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Graphing Polar Equations
Line Equation (when the equation is EQUAL to an answer) ΞΈ = b r cos ΞΈ = b r sin ΞΈ = b Circle Equation (when the radius is given or needs to be solved) r = 3 r = + 2a cos ΞΈ where a is the center r = + 2a sin ΞΈ where a is the center 12/4/2018 5:03 PM Polar Coordinates
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Example 9 Graph π½= π
π 12/4/2018 5:03 PM Polar Coordinates
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Example 10 Graph ππ¬π’π§π=βπ 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph πππ¨π¬π=π 12/4/2018 5:03 PM Polar Coordinates
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Example 11 Graph r = 3 12/4/2018 5:03 PM Polar Coordinates
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Example 12 Graph π=βππ¬π’π§π 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph π=πππ¨π¬π 12/4/2018 5:03 PM Polar Coordinates
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Calculus AP/Dual, Revised Β©2013
Graphing Polar Section 11.5 Calculus AP/Dual, Revised Β©2013 12/4/2018 5:03 PM Polar Coordinates
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Forms Introduced Polar Coordinates: (r, ΞΈ)
Rectangular/Standard Form: (x, y) 12/4/2018 5:03 PM Polar Coordinates
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Goes through the origin Radius measured from the origin
Type Graph Rules Equation Example Graphic Organizer Line Goes through the origin π½=π π ππ¨π¬ π½=π π π¬π’π§ π½=π π½= π
π , π«ππ¨π¬π½=π, π«π¬π’π§π½=βπ Circle Radius measured from the origin π=π π=Β±ππ ππ¨π¬ π½ π=Β±ππ π¬π’π§ π½ π=π, π«=π ππ¨π¬ π½, π«=βπ π¬π’π§ π½ Cardioid (Heart-Shaped) π π =π Generally, a = b π=πΒ±π ππ¨π¬ π π=πΒ±π πππ π π=π+π π¬π’π§ π½, π=π βπ ππ¨π¬ π½ LimaΓ§on (Inner Loop) π π <π βLess = Loopβ π=πΒ±π ππ¨π¬ π π=πΒ±π πππ π π=π+π π¬π’π§ π½, π=π βπ ππ¨π¬ π½ LimaΓ§on (Dimpled) π< π π <π βDimple Betweenβ π=πΒ±π ππ¨π¬ π π=πΒ±π πππ π π=π+π π¬π’π§ π½, π=π βπ ππ¨π¬ π½ LimaΓ§on (Convex) π π >π βDimple Betweenβ π=πΒ±π ππ¨π¬ π π=πΒ±π πππ π π=π+π π¬π’π§ π½, π=π βπ ππ¨π¬ π½ 12/4/2018 5:03 PM Polar Coordinates
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Graphic Organizer Type Graph Rules Equation Example Rose (cos nΞΈ)
when n = odd n is odd, n pedals Cos: 1st pedal is at 0 radian π=π ππ¨π¬ ππ½ π=π ππ¨π¬ ππ½ Rose (cos nΞΈ) when n = even n is even, 2n pedals Cos: 1st pedal is at 0 radian π=π ππ¨π¬ ππ½ π=π ππ¨π¬ ππ½ Rose (sin nΞΈ) when n = odd n is odd, n pedals Sin: 1st pedal is at π
ππ radian π=π π¬π’π§ ππ½ π=π π¬π’π§ ππ½ Rose (sin nΞΈ) when n = even n is even, 2n pedals Sin: 1st pedal is at π
ππ radian π=π π¬π’π§ ππ½ π=π π¬π’π§ ππ½ Lemniscates (cos 2ΞΈ) n is odd, n pedals Cos: loops are at 0 and Ο π π = π π ππ¨π¬ ππ π π =π ππ¨π¬ ππ Lemniscates (sin 2ΞΈ) n is even, n pedals Sin: loops are at Ο/4 and 5Ο/4 π π = π π π¬π’π§ ππ π π =π π¬π’π§ ππ 12/4/2018 5:03 PM Polar Coordinates
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Graphic Organizer Type Graph Rules Equation Example Archimedes Spiral
when a = positive a = positive CLOCKWISE π=π π½ π=ππ½ Archimedes Spiral when a = negative a = negative COUNTER CLOCKWISE π=βπ π½ π=βππ½ 12/4/2018 5:03 PM Polar Coordinates
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Example 1 Graph r = 3 + 3 sin ΞΈ r 3 6 r ΞΈ Ο/2 Ο 3Ο/2 2Ο
r ΞΈ Ο/2 Ο 3Ο/2 2Ο 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph r = 2 + 2 cos ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο 12/4/2018 5:03 PM
Ο/2 Ο 3Ο/2 2Ο 12/4/2018 5:03 PM Polar Coordinates
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Example 2 Graph r = 2 + 4 sin ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r 2 6 β2
Ο/2 Ο 3Ο/2 2Ο r 2 6 β2 12/4/2018 5:03 PM Polar Coordinates
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Example 3 Graph r = 4 + 2 cos ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r 6 4 2
Ο/2 Ο 3Ο/2 2Ο r 6 4 2 12/4/2018 5:03 PM Polar Coordinates
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Example 4 Graph r = 4 + 3 cos ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r 7 4 1
Ο/2 Ο 3Ο/2 2Ο r 7 4 1 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph r = 1 β 2 cos ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r 1 β3
Ο/2 Ο 3Ο/2 2Ο r 1 β3 12/4/2018 5:03 PM Polar Coordinates
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Example 5 Graph r = 4 cos 3ΞΈ r ΞΈ 2Ο/3 Ο 4Ο/3 2Ο r 4 β4
2Ο/3 Ο 4Ο/3 2Ο r 4 β4 12/4/2018 5:03 PM Polar Coordinates
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Example 6 Graph r = 5 cos 2ΞΈ 12/4/2018 5:03 PM
Polar Coordinates
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Your Turn Graph r = 4 cos 2ΞΈ 12/4/2018 5:03 PM
Polar Coordinates
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Example 7 Graph r2 = 16 cos 2ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r 4 β4
Ο/2 Ο 3Ο/2 2Ο r 4 β4 12/4/2018 5:03 PM Polar Coordinates
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Example 8 Graph r2 = 16 sin 2ΞΈ r ΞΈ Ο/4 Ο 5Ο/4 2Ο r 4 β4
Ο/4 Ο 5Ο/4 2Ο r 4 β4 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph r2 = 9 sin 2ΞΈ 12/4/2018 5:03 PM
Polar Coordinates
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Example 9 Graph r = 2ΞΈ r ΞΈ Ο/2 Ο 3Ο/2 2Ο r Ο 2Ο 3Ο 4Ο
Ο/2 Ο 3Ο/2 2Ο r Ο 2Ο 3Ο 4Ο 12/4/2018 5:03 PM Polar Coordinates
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Your Turn Graph r = β2ΞΈ 12/4/2018 5:03 PM Polar Coordinates
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Assignment Worksheet 2 12/4/2018 5:03 PM Polar Coordinates
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