Investigations with polar equations

Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #11 Investigation #2

How does the graph of change as a varies?

Here k = 0.1

Larger values of a tend to make the spirals grow in "width" and in "length."

How does the graph change as k varies?

Keeping all a values (and colors) as shown above, we can see the affect that the value of k has on the graph:

For k = 1

For k = 2

For k = 5

For k = 10

Try your own value of k here.

What about the graph of ?

For k = 0 we have nice concentric circles                  For k = 1, the circles are nicely tangent at (0,0)

For k = 2

For k = 5

For k = 8

How is the graph affected when a constant b is added to either of the previous equations?

Consider the graph of  for various values of b:

b = 0                                                                b = 3

b = 6                                                                            b = 9

Now consider the graph of   for various values of b (the color scheme is the same as for that above):

b = 0                                                                b = 3

b = 5                                                                            b = 8

In each case it appears that adding the constant b yields a dilation in the graph.

Lastly, let us consider the equation .

Keeping the values of a, b, and c constant (here a=b=c=1), we can change k to see its affect on the graph.

For k=0, the unit circle is graphed:

For k=1, the line y = -x + 1 is graphed.

k=2:

k=3:

k=4:

As k increases in magnitude, the number of "branches" increases.

Experiment with your own values of k here.

Now (while holding k=4) we will consider how changes in a, b, and c affect the graph.

c=2                                                                                          c=4

As c increases, the graph is dilated from the center and the graph intersects the axes at the c-values.

a=2                                                                                          a=4

As a increases, the graph is shrunk toward the center.  The intercepts appear to be 1/a on each axis.

b=2                                                                                          b=4

As b increases, the asymptotes to the curve get closer together.  The intercepts are unchanged.