PP Section 9.7

# Θ θ symmetry with respect to the y 45 60 75 90 105

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θ θ - .

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Symmetry with respect to the y -axis 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 ( 29 r , θ ( 29 r , π θ -
Symmetry with Respect to the y -axis: In a polar equation, replace by If an equivalent equation results, the graph is symmetric with respect to the line = 2 θ π θ θ π - . . Tests for Symmetry

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Symmetry with Respect to the Pole (Origin) 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 ( 29 r , θ ( 29 - r , θ
Tests for Symmetry Symmetry with Respect to the Pole (Origin): In a polar equation, replace by If an equivalent equation results, the graph is symmetric with respect to the pole. r r - .

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Example Test the polar equation below for symmetry and sketch its graph. NO cos - -1 r cos 1 r - r for r - : origin NO cos - 1 ) - cos( 1 r for - : axis - y YES cos 1 ) cos(- 1 r for - : axis - x cos 1 θ θ θ θ π θ θ π θ θ θ θ θ = + = = + = + = + = + = r θ cos 1 + = r Cardioid
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