1. Chapter 6 Additional Topics in Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 6.4 Graphs of Polar Equations

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Objectives: Use point plotting to graph polar equations. Use symmetry to graph polar equations.

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Using Polar Grids to Graph A polar equation is an equation whose variables are r and The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids like the one in the figure to graph polar equations.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Circles in Polar Coordinates

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Graphing an Equation Using the Point-Plotting Method Graph the equation with in radians. Use multiples of from 0 to π to generate coordinates for points We construct a partial table of coordinates for using multiples of Then we plot the points and join them in a smooth curve.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Graphing an Equation Using the Point-Plotting Method (continued) Graph

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Graphing an Equation Using the Point-Plotting Method (continued) Graph

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Graphing an Equation Using the Point-Plotting Method (continued) Graph We can verify that the graph is a circle by changing from polar to rectangular form. The graph is a circle with center at (0, 2) and r = 2.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Tests for Symmetry in Polar Coordinates

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Graphing a Polar Equation Using Symmetry Check for symmetry and then graph the polar equation: Symmetry with respect to the polar axis (x-axis): on the graph of the function. The polar equation does not change when is replaced with the graph is symmetric with respect to the polar axis.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Graphing a Polar Equation Using Symmetry (continued) Check for symmetry and then graph the polar equation: Symmetry with respect to the line (origin): on the graph of the function. The polar equation changes when is replaced with The graph is not symmetric with respect to the line

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Graphing a Polar Equation Using Symmetry (continued) Check for symmetry and then graph the polar equation: Symmetry with respect to the pole axis (y-axis): on the graph of the function. The polar equation changes when r is replaced with –r. The graph is not symmetric with respect to the pole.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Graphing a Polar Equation Using Symmetry (continued) Check for symmetry and then graph the polar equation: Complete a table of values for the function:

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Graphing a Polar Equation Using Symmetry (continued) The graph of is an example of a limaçon.

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Limaçons

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Graphing a Polar Equation Graph the polar equation: We first check for symmetry: Polar Axis The Line The Pole Replace with Replace r with –r and replace r with –r Equation changes and fails this symmetry test. Equation does not change. Equation changes and fails this symmetry test.

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a Polar Equation (continued) Graph the polar equation: Complete a table of values for the function:

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Example: Graphing a Polar Equation (continued) Graph the polar equation:

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Graphing a Polar Equation (continued) Graph the polar equation: The graph of is an example of a rose curve.

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Rose Curves

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Example: Graphing a Polar Equation Graph the polar equation: We first check for symmetry: Polar Axis The Line The Pole Replace with Replace r with –r and replace r with –r Equation does not change.

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Graphing a Polar Equation (continued) Graph the polar equation: Complete a table of values for the function:

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Graphing a Polar Equation (continued) Graph the polar equation:

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Graphing a Polar Equation (continued) Graph the polar equation: The graph of the polar equation is an example of a lemniscate.

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Lemniscates