10.7 Graphs of Polar Equations

There are a couple different methods one can use to graph polar equations.

Point Plotting
To show how to use this method we will use the example equation of:

Because the function of sine repeats itself a full range of values can be found simply by plugging values on the range:

By plugging in these values you should find you get the following values for each angle:

But simply plotting those points you find that the graph is simply a circle with a radius of 2 and and center at (0,2)

To check the graph above is correct you can either convert the Polar Equation to Rectangular Form or graph it using a graphing calculator (Either in polar or parametric mode)

Symmetry
Symmetry is a strategy used as a shortcut of the point plotting method. By recognizing that a graph is symmetrical with respect to either the polar axis, the pole, or the line

To test if the equation is symmetrical in any of the three ways, substitute the original polar coordinates for those coordinates that would result from a translation across the respective axis or line.
The graphs of each type of symmetry are shown below:

Symmetry with respect to the Polar Axis

Symmetry with respect to the line of

Symmetry with respect to the Pole

If symmetry is discovered fewer points must be plotted to be able to draw the entire graph. Depending of the type of symmetry you can determine which range of points to start with.

Note: Although our first example graph is symmetrical the substitutions will not show so. Thus, we must remember two quick tests for symmetry
1.The graph of the equation below is symmetric with respect to the line

2. The graph of the equation below is symmetric with respect to the polar axis

Zeros and Maximum r-Values
Knowing both were r=0 and the maximum value of r can be valuable pieces of information when graphing. There are no special ways to find the maximum values of r without using your calculator. However, using a calculator you can use either the trace or table features to determine the maximum value of r. Keep in mind that zeros and maximums may be found at more than one point.

Special Polar Graphs
Limacon: