1. HW # −17 , ,20 , ,46 , Row 1
Do Now Test for symmetry with respect to the line 𝜃= 𝜋 2 , the polar axis and the pole. 𝑟= 2 1− cos 𝜃

2. Do Now Solution 𝑟= 2 1− cos 𝜃 −𝑟= 2 1− cos (−𝜃) cos (−𝜃) = cos 𝜃
Test for symmetry with respect to the line 𝑥= 𝜋 2 , the polar axis and the pole. 𝑟= 2 1− cos 𝜃
The line 𝜃= 𝜋 2 , replace 𝑟,𝜃 by −𝑟, −𝜃
𝑟= 2 1− cos 𝜃
−𝑟= 2 1− cos (−𝜃) cos (−𝜃) = cos 𝜃
−𝑟= 2 1− cos 𝜃
no symmetry

3. Do Now Solution 𝑟= 2 1− cos 𝜃 𝑟= 2 1− cos 𝜃
2. The polar axis, replace 𝑟,𝜃 by 𝑟,−𝜃
𝑟= 2 1− cos 𝜃
𝑟= 2 1− cos (−𝜃) cos (−𝜃) = cos 𝜃
Symmetric to the polar axis
3. The pole, replace 𝑟,𝜃 by −𝑟,𝜃
𝑟= 2 1− cos 𝜃
−𝑟= 2 1− cos 𝜃
No symmetry

4. Graphs of Polar Equations

5. Objectives Graph polar equations by point plotting.
Use symmetry, zeros, and maximum r-values to sketch graphs of polar equations.
Recognize special polar graphs.

6. Example 1 – Graphing a Polar Equation by Point Plotting
Sketch the graph of the polar equation r = 4 sin . Solution: The sine function is periodic, so you can get a full range of r-values by considering values of  in the interval 0    2, as shown in the following table.

7. Example 1 – Solution
cont’d
By plotting these points, as shown in Figure 10.54, it appears that the graph is a circle of radius 2 whose center is at the point (x, y) = (0, 2).
Figure 10.54

8. Symmetry, Zeros, and Maximum r-Values

9. Example 2 – Using Symmetry to Sketch a Polar Graph
Use symmetry to sketch the graph of r = cos . Solution: Replacing (r,  ) by (r, – ) produces r = cos(– ) = cos . So, you can conclude that the curve is symmetric with respect to the polar axis.
cos(– ) = cos 

10. Example 2 – Solution
cont’d
Plotting the points in the table and using polar axis symmetry, you obtain the graph shown in Figure This graph is called a limaçon.
Figure 10.55

11. Symmetry, Zeros, and Maximum r-Values

12. Symmetry, Zeros, and Maximum r-Values
Two additional aids to sketching graphs of polar equations involve knowing the  -values for which | r | is maximum and knowing the  -values for which r = 0. For instance, in Example 1, the maximum value of | r | for r = 4 sin  is | r | = 4, and this occurs when  =  /2, as shown in Figure Moreover, when r = 0 when  = 0.
Figure 10.54

13. Example 3 – Sketching a Polar Graph
Sketch the graph of r = 1 – 2 cos . Solution: From the equation r = 1 – 2 cos , you can obtain the following. Symmetry: With respect to the polar axis Maximum value of | r |: r = 3 when  =  Zero of r : r = 0 when  =  /3

14. Example 3 – Solution
cont’d
The table shows several  -values in the interval [0,  ]. By plotting the corresponding points, you can sketch the graph shown in Figure
Figure 10.57

15. Special Polar Graphs

16. Special Polar Graphs
Several important types of graphs have equations that are simpler in polar form than in rectangular form. For example, the circle r = 4 sin  in Example 1 has the more complicated rectangular equation x2 + (y – 2)2 = 4.

17. Special Polar Graphs
Several other types of graphs that have simple polar equations are shown below. Limaçons r = a  b cos , r = a  b sin , (a > 0, b > 0)

18. Special Polar Graphs
Rose Curves n petals when n is odd, 2n petals if n is even (n  2)

19. Special Polar Graphs
Circles and Lemniscates

20. Example 5 – Sketching a Rose Curve
Sketch the graph of r = 3 cos 2. Solution: Type of curve: Rose curve with 2n = 4 petals Symmetry: With respect to polar axis, the line  = , and the pole Maximum value of | r |: | r | = 3 when  = 0, ,  , Zero of r : r = 0 when  = ,

21. Example 5 – Solution
cont’d
Using this information together with the additional points shown in the following table, you obtain the graph shown in Figure
Figure 10.58