# Page Bird

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:

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

## In our investigation of

we run into some really cool graphs.

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

### Now, if k=2

Now for k=5 and k=10

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

For this equation k=1000

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 affect the graph?Ó

I graphed the equations below to find out.

a = 1, 2, 3, 4

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

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?

A = 21, 10, 6, 5, 4

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

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.

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.

Now, letÕs investigate

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

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

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.