1. Graphing Polar Equations
Four types of graphs:
Circle
Limacons
Rose Curves
Lemniscates

2. Symmetry in Graphs Symmetry with respect to the polar axis:
graph is symmetrical over the x-axis
cosine graphs
Symmetry with respect to л/2:
graph is symmetrical over the y-axis
sine graphs
Symmetry with respect to the pole
some r2 and Ɵ
rays and angles

3. Ways to Graph Graphing Utility Plotting points From Equation Clues
Polar
Rectangular
From Equation Clues

4. Circles r = asinƟ r = acosƟ a = stretch (a/2 = radius) Example:
sin = circle across y axis r = 2sinƟ
cos = circle across x axis

5. Limacons r = a + bsinƟ r = a + bcosƟ
Symmetry on y axis Symmetry on x axis
+ = above x axis + = right of y axis
-- = below x axis -- = left of y axis

6. Types of Limacons R Ratio a/b < 1 (sign of a a/b = 1 or b not
relevant)
a/b > 2
Shape
Inner Loop
Cardioid
Dimpled
Convex
Diagram
+ sin
+ cos

7. Limacon Clues cos/sin determines: x-axis or y-axis
ratio determines: shape of graph
a + b: stretch on main axis
a: stretch on opp. axis
a – b: lower point

8. Examples of Limacons
r = 1 + 2sinƟ r = 3 + 2cosƟ

9. Rose Curves r = acos(nƟ) r = asin(nƟ) a = stretch of petals
If n is odd, n is the number of petals
If n is even, the number of petals is n x 2
Even cos – petals across both x and y axis
Odd cos – first petal across x axis
Even sin – petals across just y axis or just in quadrants
Odd sin – first petal across y axis
Positive – first petal on positive axis
Negative – first petal on negative axis

10. Rose Curve Examples r = 4cos(5Ɵ) r = 3sin(2Ɵ) r = 4cos(2Ɵ)

11. Lemniscates
r2 = 32cos2Ɵ r2 = a2sin2Ɵ

12. Finding Maximum r Values
Maximum values of r (same as stretch on main axis!)
To solve algebraically:
Find Ɵ at sin or cos of + 1 for max value
Example: r = 4 + 2cosƟ
Max value is 6 when cosƟ = 1
Ɵ = 0 and 2Л
Example: r = 4 – 2cosƟ
Max value is 6 when cosƟ = -1
Ɵ = Л

13. Finding Zeros To find zeros of the equation:
Set equation = 0 and solve
Example: r = 1 + 2cosƟ
1 + 2cosƟ = 0
2cosƟ = -1
cosƟ = -1/2
Ɵ = 2Л/3 and 4Л/3