9.7 polar graphs

equations only

test for symmetry: line θ = pi/2

replace (r, θ) with either (r, pi - θ) or (-r, -θ)

test for symmetry: the polar axis

replace (r, θ) with (r, -θ) or (-r, pi - θ)

test for symmetry: pole (origin)

replace (r, θ) with (r, pi + θ) or (-r, θ)

the graph r = f(sin(θ)) is symmetric about the ________

line θ = pi/2

the graph r = f(cosθ) is symmetric about the ________

polar axis

limacon - symmetric about the line θ = pi/2

r = a ± b(sinθ)

limacon - symmetric about the polar axis

r = a ± b(cosθ)

inner loop limacon - value of a/b

a/b < 1 (less than 1)

cardioid limacon - value of a/b

a/b = 1 (a and b are equal)

dimpled limacon - value of a/b

1 < a/b < 2 (between 1 and 2)

convex limacon - value of a/b

a/b ≥ 2 (greater than or equal to 2)

rose curve - symmetric about θ = pi/2

r = a(sin(nθ))

rose curve - symmetric about polar axis

r = a(cos(nθ))

rose curves: when n is even, there will be _____ petals

2n

rose curves: when n is odd, there will be _____ petals

n

circle - symmetric about θ = pi/2

r = a(sinθ)

circle - symmetric about polar axis

r = a(cosθ)

for r = acosθ or r = asinθ, a is the _______ of the circle

diameter

lemniscate - diagonal

r^2 = (a^2)(sin2θ)

lemniscate - horizontal

r^2 = (a^2)(cos2θ)

for a leminscate equation, a is the ________

length of one of the “petals”

limacon - when r = a + bcosθ, the majority of the graph will lie on the _______

positive x-axis

limacon - when r = a - bcosθ, the majority of the graph will lie on the _______

negative x-axis

limacon - when r = a + bsinθ, the majority of the graph will lie on the _______

positive y-axis

limacon - when r = a - bsinθ, the majority of the graph will lie on the _______

negative y-axis