1. Section 6.4 Use point plotting to graph polar equations.
Use symmetry to graph polar equations.

2. 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.

3. Circles in Polar Coordinates

4. 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.

5. Example: Graphing an Equation Using the Point-Plotting Method (continued)

6. Example: Graphing an Equation Using the Point-Plotting Method (continued)

7. Example: Graphing an Equation Using the Point-Plotting Method (continued)
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.

8. Tests for Symmetry in Polar Coordinates

9. 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.

10. 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

11. 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.

12. 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:

13. Example: Graphing a Polar Equation Using Symmetry (continued)
The graph of
is an example of a limaçon.

14. Limaçons

15. Example: Graphing a Polar Equation
Graph the polar equation:
We first check for symmetry:
Polar Axis
The Line
The Pole
Replace with
Replace with
Replace r with –r
and replace r with –r
Equation changes and fails this symmetry test.
Equation changes and fails this symmetry test.
Equation does not change.

16. Example: Graphing a Polar Equation (continued)
Graph the polar equation:
Complete a table of values for the function:

17. Example: Graphing a Polar Equation (continued)
Graph the polar equation:

18. Example: Graphing a Polar Equation (continued)
Graph the polar equation:
The graph of
is an example of
a rose curve.

19. Rose Curves

20. Example: Graphing a Polar Equation
Graph the polar equation:
We first check for symmetry:
Polar Axis
The Line
The Pole
Replace with
Replace with
Replace r with –r
and replace r with –r
Equation does not change.
Equation does not change.
Equation does not change.

21. Example: Graphing a Polar Equation (continued)
Graph the polar equation:
Complete a table of values for the function:

22. Example: Graphing a Polar Equation (continued)
Graph the polar equation:

23. Example: Graphing a Polar Equation (continued)
Graph the polar equation:
The graph of
the polar equation
is an example of
a lemniscate.

24. Lemniscates