This document discusses polar graphs and equations. It covers topics like polar curves, symmetry properties, analyzing polar curves, rose curves, limaçon curves, and other polar graphs. Specific curves discussed include the spiral of Archimedes, lemniscate curves, and examples of testing polar equations for symmetry and analyzing limaçon curves. It aims to explain polar graphs and equations which are useful in calculus.

1. 6.5
Graphs and
Polar
Equations
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Polar Curves and Parametric Curves
 Symmetry
 Analyzing Polar Curves
 Rose Curves
 Limaçon Curves
 Other Polar Curves
… and why
Graphs that have circular or cylindrical symmetry often
have simple polar equations, which is very useful in
calculus.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 2

3. Symmetry
The three types of symmetry figures to be considered will
have are:
1. The x-axis (polar axis) as a line of symmetry.
2. The y-axis (the line θ = π/2) as a line of symmetry.
3. The origin (the pole) as a point of symmetry.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 3

4. Symmetry Tests for Polar Graphs
The graph of a polar equation has the indicated symmetry
if either replacement produces an equivalent polar
equation.
To Test for Symmetry Replace By
1. about the x-axis (r,θ) (r,–θ) or (–r, π–θ)
2. about the y-axis (r,θ) (–r,–θ) or (r, π–θ)
3. about the origin (r,θ) (–r,θ) or (r, π+θ)
Copyright © 2011 Pearson, Inc. Slide 6.1 - 4

5. Example Testing for Symmetry
Use the symmetry tests to prove that the graph of
r  2sin2 is symmetric about the y-axis.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 5

6. Example Testing for Symmetry
Use the symmetry tests to prove that the graph of
r  2sin2 is symmetric about the y-axis.
r  2sin2
r  2sin2( )
r  2sin(2 )
r  2sin2
r  2sin2
Because the equations of
r  2sin2( ) and
r  2sin2
are equivalent, there is
symmetry about the y-axis.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 6

7. Rose Curves
The graphs of r  acosn and r  asin n , where n is
an integer greater than 1, are rose curves.
If n is odd there are
n petals, and
if n is even there are
2n petals.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 7

8. Limaçon Curves
The limaon curves are graphs of polar equations
of the form
r  a  bsin and r  a  bcos ,
where a  0 and b  0.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 8

9. Example Analyzing a Limaçon Curve
Show the graphs of r1  4  3cos and r2  4  3cos
are the same dimpled limaon.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 9

10. Example Analyzing a Limaçon Curve
Use a grapher's trace feature to show the following:
r1 : As  increases from 0 to 2 ,
r1  4  3cos
r2  4  3cos
the point (r1, ) begins at B and
moves counterclockwise one
time around the graph.
r2 : As  increases from 0 to 2 ,
the point (r2 , ) begins at A and
moves counterclockwise one
time around the graph.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 10

11. Spiral of Archimedes
The spiral of Archimedes is
r  
Copyright © 2011 Pearson, Inc. Slide 6.1 - 11

12. Lemniscate Curves
The lemniscate curves are graphs of polar equations
of the form
r2  a2 sin2 and r2  a2 cos2 .
Copyright © 2011 Pearson, Inc. Slide 6.1 - 12

13. Quick Review
Find the absolute maximum value and absolute
minimum value in [0,2 ) and where they occur.
1. y  2cos2x
2. y  sin2x  2
3. Determine if the graph of y  sin4x is symmetric
about the (a) x-axis, (b) y-axis, and (c) origin.
Use trig identities to simplify the expression.
4. sin(   )
5. cos  
Copyright © 2011 Pearson, Inc. Slide 6.1 - 13

14. Quick Review Solutions
Find the absolute maximum value and absolute
minimum value in [0,2 ) and where they occur.
1. y  2cos2x
max value:2 at x  0, min value:  2 at x   / 2, 3 / 2
2. y  sin2x  2
max value:3 at x   / 4,5 / 4 min value:1 at x  3 / 4, 7 / 4
3. Determine if the graph of y  sin4x is symmetric
about the (a) x-axis, no (b) y-axis, no and (c) origin. yes
Use trig identities to simplify the expression.
4. sin(   )  sin
5. cos    cos
Copyright © 2011 Pearson, Inc. Slide 6.1 - 14