9.7 Graphs of Polar Equations
Ex. 1 Graphing a Polar Equation by plotting points
Sketch the graph of r = 4 sin θ
θ=0
r=
π/6 π/4
π/3
π/2
2π/3
3π/4
5π/6
π
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You can confirm the graph found in 3 ways:
1. Convert to rectangular form: Multiply both sides by r
2. Use a polar coordinate mode on calculator
3. Convert to parametric equation and use calculator: if calculator doesn't have polar mode, you can use t as the parameter
Symmetry
1. With respect to line θ=π/2: replace (r,θ) by (r,π-θ) or (-r,-θ)
2. The polar axis: replace (r,θ) by (r,-θ) or (-r,π-θ)
3. The pole: replace (r,θ) by (r,π+θ) or (-r,θ)
You can determine the symmetry of the graph r = 4 sin θ as follows:
1. Replace (r,θ) by (-r,-θ) -r=4sin(-θ) = 4 sin θ
2. Replace (r,θ) by (r,-θ)
r = 4sin(-θ) = - 4sin θ
3. Replace (r,θ) by (-r,θ)
-r = 4sin(θ) = - 4sin θ
So the graph is symmetric to the line θ=π/2
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Ex. 2 Use symmetry to sketch the graph of r = 3 + 2 cos θ
θ = 0
π/6 π/3
π/2
2π/3
5π/6
π
r = This graph is called a limacon.
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Quick Tests for symmetry
1. The graph of r = f(sinθ) is symmetric with respect to
the line θ=π/2.
2. The graph of r = g(cosθ) is symmetric with respect to
the polar axis.
Ex. 3 p. 716 Find the maximum value of r for the
graph of r = 1- 2cosθ.
Symmetric to polar axis
Use the trace feature of your calculator to find the
maximum r-value.
r=3 occurs when θ=π. Note that (3,π) is the farthest
from the pole.
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Ex. 4 r = ecosθ ­ 2cos4θ + sin5 (θ/12)
is called the butterfly curve.
Ex. 5 Graph r = 2 cos 3θ
Special polar graphs on p. 718
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Ex. 6 r2 = 9 sin 2θ
ASSN: p. 720 1­21 odd, 23­33 odd
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