10.2 Graphing Polar Equations Day 1
SPIRAL Also called the “Spiral of Archimedes” No special rules! Typical Graph:
If we did a table, this is what it would look like 6 5.2 EX 1: θ r If we did a table, this is what it would look like 6 5.2 This is a polar grid. Famous angles are illustrated. Concentric circles represent the values of r. 4.2 3 –3 –4.2 –5.2 –6 –5.2 –4.2 –3 these points are on top of other points! 3 4.2 5.2 6
CIRCLES There are three forms for a circle. Center of circle at _______ Radius = _______ Typical Graph: Contains the _______ Tangent to _______ Center on _______ Diameter = _______ If a > 0, circle is ____ of pole If a < 0, circle is ____ of pole Typical Graph: pole pole pole k polar axis polar axis N E S W
EX 2: Radius:______ Center On: Polar Axis / Circle is N S E W of the pole 4
EX 3: θ r 2 3 3.4 3.7 4 1 0.6 0.3
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 For all the “bumps,” they hit the polar axis or (whichever is the opposite of where it is oriented) at _______ Typical Graph: ±a a a 2a larger a a a diam smaller larger inner loop a smaller a a
Limaçon with Inner Loop Cardioid Convex Limaçon Type:__________________ On: Polar Axis / Lengths:_________________ _________________________ EX 5: On: Polar Axis / EX 6: (WEST) Limaçon with Inner Loop (same as r = 2 + 2sin ) Cardioid Convex Limaçon Oriented Oriented Oriented Diam: 6 Diam: 4 Larger: 6 Inner Loop: 2 Smaller: 2
Ex 7: Determine an equation of the polar graph. r = 5 + 3sin θ Equation:_________________________________________________ Why?____________________________________________________ _________________________________________________________ Dimpled Limaçon on sin North of pole (+) larger = 8 smaller = 2 bump hits at 5 a = 5 & since a + b = 8 & a – b = 2 b = 3