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9.7 Graphs of Polar Equations Digital Lesson

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Polar Curves A Polar Curve consists of all the points (r,θ) satisfying a given equation F(r,θ) = 0. Often one can solve r from the equation and represent the polar curve in the form r = f(θ), where r represents the distance from the pole as a function of theta. Some curves are easier to describe with polar coordinates: (Circle centered at the origin) (Line through the origin)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Graphs of Polar Equations Example: Graph the polar equation r = 2cos . 123 0 2 0 –2 –1 0 1 20 r The graph is a circle of radius 1 whose center is at point (x, y) = (1, 0).

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For a polar curve that has symmetry with respect to x-axis: If (r, ) is on the graph, so is (r, – ). Examples of Polar Curves

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For a polar curve that has symmetry with respect to y-axis: If (r, ) is on the graph, so is (r, – ) or (-r, – ). Examples of Polar Curves

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Definition: Symmetry of Polar Graphs If substitution leads to equivalent equations, the graph of a polar equation is symmetric with respect to one of the following. 1. The line 2. The polar axis 3. The pole Replace (r, ) by (r, – ) or ( – r, – ). Replace (r, ) by (r, – ) or ( – r, – ). Replace (r, ) by (r, + ) or ( – r, ). Example: In the graph r = 2cos , replace (r, ) by (r, – ). r = 2cos( – ) = 2cos The graph is symmetric with respect to the polar axis. cos( – ) = cos

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Zeros and Maximum r- values Example: Find the zeros and the maximum value of for the graph of r = 2cos . 123 0 These are the zeros of r.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 You try: Analyze the graph of r = 4sin . State the symmetry, find the zeros and maximum value of.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 You try again: Analyze the graph of r = 1-2cos . State the symmetry, find the zeros and maximum value of.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 You try again: Analyze the graph of State the symmetry, find the zeros and maximum value of.

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Special Polar Graphs: Limaçon Each polar graph below is called a Limaçon. –3 –5 5 3 5 3 –3

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Special Polar Graphs: Lemniscate Each polar graph below is called a Lemniscate. –55 3 –3 –5 5 3 –3

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Special Polar Graphs: Rose Curve Each polar graph below is called a Rose curve. The graph will have n petals if n is odd, and 2n petals if n is even. –5 5 3 –3 –5 5 3 –3 a a

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9.7Classifying Polar Equations

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What do you see?

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