1. LEMNISCATE McKenzie Reimondo Emma Stoker Period B February 27, 2014

2. 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”.

3. Its Polar Equation Equations of Lemniscates where a is NOT equal to 0

4. 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

5. How the a value affects the graph (cosine) a > 1 vs. a 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

6. 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

7. How the a value affects the graph (sine) a > 1 vs. a 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

8. 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.

9. Cites Used http://sites.csn.edu/istewart/mathweb/math127/polar_equ/polar_equ.htm