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Investigate .

In this
example, a = 1 = b, and k = 1.

When a
and b are equal, we get what’s called an

“n-leaf
rose”.

Let’s see
more of these examples.

Let a = 2
= b, and k = 3

Let a = 2
= b, and k = 4

Let’s make a prediction about k. What if a = 2 = b, and k = 9? My guess is that there will be 9 leaves in the graph.

9 leaves!

Let’s try one more. What about when a = 2 = b, and k = 15?

15
leaves!

Now,
let’s investigate what happens when a = 0.

Let b = 2
and k = 2

Let b = 2
and k = 4

Let’s try
one more before making a prediction.

Let b = 2
and k = 5

Do you
see a pattern? When k is odd, there are
k leaves in the graph. When k is even,
there are 2k leaves in the graph!

Let’s
look at one more.

Let b = 2
and k = 10

20
leaves!

So let’s put this together. When a equals b, there are k leaves in the graph. When a is zero and k is odd, there are k leaves in the graph. When a is zero and k is even, there are 2k leaves in the graph!

Let’s do one final investigation. What if we change cos to sin?

Let’s
look at a = 2 = b, and k = 3

Let’s look
at a = 2 = b, and k = 4

Let a = 2
= b, and k = 9

The
graphs of sin are similar to the graphs of cos. The difference is that the leaves in the sin graphs are shifted
to the right.