Thomas Earl Ricks

Mathematics Education

 

Assignment # 2

 

“Ellipsoids and Such..”

 


 

For this Write-up, I have selected problem 10 from Assignment #2.  It reads in part:

 

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 coefficients of the xy term is close to the boundaries.

 

 

We thus begin the explorations.

 

Upon graphing the first two graphs we get:

 

 

 

We therefore observe that adding the xy term skews the circle along the line

 

y=-x

 

Curiosity would suggest we try:

 

 

which yields the graph stretched along y=x

 

 

 

 

 

We now turn our attention to other possible coefficients and obtain the following graphs for positive xy coefficients:

 

 

 

Thus the higher the coefficient, the more the graph looks like the axes.  It appears that two types of graphs are created by varying the coefficient: One piece (elliptical) graphs, and two piece (hyperbolic) graphs. It appears the split between the two graphs occurs around the coefficient 2.

 

Looking at coefficients near 2 this assumption appears correct:

 

 

The coefficients less than 2 seem to curl into an ellipse, while those greater than 2 seem to bend outward like a hyperbola.  Similar graphs are obtained for negative coefficients:

 

 

 

Thus –2 appears to be the dividing number for the negative coefficients.  The computer has trouble graphing values very close to 2 or –2, as can be seen in the graph below, which is distorted.  It has limits on its computational abilities:

 

But we can still observe that the 1.99 coefficient stays elliptical, while the 2.01 begins to branch out.  Thus we assume 2 is the limit of the boundaries.

 

 


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