10.2 Graphing Polar Equations Day 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)

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

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

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

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

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

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

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

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

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

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

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

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

Homework #1003 Pg 501 # 1–17 odd, 21, 23, 24, 27, 29, 31, 37, 41, 43, 44–47, 49–53 odd