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
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