1. 10.2 Graphing Polar Equations Day 2

2. WRT the Polar Axis (x-axis)
Yesterday, we graphed polar equations using “brute force” – making tables of values. But this is very inefficient! We can bypass having to make all these separate calculations by learning some rules. 
Symmetry – Tests for Symmetry on Polar Graphs
If the following substitution is made and the equation is equivalent to the original equation, then the graph has the indicated symmetry.
WRT the Pole (Origin)
Replace r with –r
WRT the Polar Axis (x-axis)
Replace θ with –θ
WRT the line (y-axis)
Replace θ by θ – π
or (r, θ) with (–r, –θ)

3. EX 1: Identify the kind(s) of symmetry each polar graph possesses.
Pole Polar Axis   B)
Pole Polar Axis

4.  SPIRAL
Also called the “Spiral of Archimedes”
No special rules!
Typical Graph:

5. CIRCLES There are three forms for a circle.
Center of circle at _______
Radius = _______
Typical Graph:
Contains the _______
Tangent to _______
Center on _______
Diameter = _______
If a > 0, circle is ____ of pole
If a < 0, circle is ____ of pole
 Typical Graph:
pole
pole
pole k
polar axis
polar axis N E S W

6. EX 2:
Radius:______
Center On: Polar Axis /
Circle is N S E W of the pole 3

7. LIMAÇONS French for “snail.” OR
(oriented on polar axis) (oriented on )
Limaçon with Inner Loop
When or a < b
Diameter = _______
Inner Loop = _______
Cardioid (heart-shaped)
When or a = b
Diameter
= _______ = ______
Dimpled Limaçon
When
Larger = _______
Smaller = _______
Convex Limaçon
When or a ≥ 2b
For all the “bumps,” they hit the polar axis or (whichever is the opposite of where it is oriented) at _______
Typical Graph: ±a a 2a
larger a
diam
smaller
larger
inner
loop
smaller a a

8. Limaçon with Inner Loop Cardioid Convex Limaçon
Type:__________________
On: Polar Axis /
Lengths:_________________
_________________________
EX 4:
On: Polar Axis /
EX 5:
Limaçon with Inner Loop
Cardioid
Convex Limaçon
Oriented
Oriented
Oriented
Diam: 6
Diam: 4
Larger: 6
Inner Loop: 2
Smaller: 2

9. ROSES These look like flowers…we call each loop a “petal.”
Length of each petal = ______
If b is even, there are _______ petals.
If b is odd, there are _______ petals.
(*Since the values from 0 to 2π give us the points, having b be an odd number, the values actually repeat themselves and overlap the already existing values so we do not get double the number of petals like we do with b being even.)
First peak is at _______
Peaks are ______ radians apart
(n is number of petals)
Typical Graphs: a 2b b
θ = 0

10. 5 4 4 3 θ = 0 EX 6: Length of Petals:_______ Number of Petals:_______
EX 6:
Length of Petals:_______
Number of Petals:_______
First Peak at:_______
Each Petal _______ rad apart
EX 7: 5 4 4 3
θ = 0

11. Maximum distance out is ________
LEMNISCATE These look like “figure eights.”
Oriented on _______________
Maximum distance out is ________
Typical Graph:
polar axis

12. EX 8:
Oriented On: Polar Axis /
Maximum Distance:_______   
EX 9: 2 2

13. Ex 10: Transform the rectangular equation into a polar equation and graph.
Circle
Radius 3
Center on
N of pole

14. Ex 11: Determine an equation of the polar graph.B)
r = 5 + 3sin θ
Equation:____________________
Why?_______________________
____________________________
r = 3cos 2θ
Equation:____________________
Why?_______________________
____________________________
petal graph w/ 4 petals
 2θ
Dimpled Limiçon on
peak on polar axis
 cos
larger = 8
smaller = 2
petal length is 3
 a = 3
bump hits at 5
a = 5, b = 3

15. Homework
# Pg # 1–17 odd, 21, 23, 24, 27, 29, 31, 37, 41, 43, 44–47, 49–53 odd