Topic: Complex, Parametric, and Polar Forms

Subtopic: Polar Equations and Graphs

**Overview**

Ready to learn a whole new way to graph? Instead of graphing functions in the form y=f(x), in this lesson we graph polar equations. __Polar equations__ which are (usually) given as r as a function of theta, r=r(θ). These equations relate r, the distance from the origin, to the central angle theta. As we increase theta, we spin around the origin, and while doing so the distance from the origin to the curve can change essentially pushing the curve away from the origin or pulling it in toward the origin.

You can graph polar curves electronically or manually on "polar graphing paper". Check if your grapher can graph polar equations. If you are using a graphing calculator, go to MODE and look for a graphing option called POL or POLAR. Also, you usually want to be in RADIAN mode, but it depends on your window settings. Now go to the area where you enter equations to be graphed. Instead of seeing y1, y2, etc., you should see r1, r2, etc. You are ready to go!

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 9.5.1 Convert rectangular points to polar points, and visa versa
- 9.5.2 Convert rectangular equations to polar equations, and visa versa using
**`x=rcostheta`, `y=rsintheta`, `r=sqrt(x^2+y^2)`, `theta=tan^-1(y/x)`** - 9.5.3 Plot polar points on the polar plane
- 9.5.4 Manually graph basic polar equations on the polar plane including lines and circles
- 9.5.5 Electronically graph polar equations including ellipses, lemniscates, cardioids, limacons, roses, and spirals
- 9.5.6 Identify a given polar curve as a circle, ellipse, lemniscate, cardioid, limacon, or rose

**Terminology**

Terms you should be able to define: polar plane, polar point, polar equation, lemniscate, cardioid, limacon, rose

Common Polar Curves is a simple sheet from Prof. Kate Niedzielski, Univ. of Minnesota.

**Supplementary Resources**

The four formulas (Objective 9.5.2 above) that are used to convert between polar and rectangular coordinates are connected to each other via a right triangle. Here is a sketch from MathWords:

www.mathwords.com/p/polar_rectangular_conversion_formulas.htm

Optional reading with plenty of "how to graph" examples: Graphing Five Classic Polar Curves

An interesting application of polar graphs: Polar Coordinates and Cardioid Microphones