USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING
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
where
θ = πsρ,
ρ = 1 if rs is odd and ρ = 2 if rs is even.
Shelly Berman
p. 1 of 4
Polar Roses.doc
Jo Ann Fricker
USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING
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
€ point r,θ lies on the graph, the
€ point €
r, π − θ or −r,−θ lies
If the
(
)
(
)
(
)
on the graph.
♦ Symmetry about the origin
€ point r,θ lies on the graph, the
€ point −r,€θ or r, π + θ lies on
If the
(
)
(
)
(
)
the graph.
If a graph has any two of the symmetries listed here, it also has the third.
€
Shelly Berman
€
p. 2 of 4
Polar Roses.doc
€
Jo Ann Fricker
USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING
Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0
http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html
Family of Roses
Hide Cartesian
Hide Polar
+
105°
90°
2
30°
2
1
-90°
90°
-1
180°
270°
360°
450°
0°
195°
-3
210°
225°
-5
30° 60° 90°
180°
270°
450°
360°
Hide Polar
105°
90°
-180°
2
-90°
180°
270°
450°
360°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
a = 4.00
f(θ) = a⋅sin(b⋅θ)
b = 2.00
Family of Roses
Hide Polar
90°
75°
60°
+
45°
135°
4
30°
105°
120°
5
150°
3
2
15°
30°
165°
15°
1
90°
-1
180°
270°
360°
450°
0°
195°
210°
225°
-5
30° 60° 90°
270°
450°
360°
300°
Family of Roses
Hide Polar
+
105°
90°
-180°
2
210°
330°
225°
-
315°
240°
30° 60° 90°
0° 45°
300°
255° 270° 285°
180°
270°
450°
360°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
a = 4.00
f(θ) = a⋅sin(b⋅θ)
Animate
b = 4.00
Family of Roses
Hide Cartesian
Hide Polar
+
30°
2
1
75°
60°
+
45°
150°
3
15°
90°
135°
4
165°
105°
120°
5
150°
3
345°
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
60°
+
45°
135°
4
-90°
75°
120°
5
0°
195°
-
b = 3.00
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
Hide Cartesian
450°
θ = 60°
a = 4.00
f(θ) = a⋅sin(b⋅θ)
Animate
360°
-5
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
270°
-4
255° 270° 285°
180°
180°
-3
315°
240°
-
90°
-1
180°
-2
330°
θ = 60°
0° 45°
-90°
345°
-4
30°
165°
15°
1
90°
-1
180°
270°
360°
180°
450°
0°
195°
-2
-3
210°
225°
-5
30° 60° 90°
0° 45°
180°
270°
360°
450°
540°
Animate
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
Shelly Berman
180°
270°
360°
450°
0°
195°
-3
345°
210°
330°
-4
-
315°
300°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
f(θ) = a⋅sin(b⋅θ)
a = 4.00
b = 5.00
225°
-5
-90°
30° 60° 90°
0° 45°
180°
270°
360°
450°
540°
Animate
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
p. 3 of 4
Polar Roses.doc
315°
300°
255° 270° 285°
θ = 60°
-180°
240°
-
255° 270° 285°
θ = 60°
90°
-1
180°
-2
330°
240°
-
-90°
345°
-4
-90°
300°
255° 270° 285°
+
180°
-3
-180°
30° 60° 90°
0° 45°
Hide Cartesian
60°
+
45°
165°
-2
-90°
315°
240°
Animate
1
-90°
-
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
150°
3
330°
225°
75°
135°
4
0°
345°
-
a = 4.00
120°
5
450°
60°
b = 1.00
Family of Roses
360°
210°
300°
f(θ) = a⋅sin(b⋅θ)
+
270°
-5
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
180°
195°
255° 270° 285°
Animate
Hide Cartesian
90°
-1
-3
315°
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
-180°
15°
180°
-4
240°
-
-90°
