chapter 8 – polar coordinates and parametric equations 8.2 - graphs of polar equations1

18.2 - Graphs of Polar Equations

Section 8.2 Graphs of Polar

Equations

Chapter 8 – Polar Coordinates and Parametric Equations

8.2 - Graphs of Polar Equations 2

Common Polar Curves: Circles and Spirals

8.2 - Graphs of Polar Equations 3

Common Polar Curves: Limaçons

Orientation depends on the trigonometric function (sine or cosine) and the sign of b.

sin

cos

0, 0

r a b

r a b

a b

8.2 - Graphs of Polar Equations 4

Common Polar Curves: Limaçons

Orientation depends on the trigonometric function (sine or cosine) and the sign of b.

sin

cos

0, 0

r a b

r a b

a b

8.2 - Graphs of Polar Equations 5

Common Polar Curves: Roses

sin

cos

leaved of is odd

2 leaved of is even

r a n

r a n

n n

n n

8.2 - Graphs of Polar Equations 6

Common Polar Curves: Roses

sin

cos

leaved of is odd

2 leaved of is even

r a n

r a n

n n

n n

8.2 - Graphs of Polar Equations 7

Common Polar Curves: Lemniscates

Figure-eight-shaped curves

8.2 - Graphs of Polar Equations 8

Example – pg. 553Match the polar equation with the

graphs labeled I – VI.

3. r = 3 cos

4. r = 3

5. r = 2 + 2 sin

6. r = 1 + 2 cos

7. r = sin 3

8. r = sin 4

8.2 - Graphs of Polar Equations 9

SymmetryWhen working with polar equations, it is helpful to

use symmetry when graphing.

8.2 - Graphs of Polar Equations 10

Tests for SymmetryIf a polar equation is

unchanged when we replace with – , then the graph is symmetric about the polar axis.

8.2 - Graphs of Polar Equations 11

Tests for SymmetryIf a polar equation is

unchanged when we replace r with –r, then the graph is symmetric about the pole.

8.2 - Graphs of Polar Equations 12

Tests for SymmetryIf a polar equation is

unchanged when we replace with – , then the graph is symmetric about the vertical line = /2.

8.2 - Graphs of Polar Equations 13

Examples – pg. 553Test the polar equation for symmetry with respect to

the polar axis, the pole, and the line = /2.

10. 4 8cos

12. 5cos csc

r

r

2

514.

1 3cos

16. 9sin

r

r

8.2 - Graphs of Polar Equations 14

Graphing Polar Equations

To sketch a polar curve whose graph, we plot points calculated for sufficiently many values of and join them in a continuous curve.

NOTE: This is how we first learned to graph functions in rectangular coordinates.

8.2 - Graphs of Polar Equations 15

Example – pg. 553 # 23

Sketch a graph of the polar equation.r = -2 cos

r = -2 cos

0

/6

/4

/3

/2

(2)/3

(3)/4

(5)/6

8.2 - Graphs of Polar Equations 16

Graphing Polar Equations

To sketch a polar curve whose graph, we can also sketch the graph in polar coordinates first. We can then use this graph to help us graph in the polar coordinate system.

8.2 - Graphs of Polar Equations 17

Example – pg. 553 # 25

Sketch a graph of the polar equation.r = 2 – 2 cos

8.2 - Graphs of Polar Equations 18

Examples – pg. 553Sketch a graph of the polar equation.

27. 3 1 sin

31. cos5

32. sin 4

r

r

r

2

36. 1 2cos

38. 4sin 2

39. , 0

r

r

r

8.2 - Graphs of Polar Equations 19

Example – pg. 554Match the polar equation with

the graphs labeled I-IV. Give reasons for your answers.

49. r = cos( /2)

50. r = 1/ √( )

51. r = sin

52. r = 1 + 3 cos (3 )