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

Analyzing Graphs of Polar

Equations

But first… p.548: #1 3cos 2r

04

2

3

4

5

4

3

2

7

4

r 3 30 0 0 0–3 –3

Now, do you have the graph?

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