In this assignment, we are making discoveries about polar equations. Polar equations are based on a number of variables with the main parameters of ø, the various angles of a circle, and r, the length of the radius of the circle. A polar point is of the form (r,ø) and a polar equation graphs a series of these polar points according to the function of the equation on a polar graph.

We will take a look at four specific equations:

 

, , ,and .

In looking at these equations, we will notice the difference of the graph when p takes on different values. We will also look at the different graphs when k>1, k=1, and k<-1 in each case.


The first case we will look at is the polar equation,

.

We will use the graphing calculator software to evaluate the graph of this equation. Take a look at the graph below of the equation.

This is the graph when k=1 and p=1 from 0 to 2Pi. As we can see, the graph is of a parabola opening to the right. The graph crosses the x-axis at -1 and the y-axis at -1 and 1. What would occur if P increases?

   
 k=1, p=3; crosses x-axis at -3 and crosses y-axis at -3 and 3  k=1, p=.5; crosses x-axis at -.5 and crosses y-axis at -.5 and .5

What do you think would occur if p were negative? It turns out that the parabola is facing the other way. The parabola opens to the left. One example follows.

This is the graph when k=1, p=-2. The parabola crosses the x-axis at -2 and the y-axis at -2 and 2. The parabola crosses the y-axis at the same place whether p=-2 or p=2.

Observation: As p increases, the parabola becomes wider. The parabola crosses the x-axis at the value of p and crosses the y-axis at the positive and negative value of p.

Let's now take a look at what happens when the value of k is not 1 for the graph of

.

   
 graph when p = 1, k = 2; hyperbola that crosses the y-axis at -2 and 2  graph when p = 1, k = -2; hyperbola that crosses the y-axis at -2 and 2

When k > 1, the graph is a hyperbola that crosses the y-axis at the positive and negative value of k. When k< -1, the graph is a hyperbola that crosses the y-axis at the positive and negative value of k. When k>1 and k=a you get the same graph as when k< -1 and k= -a. When -1<k<1, some changes occur in the graph. For instance, look at the following graph where k = -.75 and k = .75.

   
 k=.75  k=-.75

When -1<k<1, the graph is an ellipse. The two graphs are the same for k = a and k = -a.

Let's now look at the graph of

in purple and

in yellow

on the same coordinate axis. This is the graph when p = 1 and k = 1.

We notice that is a parabola the same as the graph of only reflected over the y-axis. The value of p affects both equations in the same way. As P increases, the parabola becomes wider. The graph crosses the y-axis at the same positive and negative values.

Looking at the equation

,

when k>1 or k< -1, the graph is of two hyperbolas. The foci are at the negative of the foci for the equation

.

.

Let's turn our attention to the graph of the equations

in purple and in yellow. The graph below is when p = 1 and k = 1.

These graphs are parabolas crossing the x-axis at the same place and crossing the y-axis at the negative value of each other. The equation gives a graph that opens downward while the equation gives a graph that opens upward. When P is increased, the opening of the graph becomes wider and when P is decreased, the graphs become slimmer.

What happens in the previous graph when k does not equal 1? The graph below is the case when k = 2. The same graph occurs when k = 2 and k = -2.

The purple graph is of the equation and the yellow graph is of the equation when p = 1 and k = 2. The graph of crosses the y-axis at the negative values of where crosses the y-axis. When using the same p and k in each equation as above, the two graphs cross the x-axis at the same place.

Conjectures: The graphs of the four equations are parabolas when k = 1, hyperbolas when k<-1 or k> 1 and ellipses when -1<k<1. The graphs of the sine and cosine function in these equations are closely related only the cosine graphs open left and right and the sine graphs open upward and downward. P changes the width of the graphs. The graphs of the two sine equations are very closely related as are the graphs of the two cosine equations as was mentioned throughout the discussion above.

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