Graphing Polar Equations - Calculus 2

Graphing polar equations is similar to graphing rectangular equations, but is also different in many ways. There are very basic polar equations that represent the graphs of lines and circles which are also commonly represented by rectangular equations, but then are some special polar equations that represent graphs of more complicated shapes that are very uncommon to see when graphing rectangular equations. These graphs include limaçons, rose curves, and lemniscates.

To graph more complicated polar equations, there are mainly two ways to go about it. One way will allow you to graph pretty much any polar equation, while the other way is more situational and rewards those who like memorization.

The first way to graph a polar equation is to determine if the equation is symmetric about the polar axis or the vertical axis of θ=π/2 (also sometimes symmetry about the pole/origin), and then use any symmetry to help plot polar coordinates. The plotted polar coordinates can then be connected to create the graph of the polar equation. Determining symmetry is a crucial part of the process as it will allow for you to plot less points and graph the polar equation more efficiently than if you were to try and graph it without making use of any symmetry.

The second way to graph a polar equation only applies to certain forms of polar equations, as this way involves being able to recognize a particular polar equation as representing a certain type of special polar graph. These special graphs include the four types of limaçons (inner loop, cardioid, dimple, and convex), rose curves, and lemniscates.  If you can remember what the polar equations for these special polar graphs look like, you can very quickly sketch the graph of these equations without the need for determining symmetry or plotting a bunch of polar coordinates. There is no easy "trick" to memorizing these polar equations and their corresponding graphs, but as you become more familiar with them in practice you may find that recognizing the form of polar equations is not as daunting of a task as it originally seems.

We discuss all of this and more in this new lesson of Calculus 2. Let's get to it!

In this lesson, you will learn:

• How to graph basic polar equations
• How to graph polar equations by plotting points & using symmetry
• How to quickly graph polar equations that represent special polar graphs

I hope you find this video to be helpful!

-Josh