USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING

Shelly Berman p. 1 of 4 Jo Ann Fricker Polar Roses.doc

GSP4 Polar Roses

The polar rose is a curve that has the shape of a petalled flower. This curve was named rhodonea by the Italian mathematician Guido Grandi between 1723 and 1728 because it resembles a rose (MacTutor Archive). The polar equation of the rose is

€

r = a sin nθ( ), or

€

r = a cos nθ( ).

If n is odd, the rose is n–petalled. If n is even, the rose is 2n–petalled.

If n = r/s is a rational number, then the curve closes at a polar angle of θ = πsρ,

where ρ = 1 if rs is odd and ρ = 2 if rs is even.

USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING

Shelly Berman p. 2 of 4 Jo Ann Fricker Polar Roses.doc

If n is irrational, then there are an infinite number of petals.

Eric W. Weisstein. "Rose." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/Rose.html

In principle, the graph of any polar equation

€

r = f θ( ) can be obtained by

setting up a table and plotting a sufficient number of points. Indeed, this is

the way a graphing calculator or a computer operates. We will use the polar

rose, and our understanding of sine and cosine functions, in order to

understand the symmetry tests for polar graphs.

♦ Symmetry about the x–axis

If the point

€

r,θ( ) lies on the graph, the point

€

r,−θ( ) or

€

−r,π − θ( ) lies

on the graph.

♦ Symmetry about the y–axis

If the point

€

r,θ( ) lies on the graph, the point

€

r,π − θ( ) or

€

−r,−θ( ) lies

on the graph.

♦ Symmetry about the origin

If the point

€

r,θ( ) lies on the graph, the point

€

−r,θ( ) or

€

r,π + θ( ) lies on

the graph.

If a graph has any two of the symmetries listed here, it also has the third.

USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING

Shelly Berman p. 3 of 4 Jo Ann Fricker Polar Roses.doc

Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0 http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅sin b⋅θ( ) a = 4.00b = 1.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

120°105° 75°

60°45°

210°

195°

165°

150°

135°

90°

300°285°255°

240°225°

30°

15°

345°

330°

315°

270°

180° 0°

-

+

-

+

-90°

-5

5

4

-4

3

-3

450°360°270°180°

2

-2

1

-190°

60°Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

90°30°-180°f θ( ) = a⋅sin b⋅θ( ) a = 4.00

b = 2.00

45°-90° 540°450°360°270°180°60°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅sin b⋅θ( ) a = 4.00b = 3.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅sin b⋅θ( ) a = 4.00b = 4.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅sin b⋅θ( ) a = 4.00b = 5.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅sin b⋅θ( ) a = 4.00b = 6.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING

Shelly Berman p. 4 of 4 Jo Ann Fricker Polar Roses.doc

Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0 http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 0.25

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 0.50

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 0.75

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 1.00

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 1.25

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate

Edit the function below to try your own.You can use parameters a and b in thefunction you create.

Family of Roses

Exploring Precalculus with Sketchpad(C) 2005 by Key Curriculum Press

--

++

90°30°-180°

-5

-4

-3

-2

-1

5

4

3

2

1

450°360°270°180°90°-90°

f θ( ) = a⋅cos b⋅θ( ) a = 4.00b = 1.50

45°-90° 540°450°360°270°180°60°

0°

θ = 60°

345°

330°

315°300°

285°270°255°240°

225°

210°

195°

180°

165°

150°

135°120°

105° 90° 75°60°

45°

30°

15°

0°

Hide PolarHide Cartesian

Animate