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**10.2 Graphing Polar Equations Day 2**

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**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)

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**EX 1: Identify the kind(s) of symmetry each polar graph possesses.**

Pole Polar Axis B) Pole Polar Axis

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

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**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

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

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**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

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**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

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**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

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** 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

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**Maximum distance out is ________**

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

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

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**Ex 10: Transform the rectangular equation into a polar equation and graph.**

Circle Radius 3 Center on N of pole

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**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

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

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Polar Coordinates and Graphs of Polar Equations Digital Lesson.

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