Graphing Polar Equations

Four types of graphs:

Circle

Limacons

Rose Curves

Lemniscates

Symmetry in GraphsSymmetry with respect to the polar axis:• graph is symmetrical over the x-axis• cosine graphsSymmetry with respect to л/2:• graph is symmetrical over the y-axis• sine graphsSymmetry with respect to the pole• some r2 and Ɵ• rays and angles

Ways to Graph

• Graphing Utility

• Plotting points– Polar – Rectangular

• From Equation Clues

Circles

r = asinƟ r = acosƟ

a = stretch (a/2 = radius) Example:

sin = circle across y axis r = 2sinƟ

cos = circle across x axis

Limacons

r = a + bsinƟ r = a + bcosƟ

Symmetry on y axis Symmetry on x axis

+ = above x axis + = right of y axis

-- = below x axis -- = left of y axis

Types of LimaconsRRatio a/b < 1

(sign of a

a/b = 1

or b not

1 < a/b < 2

relevant)

a/b > 2

Shape Inner Loop

Cardioid Dimpled Convex

Diagram + sin + sin + cos + cos

Limacon Clues

cos/sin determines: x-axis or y-axis

ratio determines: shape of graph

a + b: stretch on main axis

a: stretch on opp. axis

a – b: lower point

Examples of Limacons

r = 1 + 2sinƟ r = 3 + 2cosƟ

Rose Curves

r = acos(nƟ) r = asin(nƟ)• a = stretch of petals• If n is odd, n is the number of petals• If n is even, the number of petals is n x 2• Even cos – petals across both x and y axis• Odd cos – first petal across x axis• Even sin – petals across just y axis or just in quadrants• Odd sin – first petal across y axis• Positive – first petal on positive axis• Negative – first petal on negative axis

Rose Curve Examples

r = 4cos(5Ɵ) r = 3sin(2Ɵ)

r = 4cos(2Ɵ) r = 2sin(3Ɵ)

Lemniscates

r2 = 32cos2Ɵ r2 = a2sin2Ɵ

Finding Maximum r Values

Maximum values of r (same as stretch on main axis!)

• To solve algebraically:Find Ɵ at sin or cos of + 1 for max valueExample: r = 4 + 2cosƟ

Max value is 6 when cosƟ = 1Ɵ = 0 and 2Л

Example: r = 4 – 2cosƟMax value is 6 when cosƟ = -1

Ɵ = Л

Finding Zeros

To find zeros of the equation:

Set equation = 0 and solve

Example: r = 1 + 2cosƟ

1 + 2cosƟ = 0

2cosƟ = -1

cosƟ = -1/2

Ɵ = 2Л/3 and 4Л/3