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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 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 = 3
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.