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 “t” as our ranging variable for rotations in parametric as we have seen before, polar equations use “theta q 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 0…2p.  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 “p.”  Also, we will ask the same number of questions in different ways, “k.”


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.  


Let’s 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.