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10.2 Graphing Polar Equations Day 2

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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. A) 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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CIRCLESThere 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: pole k polar axis E W pole polar axis N S

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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 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: ±a±a smaller larger 2a aa smallerlarger inner loop diam a a

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EX 3: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ EX 4: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ EX 5: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ Limaçon with Inner Loop Diam: 6 Inner Loop: 2 Cardioid Diam: 4 Convex Limaçon Larger: 6 Smaller: 2 Oriented

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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: First peak is at _______ Peaks are ______ radians apart (n is number of petals) θ = 0 2b2b b a

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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 5 4 θ = 0 4 3

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

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EX 8: Oriented On: Polar Axis / Maximum Distance:_______ EX 9: Oriented On: Polar Axis / Maximum Distance:_______ 22

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

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

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