1. Polar Coordinates and Graphing r = directed distance = directed angle Polar Axis O Counterclockwise from polar axis to.

2. Plotting Points in the Polar Coordinate System The point lies two units from the pole on the terminal side of the angle. The point lies three units from the pole on the terminal side of the angle. The point coincides with the point.

3. Multiple Representations of Points

4. Try plotting the following points in polar coordinates and find three additional polar representations of the point:

5. Graphing a Polar Equation circle

6. Using Symmetry to Sketch a Polar Graph Symmetry with respect to the line Symmetry with respect to the Polar Axis. Symmetry with respect to the Pole

7. Tests for Symmetry The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation.

8. Using Symmetry to Sketch Thus, the graph is symmetric with respect to the polar axis, and you need only plot points from 0 to . Lima  on

9. Symmetry Test Fails Unfortunately the tests for symmetry can guarantee symmetry, but there are polar curves that fail the test, yet still display symmetry. Let’s look at the graph of Try the symmetry tests. What happens? Original Replacement New Equation All of the tests indicate that no symmetry exists. Now, let’s look at the graph. \ Not symmetric about the polar axis. \ Not symmetric about the pole.

10. Spiral of Archimedes You can see that the graph is symmetric with respect to the line

11. Quick Test for Symmetry

12. Determine the effect of “a” on the graph of. Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left. You may need to drag the graph slightly to the right.

13. Determine the effect of “a” on the graph of. Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left. You may need to drag the graph slightly to the right.

14. Determine the effect of “a” and “b” on the graph of Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left. You may need to drag the graph slightly to the right.

15. Determine the effect of “a” on the graph of Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left. You may need to drag the graph slightly to the right.

16. Determine the effect of “a” and “b’ on the graph of Click on the graph below to open an interactive page. Please note the interactive page may open with the graph to the far left. You may need to drag the graph slightly to the right.

17. Summary of Special Polar Graphs Limacon with inner loop Limacons: Cardiod (heart-shaped) Dimpled Limacon Convex Limacon

18. Rose Curves:

19. Circles and Lemniscates CirclesLemniscates

20. Analyzing Polar Graphs Analyze the basic features of Type of Curve:Rose Curve with 2b petals = 4 petals Symmetry:Polar axis, pole, and Maximum Value of | r |:| r | = 3 when Zeros of r:r = 0 when You can use this same process to Analyze any polar graph.

21. Analyze and Sketch the graph: Type of curve: Symmetry: Maximum | r | Zeros of r Convex Limacon Cardioid Limacon sym. @ polar axis none