Unit 6.5

6.5
Graphs and
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 6.1 - 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.

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)

Example Testing for Symmetry
Use the symmetry tests to prove that the graph of
r  2sin2 is symmetric about the y-axis.

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.

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.

Limaçon Curves
The limaon curves are graphs of polar equations
of the form
r  a  bsin and r  a  bcos ,
where a  0 and b  0.

Example Analyzing a Limaçon Curve
Show the graphs of r1  4  3cos and r2  4  3cos
are the same dimpled limaon.

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.

Spiral of Archimedes
The spiral of Archimedes is
r  

Lemniscate Curves
The lemniscate curves are graphs of polar equations
of the form
r2  a2 sin2 and r2  a2 cos2 .

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  

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

Unit 6.5

More Related Content

What's hot

Viewers also liked

Similar to Unit 6.5

More from Mark Ryder

Recently uploaded

Unit 6.5