Jernita
Randolph

Investigation of Polar
Equations

Here weÕll be
investigating .

LetÕs begin
with a, b, and k =1.

Now lets investigate for different values of k,
while a and b are held constant at 1.

We can see
that the number of ÒleavesÓ correlates with the value of k, in this case that
is k=2 so we have 2 leaves.

If we look at
the x-axis we see that the x-coordinate is equal to a+b,
or in this case 2.

And just
because its prettyÉ and its always nice to check multiple values;

Here we can see that both the x and y-coordinates are equal
to ± (a+b), which is still 2 and we can easily see
why this is called the n-leaf rose.

When graphing for k=1/2 we have to increase the number of
rotations (double) to get a complete graph. We can see there are 2 inverted leaves.

When graphing for k=1/4 we have to increase the number of
rotations by 4 times to get a complete graph. We can see that there are 4 leaves getting progressively
larger and superimposed on one another.

Now letÕs
investigate situations where k and b are varied and a is
held constant.

Here we have a double n-leaf
rose with k=n smaller leaves and k=n larger leaves.

Notice that the leaf values on the x and y-axes
still corresponds to ± (a+b).

As b increases the length of the second set of
leaves gets longer.

For a³3, we get a graph with a radius of a, and center (b,0)

For a²3 we get a starfish with k arms.

Here we will get a starfish with six arms.

What happens
when we exchange sine for cosine?

The graphs are the same shape but the sine graph
is rotated 90^{0} which is logical if we
consider the relationship of the sine and cosine functions.