30°
165°
-2
330°
θ = 60°
0° 45°
-90°
345°
-4
-90°
60°
+
45°
1
180°
-2
-180°
75°
150°
3
15°
90°
135°
4
165°
105°
120°
5
150°
3
Hide Polar
+
60°
+
45°
135°
4
Family of Roses
Hide Cartesian
75°
120°
5
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
f(θ) = a⋅sin(b⋅θ)
a = 4.00
b = 6.00
Jo Ann Fricker
USING GEOMETER’S SKETCHPAD TO SUPPORT MATHEMATICAL THINKING
Adapted from Exploring Precalculus with The Geometer’s Sketchpad 4.0
http://www.keypress.com/catalog/products/software/Prod_GSPModExpPrecalc.html
Family of Roses
Hide Cartesian
Hide Polar
+
105°
90°
2
30°
2
1
-90°
90°
-1
180°
270°
360°
450°
0°
195°
-3
210°
225°
-5
30° 60° 90°
270°
450°
360°
300°
-180°
Hide Polar
+
105°
90°
2
30°
30° 60° 90°
0° 45°
180°
270°
450°
360°
300°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
f(θ) = a⋅cos(b⋅θ)
a = 4.00
b = 0.50
Family of Roses
Hide Polar
105°
90°
75°
120°
60°
+
45°
135°
150°
3
2
15°
30°
165°
15°
1
90°
-1
180°
270°
360°
450°
0°
195°
210°
225°
-5
30° 60° 90°
270°
450°
360°
300°
Family of Roses
Hide Polar
+
105°
90°
-180°
2
210°
330°
225°
-
315°
240°
30° 60° 90°
0° 45°
300°
255° 270° 285°
180°
270°
450°
360°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
f(θ) = a⋅cos(b⋅θ)
Animate
Hide Polar
+
2
1
75°
60°
+
45°
150°
3
15°
90°
135°
4
30°
105°
120°
5
165°
a = 4.00
b = 1.00
Family of Roses
Hide Cartesian
150°
3
345°
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
60°
+
45°
135°
4
-90°
75°
120°
5
0°
195°
-
b = 0.75
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
Hide Cartesian
450°
θ = 60°
a = 4.00
f(θ) = a⋅cos(b⋅θ)
Animate
360°
-5
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
270°
-4
255° 270° 285°
180°
180°
-3
315°
240°
-
90°
-1
180°
-2
330°
θ = 60°
0° 45°
-90°
345°
-4
30°
165°
15°
1
90°
-1
180°
270°
360°
180°
450°
0°
195°
-2
-3
210°
225°
-5
30° 60° 90°
0° 45°
180°
270°
360°
450°
540°
Animate
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
Shelly Berman
180°
270°
360°
450°
0°
195°
-3
345°
210°
330°
-4
-
315°
300°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
f(θ) = a⋅cos(b⋅θ)
a = 4.00
b = 1.25
225°
-5
-90°
30° 60° 90°
0° 45°
180°
270°
360°
450°
540°
Animate
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
p. 4 of 4
Polar Roses.doc
315°
300°
255° 270° 285°
θ = 60°
-180°
240°
-
255° 270° 285°
θ = 60°
90°
-1
180°
-2
330°
240°
-
-90°
345°
-4
-90°
315°
255° 270° 285°
4
180°
-3
-180°
240°
5
165°
-2
-90°
330°
225°
+
1
-90°
345°
210°
Hide Cartesian
150°
3
0°
195°
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
60°
+
45°
135°
4
-90°
75°
120°
5
450°
Animate
b = 0.25
Family of Roses
Hide Cartesian
360°
-
a = 4.00
f(θ) = a⋅cos(b⋅θ)
Animate
270°
θ = 60°
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
540°
180°
-5
255° 270° 285°
180°
90°
-1
-3
315°
Exploring Precalculus with Sketchpad
(C) 2005 by Key Curriculum Press
-180°
15°
180°
-4
240°
-
-90°
30°
165°
-2
330°
θ = 60°
0° 45°
-90°
345°
-4
-90°
60°
+
45°
1
180°
-2
-180°
75°
150°
3
15°
90°
135°
4
165°
105°
120°
5
150°
3
Hide Polar
+
60°
+
45°
135°
4
Family of Roses
Hide Cartesian
75°
120°
5
Edit the function below to try your own.
You can use parameters a and b in the
function you create.
f(θ) = a⋅cos(b⋅θ)
a = 4.00
b = 1.50
Jo Ann Fricker