1. Chapter 8 – Polar Coordinates and Parametric Equations 8.2 - Graphs of Polar Equations1

2. Common Polar Curves: Circles and Spirals 8.2 - Graphs of Polar Equations2

3. Common Polar Curves: Lima çons Orientation depends on the trigonometric function (sine or cosine) and the sign of b. 8.2 - Graphs of Polar Equations3

4. Common Polar Curves: Lima çons Orientation depends on the trigonometric function (sine or cosine) and the sign of b. 8.2 - Graphs of Polar Equations4

5. Common Polar Curves: Roses 8.2 - Graphs of Polar Equations5

6. Common Polar Curves: Roses 8.2 - Graphs of Polar Equations6

7. Common Polar Curves: Lemniscates Figure-eight-shaped curves 8.2 - Graphs of Polar Equations7

8. Example – pg. 553 Match the polar equation with the graphs labeled I – VI. 3. r = 3 cos  4. r = 3 5. r = 2 + 2 sin  6. r = 1 + 2 cos  7. r = sin 3  8. r = sin 4  8.2 - Graphs of Polar Equations8

9. Symmetry When working with polar equations, it is helpful to use symmetry when graphing. 8.2 - Graphs of Polar Equations9

10. Tests for Symmetry If a polar equation is unchanged when we replace  with – , then the graph is symmetric about the polar axis. 8.2 - Graphs of Polar Equations10

11. Tests for Symmetry If a polar equation is unchanged when we replace r with –r, then the graph is symmetric about the pole. 8.2 - Graphs of Polar Equations11

12. Tests for Symmetry If a polar equation is unchanged when we replace  with  – , then the graph is symmetric about the vertical line  =  /2. 8.2 - Graphs of Polar Equations12

13. Examples – pg. 553 Test the polar equation for symmetry with respect to the polar axis, the pole, and the line  =  /2. 8.2 - Graphs of Polar Equations13

14. 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. 8.2 - Graphs of Polar Equations14

15. 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  8.2 - Graphs of Polar Equations15

16. 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. 8.2 - Graphs of Polar Equations16

17. Example – pg. 553 # 25 Sketch a graph of the polar equation. r = 2 – 2 cos  8.2 - Graphs of Polar Equations17

18. Examples – pg. 553 Sketch a graph of the polar equation. 8.2 - Graphs of Polar Equations18

19. 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 = 1 + 3 cos (3  ) 8.2 - Graphs of Polar Equations19