*Polar
Equations*

* *

*By Courtney
Cody*

In this assignment, we will investigate the following polar equations for different
values of *p*:

for *k* > 1, *k *= 1, and *k*<1.

LetÕs begin by
considering just the first polar equation, , for different values of *p and k.*

First weÕll
consider the general case of this polar equation *p = k* = 1:

By studying the
graph, we first notice that our equation for *r* is not a function since it does not pass
the vertical line test. It does,
however, look like a sideways parabola opening up to the right. To confirm this observation, letÕs
observe the graph when *k*
= 1 and *p = *4.

Once again, we
see that the graph when *k* = 1 and *p*
is varied is a parabola, only this second graph is wider. Hence it appears that *p* is a scalar and that the *k* = 1 causes the graph to be a
parabola. Thus, the equation, , where *k *= 1 and *p*
is varied is the polar equation of a parabola.

Next weÕll
consider the case of this same polar equation *k* < 1 and *p* is varied. Consider the following graphs for *k* = 0.5, *p* = 1; *k* = 0.5, *p* = ; and *k* = 0.5, *p*
= -2, respectively.

By observing the
graphs when *k* < 1
and *p* is varied we see
that it is an ellipse. Another aspect
of the graph that we can see is the furthest distance a point on the ellipse is
away from the origin is *p* units. Additionally
the y-intercepts are always .

Once again weÕll
consider this same polar equation but for the case when *k* > 1 and *p* is varied. Consider the following graphs for *k* = 1.1, *p* = 1; *k* = 1.1, *p* = ; and *k* = 1.1, *p*
= -2, respectively.

By observing the
graphs when *k* >1
and *p* is varied we see
that it is a hyperbola and its asymptotes.

Now we will
consider the second polar equation, , for different values of *p and k.*

First weÕll
consider the general case of this polar equation *p = k* = 1:

By studying the graph,
we notice that it looks like a sideways parabola opening up to the left. To confirm this observation, letÕs
observe the graph when *k*
= 1 and *p = *4.

Once again, we
see that the graph when *k* = 1 and *p*
is varied is a parabola, only this second graph is wider. Hence it appears that *p* is a scalar and that the *k* = 1 causes the graph to be a
parabola. Thus, the equation, , where *k *= 1 and *p*
is varied is the polar equation of a parabola.

Next weÕll
consider the case when *k*
< 1 and *p* is varied
for the second polar equation.
Consider the following graphs for *k* = 0.5, *p* = 1; *k* =
0.5, *p* = ; and *k* = 0.5, *p*
= -2, respectively.

By observing the
graphs when *k* < 1
and *p* is varied we see
that it is an ellipse similar to the one for the first polar equation. In addition, we can see the furthest
distance a point on the ellipse is away from the origin is once again *p* units and the y-intercepts are always .

Seeing as how the
conic sections have been of the same type for the different polar equations
thus far, what do we expect for the case when *k* > 1 and *p* is varied? For the previous polar equation we graphed a hyperbola. To test our hypothesis, weÕll consider
the following graphs for *k* = 1.1, *p*
= 1; *k* = 1.1, *p* = ; and *k* = 1.1, *p*
= -2, respectively.

By observing the
graphs when *k* >1
and *p* is varied we see
that it is a hyperbola and its asymptotes, as we expected!

For the next two
polar equations, we will investigate the effect on the graphs from replacing with . We will begin
by considering the second polar equation, , for different values of *p and k.*

First weÕll
consider the general case of this polar equation *p = k* = 1:

By studying the
graph, we notice that it looks like a parabola opening up. This is the same parabola as in the
first polar equation only rotated 180 degrees. To confirm this observation, letÕs observe the graph when *k* = 1 and *p = *4.

Once again, we
see that the graph when *k* = 1 and *p*
is varied is a parabola, only this second graph is wider. Hence it appears that *p* is a scalar and that the *k* = 1 causes the graph to be a
parabola. Thus, the equation, , where *k *= 1 and *p*
is varied is also a polar equation of a parabola.

Next weÕll
consider the case when *k*
< 1 and *p* is varied
for the third polar equation. Consider
the following graphs for *k* = 0.5, *p*
= 1; *k* = 0.5, *p* = ; and *k* = 0.5, *p*
= -2, respectively.

By observing the
graphs when *k* < 1
and *p* is varied we see
that it is an ellipse that is elongated vertically. In addition, we can see the furthest distance a point on the
ellipse is away from the origin is once again *p* units. In this case, the x-intercepts are always .

Seeing as how the
conic sections have been of the same type for the different polar equations
thus far, what do we expect for the case when *k* > 1 and *p* is varied? For the previous polar equations we graphed a hyperbola
opening to the left and right.
Since our ellipse rotated 180 degrees when we replaced with , we expect our hyperbola to rotate similarly. To test our hypothesis, weÕll consider
the following graphs for *k* = 1.1, *p*
= 1; *k* = 1.1, *p* = ; and *k* = 1.1, *p*
= -2, respectively.

By observing the
graphs when *k* >1
and *p* is varied we see
that it is a hyperbola opening upward and downward and its asymptotes, as we
expected!

For the fourth
polar equation, we will consider the second polar equation, , for different values of *p and k.*

First weÕll
consider the general case of this polar equation *p = k* = 1:

By studying the
graph, we notice that it looks like a parabola opening downward. This is the same parabola as in the
second polar equation only rotated 180 degrees. To confirm this observation, letÕs observe the graph when *k* = 1 and *p = *4.

Once again, we
see that the graph when *k* = 1 and *p*
is varied is a parabola, only this second graph is wider. Hence it appears that *p* is a scalar and that the *k* = 1 causes the graph to be a
parabola. Thus, the equation, , where *k *= 1 and *p*
is varied is also the polar equation of a parabola opening downward.

Next weÕll
consider the case when *k*
< 1 and *p* is varied
for the fourth polar equation.
Consider the following graphs for *k* = 0.5, *p* = 1; *k* =
0.5, *p* = ; and *k* = 0.5, *p*
= -2, respectively.

By observing the
graphs when *k* < 1
and *p* is varied we see
that it is an ellipse that is elongated vertically. In addition, we can see the furthest distance a point on the
ellipse is away from the origin is once again *p* units. In this case, the x-intercepts are always .

Using the
patterns we have observed for the different polar equations thus far, what do
we expect for the case when *k* > 1 and *p* is varied for the fourth polar equation? For the previous polar equations we
graphed a hyperbola opening to the left and right. Since our ellipse rotated 180 degrees when we replaced with , we expect our hyperbola to rotate similarly. To test our hypothesis, weÕll consider
the following graphs for *k* = 1.1, *p*
= 1; *k* = 1.1, *p* = ; and *k* = 1.1, *p*
= -2, respectively.

By observing the
graphs when *k* >1
and *p* is varied we see
that it is a hyperbola opening upward and downward and its asymptotes, as we
expected!