Thomas Earl Ricks

Mathematics Education

Assignment # 11

**ÒPolar EquationsÓ**

Assignment #11

Investigation #3

Investigate with different values of p:

For k > 1, k =1, k < 1.

Note: The
parameter k is called the ÒeccentricityÓ of these conics. It is usually called ÒeÓ but for many
software programs e is a constant and cannot be set as a variable.

We will begin our investigations by looking at all four
graphs in unison for a set value of p, and varying the eccentricity parameter
k. Then we will investigate one
graph in more depth.

Setting p = 1 and k = 1 yields:

Notice that each appears to be a parabola, with the
sine equations being up and down facing parabolas, and the cosine graphs left
and right.

Dropping k < 1 yields ellipse shapes of various
kinds.

For p = 1 and k = .7 we get:

Dropping k even further has the ellipses approach a
circle.

For p =1 and k = .3 we get:

Thus as k approaches zero, the graphs not only shrink
in size, but approximate a circle more and more. Let us zoom in to observe the more circular shape:

What would you predict to happen if we raised k above
1?

If you guessed a hyperbola, you are right!

We will observe just the first graph as it will be
confusing to interpret four overlapping hyperbolas all at once.

If p = 1 and k = 1.3 we get:

As we increase k, the hyperbola becomes more and more
ÒsteepÓ as in the example below where k = 2

We could observe the change in the shape of the graphs
as k changes from below 1 to above one in the graph below:

Or if you wish to see a movie as the value of k varies
between .1 and 2, click here. The graph with k =1 is given in purple
as a reference graph.

Now let us investigate what happens to a specific
graph as p is varied. Can you
create a rough guess of what you think will happen? Will it shift, expand, contract?

Let us one polar equation and vary p (holding k
constant at k = 1):

We will graph various values for p. When we do this we observe:

It appears that the p values changes the ÒsteepnessÓ of
the graph, at least for a parabola.
What about for an ellipse?
Thus for a value of k < 1, say k = .5, we vary the p value for
several graphs and obtain:

It appears that the right side of the ellipse goes
through the p value on the x-axis.

Can you explore hyperbolas and determine what varying
p does to them?

Have fun!

For more information on eccentricity, click here.

For more information on how to derive these polar
equations from the rectangular conic formulas, click here.