The number of leaves
is still **k**. Click HERE to change **a**, **b**, and **k** (with **a**
= **b**) and observe the graphs. Make sure to try **a**
and **b** where they are not integers, but any real number.

Now, let's let **a** = 0 and graph the polar
equation *r = b cos (k*** q)**.
How are these graphs different than

We see that when **a** = 0, the number of
leaves is still dependent on **k**, but it is also dependent
on whether **k** is even or odd. When **k** is odd, we get
a k-leaf rose. When **k** is even, we get a 2k-leaf rose. We
can predict that if **k** = 7, we will get a 7-leaf rose and
if **k** = 8, we will get a 16-leaf rose. Click HERE to change **b** and **k** and observe the graphs.
Let's look at several graphs of *r = b cos (k*** q)** where

Again, we see that these graphs are similar
to the set of graphs above, but these are larger. These graphs
are dilations of the previous set of graphs by a scale factor
of **b**. Now, let's explore what happens to the graphs when
we replace cosine with sine.

First, we will look at the graphs of *r
= a + b sin (k*** q)** where

When cosine is replaced by sine, the graph
of the polar equations rotates by radians.
When **k** = 1, the graph rotates by
radians. When **k** = 2, the graph rotates by radians.
When **k** = 3, the graph rotates by radians.
Following the pattern, the graph will rotate radians
when cosine is replaced with sine.

Again, if we change **a** and **b** and
leave them equal, the only difference will be the dilation of
the new graphs.

In this exploration, we restricted **k**
to be an integer. Click HERE to further explore the graphs of the polar equations
*r = a + b cos (k*** q)** and