9.7 Graphs of Polar Equations Digital Lesson. HWQ Convert the polar equation to rectangular form....

9.7 Graphs of Polar Equations

Digital Lesson

Warm-Up

• Find the maximum and minimum values of

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

) cosa f x x

Warm-Up #2

• Find the maximum and minimum values of


) 1 2sinb f x x

Polar Curves

r a

A Polar Curve consists of all the points (r,θ) satisfying a given equation F(r,θ) = 0. Often one can solve r from the equation and represent the polar curve in the form r = f(θ), where r represents the distance from the pole as a function of theta.

Some curves are easier to describe with polar coordinates:

(Circle centered at the origin)

o (Line through the origin)


Example:Graph the polar equation r = 2cos .

2

32

1 2 3 0

2

0

–2

–101

20r

6

3

2

23

56

76

32

116

2

3

3

3

3 The graph is a circle of radius 1 whose center is at

point (x, y) = (1, 0).

For a polar curve that has symmetry with respect to x-axis: If (r, ) is on the graph,

r2cosr

r

so is (r, -).

Examples of Polar Curves

For a polar curve that has symmetry with respect to y-axis: If (r, ) is on the graph,

r

2sinr

r

so is (r, -)

or (-r, -).

Examples of Polar Curves


Example:Find the zeros and the maximum value of for the graph of r = 2cos .

2

32

1 2 3 0

0 when 3 and .2 2

r

These are the zeros of r.

The maximum value of r is 2. It occurs when r 0, and 2 .

on 0,2 ,r


You try: Analyze the graph of r = 4sin. State the symmetry, find the zeros and maximum value of .

Graph has symmetry to =2

0 0, ,r at n

34 4, , 4,2 2

Max r

r on 0,2 ,


You try again: Analyze the graph of r = 1-2cos. State the symmetry, find the zeros and maximum value of .

Graph has symmetry to the polar axis

50 3 3

r at and

3 3,Max r

r on 0,2 ,


You try again: Analyze the graph ofState the symmetry, find the zeros and maximum value of .

Graph has symmetry to the y-axis

5 70 4 4

r at and

2 2 4 @2

Max r

r 2 2 2sinc h

r 2 2 2sinc hr

on 0,2 ,


Each polar graph below is called a Limaçon.

1 2cosr 1 2sinr

–3

–5 5

3

–5 5

3

–3


Each polar graph below is called a Lemniscate.

2 22 sin 2r 2 23 cos 2r

–5 5

3

–3

–5 5

3

–3


Each polar graph below is called a Rose curve.

2cos3r 3sin 4r

The graph will have n petals if n is odd, and 2n petals if n is even.

–5 5

3

–3

–5 5

3

–3

a

a

9.7 Classifying Polar Equations

2sin 2.150 16

r

What do you see?


5cos 2cos 4 sin

120 2

r e

What do you see?


Homework

9.7 Pg. 689 7-23 odd


Quiz Today 9.5/9.6

Homework: Ch. Review pg. 700 45-95 odd

9.7 Graphs of Polar Equations Digital Lesson. HWQ Convert the polar equation to rectangular form....

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