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Polar Equations

**Graphing Calculator 3.2 **and **xFunction **are suggested for these investigations. **Some
of them could be done with a TI-83 or similar.**

1. Investigate

Note:

á When a and b are equal, and k is an integer, this is one textbook version of the " n-leaf rose."

á

Compare with

for various k. What if . . . cos( ) is replaced with
sin( )?

First, letÕs discuss polar
equations basics.

**Definition
of Polar Coordinates**

To define polar coordinates, we first fix an **origin** O and an **initial ray** from O.

Each point
P can be located by assigning to it a polar coordinate pair (r, ¿), in which
the first number, r, gives the directed distance from O to P and the second
number, ¿, gives the directed angle from the initial ray to the segment OP:

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Now that we have a polar
coordinate review, letÕs keep going.

we run into
some really cool graphs.

LetÕs start
simple. Values of **a and b are 1** for the next few exercises.

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**Now for k=5 and k=10**

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So for k=5 there are five pedals and for k = 10 there are ten pedals.

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**For this equation k=1000**

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**Thus far, a=b=1. LetÕs change the values of a and b and
see what happens.**

**We know from before that
the value of k determines the number of pedals. **

**The next question is, ÒHow
does a**

I graphed the equations below to find out.

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**a = 1, 2, 3, 4**

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**Based on this, it appears
that the graphs cross the y-axis in two places at positive (a-1) and negative (a-1).**

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**To double check, I used
larger values of a, and found that this works out.**

**So, what do you think the
equations of the graphs below are?**

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**A = 21, 10, 6, 5, 4**

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The next
question to answer is, Òwhat role does the b play in the graph of the equation?Ó

**The graph below is for a=
1, b= 2, 3, 4, 5, and 6, and k=2**

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One of the biggest relationships is in the length of the
pedals. Also, there are not just
vertical pedals but also horizontal pedals. The height of the vertical pedals can be found by **b-1**. The
height of the horizontal pedals can be found by **b+1. **

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**This can be generalized to include graphs when a is
greater than one. The height of
the vertical pedals can be found by b-a if a<b or a-b if b<a. The height of the horizontal pedals can
be found by a+b.**

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**Now, letÕs investigate**

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**Here b=1, 2, 5, and 10 and
k=1. So this seems to be a circle
tangent to the y-axis at the origin, with diameter equal to the value of b**

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**Next, letÕs graph the same
b values, but change k.**

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**This is where it gets interesting,
and so I finally smartened up and graphed the equation so that k varies.**

**Click **here** and drag on the n-value to see how the
graph changes. So at k=1 we start with a circle, and then as k increases, the
number of pedals go through a cycle based on the rotation. **

**When you change the cos to
sin, the graphs are very similar.
They are not reflections of each other, however. There appears to be a 90 degree
rotation between the two. Click **here** to compare the
two equations and their graphs.**

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