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10.2 Graphing Polar EquationsDay 2
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 – πor (r, θ) with (–r, –θ)
2
EX 1: Identify the kind(s) of symmetry each polar graph possesses.
A)
Pole Polar Axis B)
Pole Polar Axis
5 10cosr
2
2 36cos 2r
2
5 10cos 5 10cos( ) 5 10cos( )
5 10cos 5 10cos
rr r
r r
2 2 2
2 2 2
( ) 36cos 2 36cos 2( ) ( ) 36cos 2( )
36cos 2 36cos 2 36cos 2
r rr
r r r
SPIRAL Also called the “Spiral of Archimedes” No special rules! Typical Graph:
r k
CIRCLES There are three forms for a circle.
Center of circle at _______
Radius = _______
Typical Graph:
Contains the _______
Tangent to _______
Center on _______
Diameter = _______
Radius = _______
If a > 0, circle is ____ of pole
If a < 0, circle is ____ of pole
Typical Graph:
Contains the _______
Tangent to _______
Center on _______
Diameter = _______
Radius = _______
If a > 0, circle is ____ of pole
If a < 0, circle is ____ of pole
Typical Graph:
r k cosr a sinr a pole
kpole
2
polar axisa
EW
polepolar axis
2 a
NS
2a
2a
EX 2: Radius:______ Center On: Polar Axis / Circle is N S E W of the pole
6cosr
2
3
LIMAÇONS French for “snail.”
OR (oriented on polar axis) (oriented on )
cos sinr a b r a b 2
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
Larger = _______
Smaller = _______
For all the “bumps,” they hit the polar axis or (whichever is the opposite of where it is oriented) at _______
Typical Graph:
Typical Graph: Typical Graph: Typical Graph:
1ab 1a
b 1 2ab 2a
b
2
b a
b aa b 2 a
a b
a b
a b
a b
±a
smaller
larger2a
a a
smaller largerinnerloop
diam
a
a
EX 3:
Type:__________________ On: Polar Axis / Lengths:_________________ _________________________
EX 4:
Type:__________________ On: Polar Axis / Lengths:_________________ _________________________
EX 5:
Type:__________________ On: Polar Axis / Lengths:_________________ _________________________
2 4cosr 2 2sinr 4 2cosr
Limaçon with Inner Loop
2 2
2
Diam: 6
Inner Loop: 2
Cardioid
Diam: 4
Convex Limaçon
Larger: 6
Smaller: 2
Oriented Oriented Oriented
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:
First peak is at _______
Peaks are ______ radians apart(n is number of petals)
cosr a b sinr a b
θ = 02n
2b
2n
2bb
a
EX 6:
Length of Petals:_______ Number of Petals:_______ First Peak at:_______
Each Petal _______ rad apart
EX 7: Length of Petals:_______ Number of Petals:_______ First Peak at:_______
Each Petal _______ rad apart
5cos 2r 4sin 3r
54
θ = 0
2
43
23
6
LEMNISCATE These look like “figure eights.”
Oriented on _______________
Oriented on _______________
Maximum distance out is ________
Typical Graph: Typical Graph:
2 2 cos 2r a 2 2 sin 2r a
polar axis 4
2a a
EX 8: Oriented On: Polar Axis / Maximum Distance:_______
EX 9: Oriented On: Polar Axis / Maximum Distance:_______
2 4cos 2r 2 4sin 2r
4 4
2 2
Ex 10: Transform the rectangular equation into a polar equation and graph.
2 2
2 2
2
6 0
6
6 sin
6sin
x y y
x y y
r r
r
CircleRadius 3Center onN of pole
2
Ex 11: Determine an equation of the polar graph.
A) B)
Equation:____________________Why?_______________________________________________________________________________
Equation:____________________Why?_______________________________________________________________________________
r = 3cos 2θpetal graph w/ 4 petalspeak on polar axispetal length is 3
Dimpled Limiçon 2 on
larger = 8 smaller = 2bump hits at 5 a = 5, b = 3
r = 5 + 3sin θ
2θ cos
a = 3
Homework
#1003 Pg 501 # 1–17 odd, 21, 23, 24, 27, 29, 31, 37, 41, 43, 44–47, 49–53 odd