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“Eccentricity”

Is that a dance move??

 

By: Carly Cantrell

 

 

This investigation will use a graphing software to compare the below equations to try and figure out what “eccentricity” is all about.

 

The following will be explored:

 

 


 

When p=1 and k=1

 

c c

 

c c

 

 

 

The graphs have shown us very similar parabolas… keep that in mind!

The parabolas only differ by a determined transformation.

 


 

When p=1 and k=2

 

c c

 

c c

 

 

 

The graphs have shown us very similar hyperbolas… Hmmmmmm.

There are no long similar parabolas, but very similar hyperbolas.

 

 


 

When p=1 and k=

 

c c

 

c c

 

 

The graphs have shown us very similar ellipses…

I am beginning to see a pattern.

 


 

 

Have you caught on?

Eccentricity is not a dance move.

Rather, it can be thought of as a measure of how much the conic section deviates from being circular.

 

Specifically,

The eccentricity of a parabola is 1.

The eccentricity of a hyperbola is greater than 1.

And the eccentricity of an ellipse is less than 1 but greater than 0.

 

What does that make the eccentricity of a circle?

Well, a circle does not deviate from a circle very much, so it is 0.

 

 


 

Below are more intriguing examples:

 

c

 

 

c

 

An imporatant note: As theta was affected by 45 degrees, the ellipse roatated to sit on poles along the line y=x.

 

 

c

 

 

 

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