HamiltonHardisonÕs Exploration of

Assignment 2:  An Exploration of a Quadratic Relation

From Jim WilsonÕs Assignment Page:  Graph .  Now, on the same axes, graph

Describe the new graph. Try different coefficients for the xy term. What kinds of graphs do you generate? What coefficients mark the boundaries between the different types of graphs? How do we know these are the boundaries? Describe what happens to the graph when the coefficient of the xy term is close to the boundaries.

It is clear that the graph of  is a circle.

Graphing  gives an ellipse whose major axis appears to be on the line . 

By graphing  for n = .2, .4, .6, É , 1.6, 1.8,  We see that we have a series of ellipses.  Each has itÕs major axis lies along the line  .  As n increases, the ellipse becomes longer and thinner.

When n=2, we have a very different graph.

This appears to be two parallel lines.  Algebraically we have:
  By solving for (x+y); We have  or .  This confirms the conjecture that the graph is no longer an ellipse, but two parallel lines.

As the coefficient of the xy term moves beyond 2; we notice a different class of graphs. The graph of  for n = 2.2, 2.4, 2.6, É , 3.6, 3.8 is shown below.  

Each curve appears to be a hyperbola whose foci lie along the line .  As n gets further away from 2, the curvature of the graph becomes more exaggerated.  By examining the graph of , we can somewhat safely determine that all coefficients of the xy term greater than 2 will produce hyperbolas.  (Note the difference in scale)

If negative values are considered, the results can be summarized in the graph and table below:




Parallel Lines








Parallel Lines




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