## Second Degree Equations

#### by Kent Wiginton

In assignment 2 problem 10 we are asked to look at the graph of the following function:

Then on the same axis we are asked to graph:

Let's consider what the graph would look like when we decide to vary the coeffecient of the xy term.

Click here for a little movie to see how the graph will change.
What do you notice about the shape as n gets close to 2?  Is it still an ellipse?

As we begin to vary n from zero to three we can see the circle begin to flatten out to an ellipse and then when n = 2 we will see two parallel lines as the ends of the ellipse go out to infinity.  As n passes 2 then the graph begins to take on a new shape.  It begins to look like a hyperbola.

The Red circle is given by the quation: .

The parallel Purple lines are given by the equation: .

The Blue graph is the hyperbola given by the equation:

Just by changing the coefficent of the middle term we can create very different graphs. A great way to see them all is by the three-dimensional graph. Remember that we just looked at one slice of this surface.

Take a look

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