6.5
Graphs of
Polar
Equations
Copyright © 2011 Pearson, Inc.
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 6.5 - 2
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 6.5 - 3
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 6.5 - 4
Example Testing for Symmetry
Use the symmetry tests to prove that the graph of
r = 2sin 2q is symmetric about the y-axis.
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 5
Example Testing for Symmetry
Use the symmetry tests to prove that the graph of
r = 2sin 2q is symmetric about the y-axis.
r = 2sin 2q
Because the equations of
-r = 2sin 2(-q )
-r = 2sin 2(-q ) and
-r = 2sin(-2q )
r = 2sin 2q
-r = -2sin 2q
are equivalent, there is
r = 2sin 2q
symmetry about the y-axis.
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 6
Rose Curves
The graphs of r = acos nq and r = asin nq , 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.
= 3sin4
curve r
Slide 6.5 - 7
Limaçon Curves
The limaçon curves are graphs of polar equations
of the form
r = a ± bsin q
and r = a ± bcosq ,
where a > 0 and b > 0.
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 8
Example Analyzing a Limaçon Curve
Show the graphs of r1 = 4 + 3cosq and r2 = -4 + 3cosq
are the same dimpled limaçon.
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 9
Example Analyzing a Limaçon Curve
Use a grapher's trace feature to show the following:
r1 : As q increases from 0 to 2p ,
the point (r1 ,q ) begins at B and
moves counterclockwise one
r1 = 4 + 3cosq
r2 = -4 + 3cosq
time around the graph.
r2 : As q increases from 0 to 2p ,
the point (r2 ,q ) begins at A and
moves counterclockwise one
time around the graph.
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 10
Spiral of Archimedes
The spiral of Archimedes is
r =q
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 11
Lemniscate Curves
The lemniscate curves are graphs of polar equations
of the form
r 2 = a 2 sin 2q
Copyright © 2011 Pearson, Inc.
and r 2 = a 2 cos 2q .
Slide 6.5 - 12
Quick Review
Find the absolute maximum value and absolute
minimum value in [0,2p ) and where they occur.
1. y = 2cos 2x
2. y = sin 2x + 2
3. Determine if the graph of y = sin 4x is symmetric
about the (a) x-axis, (b) y-axis, and (c) origin.
Use trig identities to simplify the expression.
4. sin(q - p )
(
5. cos q - p
)
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 13
Quick Review Solutions
Find the absolute maximum value and absolute
minimum value in [0,2p ) and where they occur.
1. y = 2cos 2x
max value:2 at x = 0, p min value:- 2 at x = p / 2, 3p / 2
2. y = sin 2x + 2
max value:3 at x = p / 4,5p / 4 min value:1 at x = 3p / 4, 7p / 4
3. Determine if the graph of y = sin 4x 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(q - p )
- sinq
5. cos q - p
- cosq
(
)
Copyright © 2011 Pearson, Inc.
Slide 6.5 - 14