1. Section 6-5 symmetry for polar graphs analyzing a polar graph finding maximum r-values rose curves limaçon curves other polar graphs

2. Symmetry For Polar Graphs x-axis:

3. Symmetry For Polar Graphs y-axis:

4. Symmetry For Polar Graphs origin:

5. Test For x-axis Symmetry insert the following values into the equation and then simplify, if either case reduces to the same as the original then it has x-axis symmetry Example: test our earlier example

6. Test For y-axis Symmetry insert the following values into the equation and then simplify, if either case reduces to the same as the original then it has y-axis symmetry

7. Example for y-axis: r = 4 + 4sinθ

8. Test For origin Symmetry insert the following into the equation and then simplify, if either case reduces to the same as the original then it has origin symmetry

9. Analyzing Polar Graphs analysis of a polar graph is not as extensive as with functions domain = possible θ’s (usually all reals) range = r values boundedness (varies) continuity (usually continuous) symmetry (just did this)

10. Maximum r-values Two ways to find the range (including the maximum r-values) since sinθ and cosθ must be between –1 and 1, plug in these values to see what happens to r change the equation to y= format and graph the function, find the max and min of the graph

11. Example: when cosθ = -1 this becomes 1 when cosθ = 1 this becomes 5 thus, the range is [1, 5] and the max r-value is 5

12. Rose Curves format if n is even  2n petals if n is odd  n petals a is the length of the petals with cosθ then x-axis symmetry with sinθ then y-axis symmetry sometimes origin symmetry

13. More examples of rose curves

14. Limaçon Curves format range is [a – b, a + b] with cosθ then x-axis symmetry with sinθ then y-axis symmetry shape depends on a and b

15. More examples of limaçon curves:

16. Other Polar Graphs Spiral of Archimedes: r = θ Lemniscates: