Samantha Driskill 

Footnote 18 

5/7/08 

Families of Surfaces part II  - click here for the Maple 11 Version

Prompt 2:  

              Use a computer to investigate the family of surfaces .  In particular, you shoudl determine the transitional values of c for which the surface changes form one type of quadradic surface to another. 

Reply: 

              To first begin to answer this question, I needed to know the quadradic surface of the original function when c = 1 so that I can compare any differences to this original graph.  Below is the graph, an eliptic paraboloid extended along the z-axis.  

Now with this graph in mind, I can pick up on the changes in the quadradic surface as the constant "c" within the eqaution changes value.  To being testing out values, I picked my first value of c = 1/2.  With this value of c, the function didn't change quadradic shapes, but the diameter of the eliptic parabolid did decrease. 

The next two values that I explored were when c = 1/8 down to values of c = 1/93 and I found that as soon as c ≤ 1/8, the graph of the function z stayed relatively constant to the point where I couldn't tell if the diameter was shrinking or not because the values oc c had an extremely little affect on the function. 

So now that I have determined that the graph of the function as c appproaches zero is an elliptic paraboloid, I now will explore the values of c that are greater than 1.  The first value I began with was c = 2 and I found that the the quadradic surface had changed to resemble a parabolic cylinder along the z-axis. 

The next value I tried was c = 2.5.  I found that the outward bottom folds of the parabolic cylinder were starting to droop, the sides were starting to curve, ultimately, resembling a hyperbolic paraboloid.  With this new value of "c" it seems that the quadradic surface again changed. 

To make sure that that the quadradic surface did in fact change again from a parabolic cylinder to a hyperbolic paraboloid, I graphed c = 2.1, 3, 4, and 10 and found the same tendancy to curve and droop, meaning that when c= 2 the quadradic surface of the function is a parabolic cylinder.  Additionaly, so far I can theorize that when c > 2, the resulting quadradic surface is a hyperbolic paraboloid. Below is the resultig graph for when c = 2.1, 3, 4, and 10. 

However, at c = 20, the quadradic surface again changes from that of a hyperbolic paraboloid to a hyperbolic cylinder.  So now the rule has been defined that in the equation when 2 < c < 20 the quadradic surface is a hyperbolic paraboloid. Now, we can move on to define the boundries for c when the function's quadradic surface is a hyperbolic cylinder. Below I have included some graphs where c ≥ 20 for the values of c = 20, 32, 54, and 100. 

So after graphing values of c ≥ 20, I found that the quadradic surface did not change and only became more and more defined as a regular hyperbolic clyinder surface. Below is a graph with a large value of "c" to check if the graph continued on the same change trend: 

Additionally, to make sure that a "-c" value wouldn't produce differing quadradic surfaces as compared to their positive counterparts, I graphed several functions and found that, besides changing the direction of the graph, the translations values of "c" that changed the quadradic surfaces were the same.