Investigations of Polar Equations


Jennifer Shea

Problem: To investigate several graphs of polar equations to determine what polar coordinates are and what information they provide us with.

Investigation/ Exploration:

The polar coordinates are :


where r is known as the radical coordinate


is known as the angular coordinate

Polar coordinates can be related back to the Cartesian coordinates (x,y) in the following manner:

The angular polar coordinate is equivalent to the inverse tangent of y/x in the cartesian coordinates


I began my investigation into polar equations by graphing:

for several different values of a.

This graph is for a = 1.

This graph measures a distances r from the origin and then measures the angle counterclockwise from the x-axis at 0 degrees. Since in this case my range for theta is from 0 to 15, I am taking graphs form 0 degrees to a little less than 5 pi. Notice that the number of spirals accounts for our larger range of theta. Each time one goes completely around the x-axis he has traveled 2pi radians.


For a = 2, notice that this graph expands about the vertical axis:

This makes sense. Since the range of theta is from 0 to 15 (a little less than 5 pi), and a=2, a is going to cross the x - axis at the following points, 4pi, and 8pi. Notice that r, in this case becomes larger.

Notice you can find the length of r, by forming a right triangle and using the pythagorean theorem:

As theta gets larger the distance r from the origin of the graph appears to be larger.

Notice that if a is negative, in this case a = -1, then the graph of:

looks as follows:

Notice that the graph is almost the same as when a =1 except it goes in the opposite direction. The graph starts at the third quadrant instead of the first quadrant. This makes sense because as the angle of theta will now be negative when you have traveled a 2pi, or one full distance from two points on the x-axis.

Click HERE to see a graph as a changes from -3 to 3, while the range of theta remains from 0 to 15.

I then investigated the graph:

when a= 1, for the range of theta from 0 to 15 the graph looks as follows:

Notice, that this graph does not start at the origin, because when theta is equal to 0, r=1/0.

Notice also in this case the larger theta is the smaller r is and the smaller theta is the larger r is. This makes sense.

Notice what happens as I change a to 2. The graph looks as follows:

This graph makes sense because 2 over a very small number is larger than one over a very small number so this graph appears to stretch out along the vertical axis.

Click HERE to view this graph as a ranges from -3 to 3.

Notice that these graphs begin in quadrants II and IV instead of I and III.

Another intersting graph to investigate is :

Here is this graph for theta ranging from 0 to 6.28 (about 2 pi):

Notice that if I zoom out far enough this appears to be part of a line segment.

This makes sense because if I I solve my equation for 1, I obtain the following:

If I expand this I obtain:

Recall that in Cartesian coordinates this is equivalent to:

Looking at this graph one can see that my graph is a line segment of the line y-x = 1:

It makes sense that it would be a line segment of this graph because:

is restricted from 0 to 2pi, in this case.

Now, I would like to continue my investigations of polar graphical representations by looking at the following:

The graph looks as follows:

This is very interesting I wonder how the following graph will compare with this:

Click HERE to see all three graphed together.

Click HERE to see as graph varied for n from -3 to 3 with theta ranging from 0 to 15.

This does appear to give the following information:

The closer you get to 0, the more the graph comes down into a spiral and finally into a circle at 0, as the graph gets farther away from 0, the graph appears to form more lines.

As n gets larger the more it appears that the graph is tracing along the length of the radius from the origin in several directions..

I now want to investigate this graph of the following polar equation:

This is the graph as theta ranges from 0 to 2pi.

Click HERE to see an animation for:

as n ranges from -3 to 3.

Now look at the graph of the following equation:

Here is what the graph looks like:

Notice that this graph is a circle. Theta in this case is ranging from 0 to 2pi.

Click HERE to see this graph as n ranges from -3 to 3.

That this is the graph of a circle makes a lot of sense. This makes sense because the graph of :

is a circle with that goes through the origin and the point (2,0)

When I subtract one from this it simply translates the circle. This graph makes sense because in looking at the graph of:

I can by looking at specific values of theta determine that this must be a circle. This must be a circle because as various values of theta are calculated the graph traces around the circle, and as theta crosses the y-axis it is mapped back into the first or the third quadrant.

Here is the graph of:

Now look at the graph of the following equation:

The asymptotic line is graphed as part of this curve due to the parametric nature of the polar coordinates. Notice that the secant function is 1 over the cosine function.

Note that if I graph the following:

with the other graph I obtain a graph that looks like the follwing:

These graphs give portions of a four leaf rose.

What happens if I add the following two graphs:

This does not give the other part of a four leaf rose.

If I graph the following two equations:

Notice that in order to get the other two parts of the four leaf rose, the following equations must be graphed:

It is interesting here to note that the other parts of the four leaf rose are obtained by taking the opposite of my two equations.

Click HERE to see the graph of the equation :

for theta ranging from 0 to 2pi, and n ranging from -3 to 3.

I know want to investigate the graph of the following equation:



Notice that this graph looks similar to the graphs of this following two equations:



This seems to make some sense because I do notice that I have a cosine function and sine function in the denominator of my equation, 1/cos(x) and 1/sin(x) are my secant and cosecant functions.

The four leafs of this rose are formed by graphing the follwing equations:





Notice that this involves looking at the opposite of two of these graphs as well.

I also looked at the graph of the following equation:



Here is what the graph looks like:



This graph appears to be related to the graphs of the other curves we have been investigating previously.

What if I replace theat by some f(theta) such as two theta, .5 theta, or 3 theta in some of the graphs that I have recently looked at?

Click HERE to see an animation of the graph:


as n ranges from 0 to 8.

This is really interesting because the graph seems to go from something very simply to something very complex.

Click HERE to see an animation of the graph:


as n ranges from -5 to 5. Note that as the absolute value of n gets larger the number of roses in my graph gets larger. This occurred with the other graph as well.

Click HERE to see an animation of the graph:


as n ranges from -5 to 5. Notice that this does the same thing as well. There does appear to be a pattern here.

This does appear to make sense because the greater n is the farther it is in between two plotted points for r, so in these cases for the same range of theta more points or roses will be plotted.

These graphs and discussions represent my investigation into graphing polar equations.

Author & Contact:

Jennifer Shea

e-mail me