19
9.7 Graphs of Polar Equations Digital Lesson
| Date post: | 19-Jan-2018 |
| Category: |
Documents |
| Upload: | magnus-burns |
| View: | 217 times |
| Download: | 0 times |
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4
) 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)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10
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
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11
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 ,
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12
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 ,
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13
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 ,
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14
Each polar graph below is called a Limaçon.
1 2cosr 1 2sinr
–3
–5 5
3
–5 5
3
–3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15
Each polar graph below is called a Lemniscate.
2 22 sin 2r 2 23 cos 2r
–5 5
3
–3
–5 5
3
–3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16
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?
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19
5cos 2cos 4 sin
120 2
r e
What do you see?
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20
Homework
9.7 Pg. 689 7-23 odd
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21
Quiz Today 9.5/9.6
Homework: Ch. Review pg. 700 45-95 odd
