unit 6.5
DESCRIPTION
Unit 6.5TRANSCRIPT
Copyright © 2011 Pearson, Inc.
6.5
Graphs and
Polar
Equations
Copyright © 2011 Pearson, Inc. Slide 6.1 - 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 - 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 - 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 - 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 - 6
Example Testing for Symmetry
r 2sin2
r 2sin2( )
r 2sin(2 )
r 2sin2
r 2sin2
Use the symmetry tests to prove that the graph of
r 2sin2 is symmetric about the y-axis.
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 - 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 - 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 - 9
Example Analyzing a Limaçon Curve
Show the graphs of r1 4 3cos and r
2 4 3cos
are the same dimpled lima�on.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 10
Example Analyzing a Limaçon Curve
r1 4 3cos
r2 4 3cos
Use a grapher's trace feature to show the following:
r1
: As increases from 0 to 2 ,
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 - 11
Spiral of Archimedes
The spiral of Archimedes is
r
Copyright © 2011 Pearson, Inc. Slide 6.1 - 12
Lemniscate Curves
The lemniscate curves are graphs of polar equations
of the form
r 2 a2 sin2 and r 2 a2 cos2.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 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 - 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