**Visiting Polar
Equations**

**By: Damarrio Holloway**

**Summer 2006**

Assignment 11

Problem #3

We have just recently visited with parametric equations and
now we will go next door and introduce the Polar Equation family. Here, we will specifically investigate
with different values of **p** for the following
polar equation family:

The characteristics of this family are as follows:

*Figure 1*

My initial graph displays a series of parabolas that all
intersect with each other. The set
parameters for my p-values and k-values are equal to 1. Instead of Ò**t**Ó as our ranging variable for rotations in
parametric as we have seen before, polar equations use Ò**theta ****q****Ó** as our variable. Since we are using sin and cosine functions, our values will
be in terms of p. In order
for our functions to graph a complete rotation and maintain their sanity, we
need their parameters set from 0É2p. If not, they will only be half as sane
as displayed below:

We will now conduct a psychological experiment to test how
they answer a specific number of questions, called Ò**p**.Ó
Also, we will ask the same number of questions in different ways, Ò**k**.Ó

As we saw in figure 1, the family has a normal response to one question. They do seem to have a distinct, yet common way of response in the shape of a parabola, illustrating the distances from the diretrix. Together, their answers generate a locus points on the conic, which will give me the information I need to determine their sanity.

LetÕs explore their reaction when asked 1 question different ways.

1 Question asked at .5 speed, k<1. We see here that the sin ellipse (blue and green) intersect the x-axis at 0.5 and the cosine functions intersect the x-axis at 1, which are their foci.

1 Question asked at .3 speed, k<1

Asked 1 question twice as fast, where k=2, the family yields hyperbolic answers. Here, their focus is illustrated by their respective asymptote.

When asked 1 question 4 times as fast, where k>1.

When asked 2 questions at normal speed, p =+2, the parabolas widen, yielding further foci points and directrix. This tells me that their minds are opening up and their answers are becoming more thorough.

When asked 2 questions backwards at normal speed, p = -2, their total train of thought was completely reversed.

After interviewing this family my results showed that they are a normal, close-nit family, and that I am completely insane.

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