Chapter 8 – Polar Coordinates and Parametric Equations Graphs of Polar Equations1

Common Polar Curves: Circles and Spirals Graphs of Polar Equations2

Common Polar Curves: Lima çons Orientation depends on the trigonometric function (sine or cosine) and the sign of b Graphs of Polar Equations3

Common Polar Curves: Lima çons Orientation depends on the trigonometric function (sine or cosine) and the sign of b Graphs of Polar Equations4

Common Polar Curves: Roses Graphs of Polar Equations5

Common Polar Curves: Roses Graphs of Polar Equations6

Common Polar Curves: Lemniscates Figure-eight-shaped curves Graphs of Polar Equations7

Example – pg. 553 Match the polar equation with the graphs labeled I – VI. 3. r = 3 cos  4. r = 3 5. r = sin  6. r = cos  7. r = sin 3  8. r = sin 4  Graphs of Polar Equations8

Symmetry When working with polar equations, it is helpful to use symmetry when graphing Graphs of Polar Equations9

Tests for Symmetry If a polar equation is unchanged when we replace  with – , then the graph is symmetric about the polar axis Graphs of Polar Equations10

Tests for Symmetry If a polar equation is unchanged when we replace r with –r, then the graph is symmetric about the pole Graphs of Polar Equations11

Tests for Symmetry If a polar equation is unchanged when we replace  with  – , then the graph is symmetric about the vertical line  =  / Graphs of Polar Equations12

Examples – pg. 553 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line  =  / Graphs of Polar Equations13

Graphing Polar Equations To sketch a polar curve whose graph, we plot points calculated for sufficiently many values of  and join them in a continuous curve. NOTE: This is how we first learned to graph functions in rectangular coordinates Graphs of Polar Equations14

Example – pg. 553 # 23 Sketch a graph of the polar equation. r = -2 cos   0  /6  /4  /3  /2 (2  )/3 (3  )/4 (5  )/6  Graphs of Polar Equations15

Graphing Polar Equations To sketch a polar curve whose graph, we can also sketch the graph in polar coordinates first. We can then use this graph to help us graph in the polar coordinate system Graphs of Polar Equations16

Example – pg. 553 # 25 Sketch a graph of the polar equation. r = 2 – 2 cos  Graphs of Polar Equations17

Examples – pg. 553 Sketch a graph of the polar equation Graphs of Polar Equations18

Example – pg. 554 Match the polar equation with the graphs labeled I-IV. Give reasons for your answers. 49. r = cos(  /2) 50. r = 1/ √(  ) 51. r =  sin  52. r = cos (3  ) Graphs of Polar Equations19