Polar Equations

I will investigate the polar equation, ,
for different values of a, b, and k.

LetŐs
start by graphing this polar equation when a, b, and k are all equal to 1.

Now
letŐs keep a and b equal to 1 and change the value of k to 2.

It looks like there are two leaves on the
graph. These leaves cross the x axis at the origin and at 2 and -2.

Now, letŐs keep a and b equal to 1, but
change k to a value of 3.

Notice,
now there are three leaves and they all intersect at the origin.

Now,
letŐs try a larger value for k and see what happens. LetŐs try k=15.

Now,
I have 15 leaves. It looks like k changes the number of leaves.

Now,
letŐs try k=100.

Wow,
look at that, it almost resembles a flower.

So,
when a and b are equal, and k is an integer, the function forms an Ňn-leaf roseÓ.

Now,
letŐs look at the function , when
k=0 and b=1.

This
is the graph of the cosine function on the polar coordinate map. There is only
one leaf.

Now,
letŐs try k=2 and let b remain 1.

The graph now has 4 leaves. Maybe the
number of leaves is now 2k. LetŐs see.

LetŐs try k=3 to see if we get 6 leaves.

Intersecting,
I only get 3 leaves. So the above conjecture is not correct. Let's try k=4.

Well,
now I have 8 leaves. Maybe the
leaves being equal to 2k only works when k is even. Let's try when k= 6.

Yes,
my above conjecture seems to be correct. When k is odd the number of leaves is
k. But when k is even the number of leaves is equal to 2k. Let's try one more
odd value just to be sure. LetŐs try k=9.

Yes,
there are only 9 leaves, I was correct.

Now,
lets change the value of b, while k remains constant. Let b=2 and k=2.

LetŐs
try a few more before I make a conjecture.

What
about when b= 3 and b=6.

It
appears that the value of b effects where the function intersect both axis. For
instance, when b = 6, illustrated above, the function intersects the x and y
axis at 6 and -6 .

Now,
letŐs see what happens if I graph the function , and how this compares with the graphs
of .

LetŐs
see what happens when k=0 and b=1. This yield no graph at all. LetŐs change k
to 1.

This
graph has only one leaf.

LetŐs
see what happens when k=2.

This
looks like this might follow the same pattern as the cos function.

LetŐs
try another even value and then a couple of odd values for k.

First,
letŐs try k=4. Will there be 8 leaves?

Yes,
so the k must determine the number of leaves as before. So, there will be 2k
leaves.

Now,
what about the odd values, letŐs try k=5. Will there now be 5 leaves?

Yes,
letŐs try one more, k=11.

So, there are 11 leaves. It looks like the
graph of the cosine polar function is the same as the graph of the sine polar
function. But, I have to try one more case. What happens when the value of b is
altered and k remains constant. LetŐs try b=2 and k=2.

I
see that the b value does expand the function. But there seems to be a
rotational shift. LetŐs try one more

b=
4 and k=2.

Yes,
the graph is a definite rotation from the cosine function. It looks to be a 90
degree rotation.