March 25, 2015
(6.5) Graphs of Polar Equations
Objective: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Why: Graphs that have circular symmetry often have simple polar
equations, that can be very helpful in calculus.
March 25, 2015
y = cos x
y = sin x
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
r = cos θ
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
r = sin θ
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
r = 3cos θ
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
Algebraic Relationship
r = cos θ
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
r = sin θ
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Symmetry:
1. line θ =
π
(r,θ)
2
2. polar axis
(r,θ)
3. the pole
(r,θ)
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Use polar symmetry tests to determine if the graph is
symmetric about the x-axis, the y-axis, or the origin.
r = 4 sin 3θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Maximum r-value: value occurring at a point the maximum
distance from the pole.
*To find: find values of |r| as opposed to the directed
distance r.
Find the maximum r-value of the following graphs.
1. r = 3 + 2cos θ
2. r = 4 - sin θ
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Analyze the graph of the polar curve.
r = -3 cos 4θ
Domain:___________________
Range:____________________
Continuity:_________________
Boundedness: ______________
Symmetry: ________________
Maximum r-value: ___________
Asymptotes:_______________
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
Rose Curves:
r = a cos nθ
r = a sin nθ
n is even: 2n petals
n is odd: n petals
where n is an integer > 1
Limacon Curves: r = a -+ b cos θ
r = a -+ b sin θ
where a > 0, b > 0
Lemniscates:
r2 = a2sin 2θ
r2 = a2cos 2θ
Spiral of Archimedes:
r=θ
r-value max = a + b
March 25, 2015
Obj: To learn about polar curves, symmetry, rose curves,
limacon curves, and lemniscates.
HW:
(HR): (6.5) Pg.500: 1, 2, 9-13, 17, 25, 27, 29