10.2 Graphing Polar Equations Day 1 Today’s Date: 2/24/16

SPIRAL Also called the “Spiral of Archimedes” No special rules! Typical Graph:

θr 0 EX 1: –3 – – – –3 –6 6 these points are on top of other points!  If we did a table, this is what it would look like This is a polar grid. Famous angles are illustrated. Concentric circles represent the values of r.

CIRCLESThere are three forms for a circle. Center of circle at _______ Radius = _______ Typical Graph: Contains the _______ Tangent to _______ Center on _______ Diameter = _______ Radius = _______ If a > 0, circle is ____ of pole If a < 0, circle is ____ of pole Typical Graph: Contains the _______ Tangent to _______ Center on _______ Diameter = _______ Radius = _______ If a > 0, circle is ____ of pole If a < 0, circle is ____ of pole Typical Graph: pole k polar axis E W pole polar axis N S

EX 2: Radius:______ Center On: Polar Axis / Circle is N S E W of the pole 4

θr EX 3:

LIMAÇONS French for “snail.” OR (oriented on polar axis) (oriented on ) Limaçon with Inner Loop When or a < b Diameter = _______ Inner Loop = _______ Cardioid (heart- shaped) When or a = b Diameter = _______ = ______ Dimpled Limaçon When Larger = _______ Smaller = _______ Convex Limaçon When or a ≥ 2b Larger = _______ Smaller = _______ For all the “bumps,” they hit the polar axis or (whichever is the opposite of where it is oriented) at _______ Typical Graph: Typical Graph: ±a±a smaller larger 2a aa smallerlarger inner loop diam a a

EX 4: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ EX 5: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ EX 6: Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ Limaçon with Inner Loop Diam: 6 Inner Loop: 2 Cardioid Diam: 4 Convex Limaçon Larger: 6 Smaller: 2 Oriented (WEST)

Symmetry – Tests for Symmetry on Polar Graphs If the following substitution is made and the equation is equivalent to the original equation, then the graph has the indicated symmetry. WRT the Pole (Origin) Replace r with –r WRT the Polar Axis (x-axis) Replace θ with –θ WRT the line (y-axis) Replace (r, θ) with (–r, –θ) Or θ by θ – π

EX 7: Identify the kind(s) of symmetry each polar graph possesses. A) Pole Polar Axis B) Pole Polar Axis

Ex 8: Determine an equation of the polar graph. Equation:_________________________________________________ Why?____________________________________________________ _________________________________________________________ Dimpled Limiçon on larger = 8smaller = 2 bump hits at 5a = 5, b = 3 r = 5 + 3sin θ North of pole  (+)

Homework # Day 1 Worksheet