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Symmetry in Graphs Symmetry in Graphs of Polar Equationsof Polar Equations
On to Sec. 6.5a!!!On to Sec. 6.5a!!!
Recall our three types of symmetry…
θ
,θr
θ
1. The x-axis (polar axis) as a line of symmetry
x
y
, θ , π θr r
Recall our three types of symmetry…
π θ ,θr
θ
2. The y-axis (the line ) as a line of symmetry
x
y , π θ , θr r
θ π 2
Recall our three types of symmetry…
θ π
,θr
θ
3. The origin (the pole) as a point of symmetry
x
y
,θ ,θ πr r
Symmetry Tests for Polar Graphs
,θr
,θr
,θ or ,θ πr r
The graph of a polar equation has the indicated symmetry ifeither replacement produces an equivalent polar equation.
To Test for Symmetry Replace By
1. about the x-axis
2. about the y-axis
3. about the origin
,θr , θ or , π θr r
, θ or , π θr r
Guided PracticeUse symmetry tests to determine if the graph if symmetricabout the x-axis, y-axis, or origin. Confirm your answersgraphically.
4sin3θr
Symmetric about the Symmetric about the yy-axis-axis
(check the graph first???)
Guided PracticeUse symmetry tests to determine if the graph if symmetricabout the x-axis, y-axis, or origin. Confirm your answersgraphically.
1 2cosr Symmetric about the Symmetric about the xx-axis-axis
Guided PracticeUse symmetry tests to determine if the graph if symmetricabout the x-axis, y-axis, or origin. Confirm your answersgraphically.
1 3sinr Symmetric about the Symmetric about the yy-axis-axis
Guided PracticeUse symmetry tests to determine if the graph if symmetricabout the x-axis, y-axis, or origin. Confirm your answersgraphically.
7sin3r Symmetric about the Symmetric about the yy-axis-axis
Guided PracticeUse symmetry tests to determine if the graph if symmetricabout the x-axis, y-axis, or origin. Confirm your answersgraphically. 2
1 cosr
Symmetric about the Symmetric about the xx-axis-axis
We can “analyze” the graphs of polar equations in much thesame way as we have for rectangular equations…
As before, we check:
DomainDomain RangeRange SymmetrySymmetry
ContinuityContinuity BoundednessBoundedness
AsymptotesAsymptotes
Maximum r-valueMaximum r-value (this is the new one…)
Let’s focus on that last one for a bit…
Find the maximum r-value of 2 2cosθr
Maximum r-value = 4Maximum r-value = 4
Check algebraically and graphically…
Let’s focus on that last one for a bit…
Identify the points on the graph offor that give maximum r-values0 θ 2π
3cos 2θr
3,03π
3,2
3, ππ
3,2
Guided PracticeGuided PracticeAnalyze the graph of the given polar curve.
Domain:Domain:
8r , Range:Range: 8r
Symmetry:Symmetry: About the x-axis, y-axis, and origin
Continuity:Continuity: Continuous
Boundedness:Boundedness: Bounded
Asymptotes:Asymptotes: None
Maximum Maximum rr-value:-value: 8
The graph???
Guided PracticeGuided PracticeAnalyze the graph of the given polar curve.
Domain:Domain:
6 6 Range:Range: ,
Symmetry:Symmetry: About the origin
Continuity:Continuity: Continuous
Boundedness:Boundedness: Unbounded
Asymptotes:Asymptotes: None
Maximum Maximum rr-value:-value: None
The graph???
One Last DefinitionOne Last DefinitionThe graphs of
and
where n is an integer greater than 1, are rose curvesrose curves
cos nθr a
sin nθr a
If n is odd there are n petals;if n is even there are 2n petals
Analyze the graph:
The graph??? How many petals???The graph??? How many petals???
Domain:Domain: , Range:Range: 3,3Symmetry:Symmetry: About the x-axis, y-axis, and origin
Continuity:Continuity: Continuous
Boundedness:Boundedness: Bounded
Asymptotes:Asymptotes: None
Maximum Maximum rr-value:-value: 3
3sin 4r