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Polar Equations

 

By Lauren Lee

 

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Investigate†† .

 

 

 

 

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

When a and b are equal, we get whatís called an

ďn-leaf roseĒ.

 

 

 

Letís see more of these examples.

Let a = 2 = b, and k = 3

 

 

 

 

 

 

 

Let a = 2 = b, and k = 4

 

 

 

 

 

 

 

 

Letís 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!

 

 

 

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

 

 

 

 

15 leaves!

 

 

 

 

 

Now, letís investigate what happens when a = 0.

 

 

 

 

Let b = 2 and k = 2

 

 

 

 

 

 

Let b = 2 and k = 3

 

 

 

 

 

Let b = 2 and k = 4

 

 

 

 

 

 

Letís 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!

 

 

Letís look at one more.

 

Let b = 2 and k = 10

 

 

 

 

20 leaves!

 

 

 

So letís 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!

 

 

 

 

Letís do one final investigation.What if we change cos to sin?

 

 

 

 

Letís look at a = 2 = b, and k = 3

 

 

 

 

 

Letís 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.

 

 

 

 

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