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

Try
your own here.

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.