Today in Precalculus
•Go over homework•Notes: Graphs of Polar Equations•Homework
Rose Curves1. r = 4sin3θ 2. r = 4cos3θ
3. r = 4sin2θ 4. r = 4cos2θ
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Conclusions General form of the equations:
r = acosnθr = asinnθ
Number of petals: if n is odd, n petalsif n is even, 2n petals
Position of petals: cos: if n is odd, one petal on positive x-axis if n is even, petals on each axissin: if n is odd, one petal on half of y-axis if n is even, no petals on axis
Length of petal is a
Limaçon Curves5. r = 2 – 3sinθ 6. r = 2 + 2sinθ
7. r = 3 + 2cosθ 8. r = 2 – 1cosθ
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Conclusions General form of the equations:
r = a ± bcosθ
r = a ± bsinθ
when 1, there is an inner loopa
b
when 1, touches the origin "cardioida
b
when 1 2, called a "dimpled limacon"a
b
when 2, it is a "convex limacon"a
b
Analyzing polar graphs• The domain is the set of possible inputs for • The range is the set of outputs for r. The
domain and range can be read from the “trace” or “table” features on your calculator.
• The maximum r-value is the maximum distance from the pole. This can be found using trace, or by knowing the range of the function.
• Symmetry can be about the x-axis, y-axis, or origin, just as it was in rectangular equations.
• Continuity, boundedness, and asymptotes are analyzed the same way they were for rectangular equations.
r = 4sin3θDomain: All reals
Range: [-4, 4]
Maximum r-value: 4
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
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y
r = 4cos3θDomain: All reals
Range: [-4, 4]
Maximum r-value: 4
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
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y
r = 2 – 3sinθDomain: All reals
Range: [-1, 5]
Maximum r-value: 5
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
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r = 2 +2sinθDomain: All reals
Range: [0, 4]
Maximum r-value: 4
Symmetry: y-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
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y
r = 3 +2cosθDomain: All reals
Range: [1,5]
Maximum r-value: 5
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
x
y
r = 2 – 1cosθDomain: All reals
Range: [1,3]
Maximum r-value: 3
Symmetry: x-axis
Continuity: continuous
Boundedness: bounded
Asymptotes: none
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Homework
Worksheet
Quiz Monday, April 13