Graphing Polar Equations Four types of graphs: Circle Limacons Rose Curves Lemniscates

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

Ways to Graph Graphing Utility Plotting points From Equation Clues 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 Limacons R Ratio a/b < 1 (sign of a a/b = 1 or b not relevant) a/b > 2 Shape Inner Loop Cardioid Dimpled Convex Diagram + sin + 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)

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 value Example: r = 4 + 2cosƟ Max value is 6 when cosƟ = 1 Ɵ = 0 and 2Л Example: r = 4 – 2cosƟ Max 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