1. Section 6.4 Graphs of Polar Equations
MA.PC Graph equations in the polar coordinate plane.

2. Graphing a Polar Equation by Point Plotting
A polar equation is an equation whose variables are 𝑟 and 𝜃. The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids to graph polar equations.

3. This type of graph is called a ROSE WITH 4 PETALS.𝜽
𝒓=𝟐 cos (𝟐𝜽)
2 1 =2
𝜋 6 =1
𝜋 4
2 0 =0
𝜋 3
2 − 1 2 =−1
𝜋 2
2 −1 =−2
Let's let each unit be 1/2.

4. Classical Curves
Circle
Rose
Lemniscate
Limacon
Cardioid

5. CIRCLES
Equations of circles would look like one of the following: 𝒓=𝒂 cos 𝜽 𝒓=𝒂 sin 𝜽

6. ROSE
Equations of rose curves would look like one of the following: 𝒓=𝒂 cos 𝑛𝜃 𝒓=𝒂 sin 𝑛𝜃 Where 𝑛 even has 2𝑛 petals and 𝑛 odd has 𝑛 petals.

7. LEMNISCATE
Equations of lemniscates would like one of the following: 𝒓 𝟐 = 𝒂 𝟐 cos 2𝜃 𝒓 𝟐 = 𝒂 𝟐 sin 𝟐𝜽 These graphs will pass through the pole and are propeller shaped.
Unit is 1/4

8. LIMAÇON
Equations of limaçons would like one of the following: 𝒓=𝒂+𝒃 cos 𝜽 𝒓=𝒂+𝒃 sin 𝜽 If 𝑎 < 𝑏 there is an inner loop. If 𝑎 > 𝑏 there is an indentation.
Unit is 1/2

9. CARDIOD
Equations of cardiods would like one of the following: 𝒓=𝒂+𝒂 cos 𝜽 𝒓=𝒂+𝒂 sin 𝜽 All graphs of cardiods pass through the pole.
Unit is 1/4

10. Example
Graph the equation 𝑟=3+ cos 𝜃

11. Example
Graph the equation 𝑟= sin 2𝜃

12. Example
Graph the equation 𝑟=4(1− cos 𝜃 )

13. Example
Graph the equation 𝑟 2 = cos 2𝜃

15. Homework
Worksheet