Download - Lemniscate
LEMNISCATEMcKenzie Reimondo
Emma Stoker
Period B
February 27, 2014
What is it?
• A lemniscate graph takes on the shape of a figure eight
• It is similar to a rose graph, but it differs because a lemniscate graph only has two “petals”.
Its Polar Equation
Equations of Lemniscates where a is NOT equal to 0
How the a value affects the graph (cosine)• Positive a value vs. negative a value (in a cosine equation):
The red graph has the equation: r= sqrt(2cos(2t))
The green graph has the equation:
r= sqrt(-2cos(2t))
A positive a value makes
the graph lie on the x-axis
A negative a value flips the graph and
makes it lie on the y-axis
r(t)=sqrt(2cos(2t))
r(t)=sqrt(-2cos(2t))
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
How the a value affects the graph (cosine)•a > 1 vs. a < 1 and larger vs. smaller a values that are > 1 (in a cosine equation):
•The red graph equation: r=sqrt(.5cos(2t))
•The pink graph equation: r=sqrt(2cos(2t))
•The green graph equation: r=sqrt(3cos(2t))
•The blue graph equation: r=sqrt(4cos(2t))
•As the a value increases the loop of the petal
expands and as the a value decreases the loop condenses
r(t)=sqrt(.5cos(2t))
r(t)=sqrt(2cos(2t))
r(t)=sqrt(3cos(2t))
r(t)=sqrt(4cos(2t))
-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
x
y
How the a value affects the graph (sine)
•Positive a value vs. negative a value (in a sine equation):
•The black graph equation: r=sqrt(5sin(2t))
•The purple graph equation: r=sqrt(-5sin(2t))
•The positive a value makes the graph’s
loops land in the first and third quadrant
•The negative a value makes the
loops land in the second and fourth quadrant
r(t)=sqrt(5sin(2t))
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
x
y
r(t)=sqrt(-5sin(2t))
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
3.5
4
x
y
How the a value affects the graph (sine)
•a > 1 vs. a < 1 and larger vs. smaller a values that are > 1 (in a sin equation):
• The green graph equation:
r= sqrt(.5sin(2t))
• The red graph equation: r=sqrt(2sin(2t))
• The blue graph equation: r=sqrt(3sin(2t))
• The gray graph equation: r=sqrt(4sin(2t))
• The less the a value is the more
condensed the graph is and the more the a value is the more the loops expand
r(t)=sqrt(2sin(2t))
r(t)=sqrt(3sin(2t))
r(t)=sqrt(4sin(2t))
r(t)=sqrt(.5sin(2t))
-3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
2.5
3
x
y
Cosine vs. Sine
• Cosine lemniscate graphs lie on either the x or y-axis depending on the positive or negative a value.
• Sine lemniscate graphs are on an angle and lie in two quadrants that are diagonal from each other; the positive or negative a value affects which two quadrants the graph lies in.
• Both cosine and sine graphs go through the pole.
Cites Used
• http://sites.csn.edu/istewart/mathweb/math127/polar_equ/polar_equ.htm