1. Section 10.8 Notes

2. In previous math courses as well as Pre-Calculus you have learned how to graph on the rectangular coordinate system. You first learned how to graph using a table. Then you learned how to use intercepts, symmetry, asymptotes, periods, and shifts to help you graph. Graphing on the polar coordinate system will be done similarly.

3. Graphing by Plotting Points

4. Let’s start from the beginning with a table to graph a curve. When graphing using a table use θ’s that will give you an “r” that is an integer or a terminating decimal.

5. Example 1

6. Graph r = 6cos θ using a table. What θ’s would you use? Which points are the exact same points? Let’s graph these points on a polar graph. θ r

8. If we look at how the points are arranged, it appears that this is a circle whose center is at (3, 0) with a radius of 3.

9. The graph from the previous example is a circle whose center is not at the pole. The equations for this graph are r = a cos θ with center at (½a, 0) r = a sin θ with center at a = the diameter of the circle

10. Graphing using Symmetry

11. There are three types of symmetry that are used to graph on the polar coordinate system.

12. Replace (r, θ) by (r,  −  ) or (-r, -  )

13. Replace (r, θ) by (r, -  ) or (-r,  −  )

14. Replace (r, θ) by (r,  +  ) or (-r,  )

15. Quick Tests for Symmetry in Polar Coordinates

16. Example 2

17. Use symmetry to sketch the graph r = 2 + 4sin  This graph has symmetry with We will only look at points in the 1 st and 4 th quadrants and then use symmetry.  r

18. Now we will use symmetry to find points in the 2 nd and 3 rd quadrants. Now draw the graph.

19. The graph in the previous example is called a limaçon with a loop. The equations for a graph of a limaçon with and without a loop are r = a ± b cos  r = a ± b sin  (a > 0, b > 0) If a > b, then there is no loop. If a < b, then there is a loop.

20. End of 1 st Day The two polar graphs we did today are 1.a circle whose center is not at the pole. 2.A limaçon with and without a loop.

21. 2 nd Day

22. Today we will look at two more polar graphs. 1.Cardioid r = a ± b sin  r = a ± b cos  where a = b

23. Example 3

24. Graph r = 4 + 4cos  This graph is symmetric to the polar axis. To graph this cardioid we are going to use another aid for graphing. This aid is finding the maximum value |r| and the zeroes of the graph.

25. To find the maximum value of |r| we must find the |r| when our trig function is equal to 1. In this example cos  = 1  = 0

26. So our maximum value of |r| is r = 4 + 4cos(0) r = 8 Now we need to find the zeroes of r. This will be where the polar equation is equal to 0.

27. In the example, 4 + 4cos  = 0 Now graph the maximum point and the zero point.

29.  r r = 4 + 4cos θ

31. Using symmetry we can graph the points in the 3 rd and 4 th quadrants. Now we can draw the graph.

33. Now the 2 nd graph: 2.Rose Curves r = a sin n  r = a cos n 

34. If the trig ratio is cosine and n is odd the graph is symmetric with the polar axis. If the trig ratio is sine and n is odd then the graph is symmetric with When n is even, then both graphs are symmetric with the pole, polar axis, and

35. Furthermore, the number of petals on the curve depends on whether n is even or odd. If n is odd, then there are n petals on the rose curve. If n is even, then there are 2n petals on the rose curve.

36. Example 4

37. Graph r = 4sin 2 . The graph will have 4 petals. It will be symmetric with the pole, polar axis, and

38. Maximum value of |r| = 4 when sin 2  = ±1. The zeroes are found when sin 2  = 0.

39. r = 4sin 2 

40. We can graph the rose curve using this information and finding one point where

41. r = 4sin 2 

42. Using symmetry we can now find other points to help us graph the petals. The rose curves also have symmetry on the line containing the maximum point.

43. r = 4sin 2 

44. The polar graphs that you need to know are: 1.A circle whose center is the pole 2.A circle whose center is not the pole 3.A limaçon with and without a loop 4.A cardioid 5.A rose curve