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