Samantha Driskill 

Footnote 18 

5/7/08 

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

Prompt : Use a computer to invesitgate the family of functions for .  How does the shape of the graph depend on the numbers "a" and "b"? 

Response:  So to first begin to answer this question, I decided to keep both variables a and b = 1.  This way I could determine the basic quadradic surface so that I could compare it to the other graphs where I changed both the variables "a" and "b".  Below is the resulting graph. It looks closest to the quadradic surface of a eliptic paraboloid and reminds me of a upside-down bell flower. 

So now I am able to move on to manipulate either "a" or "b" in my equation to see what that change will have on my original graph.  The first varaible that I decided to manipulate is a = 2.  When a = 2 the bell shape is distorted and the sides along the y-axis start to droop toward zero, as shown below: 

So then I again set a = 1 and set y = 2 to see if manipulating y would have the inverse effect it did when y = 1 and x = 2.  I found that yes, in fact, it did have the inverse effect.  The bell shape is distorted with the sides along the x-axis starting to droop down to zero.  Below is the graph: 

But now what happens if both of the variables "a" and "b" are equal to each other, but not = 1? In this graph, I made both a and b = 2 and found my original bell flower shape, only with a greater height and greater diameter. Below is the graph: 

So now I have determined that when "a" and "b" are the same value, the graph resembles the original bell shape, only varrying in size of height and diameter depending the the choosen values of "a" and "b".  Now I want to determine if keeping one variable constant, like y = 1, while increasing the numerical value of "a", will that then continue to show the same trend as before with the drooping sides?  Thus, I decided to graph x = 6 and 27 while y = 1 and discovered that yes, the graph does continue to follow the trend I noticed before.  As the value "a" increases, the lower the sides of the bell along the y-axis approach zero.  Below are the two graphs: 

Now that I have determined the trend of change as "a" increases while "b" stays constant for values of "a" and "b" that are greater than or equal to one, I wanted to see if there was any significant difference in trend of change as a < 1.  The first value that I tried was a = 1/2 and y = 1and found that instead of the sides drooping along the y-axis, they drooped along the x-axis.  I found the same inverse trend in a = 1 and b = 1/2 (the sides along the y-axis drooped instead of along the x-axis). Below are the two graphs: 

Now I have pretty much determined that if the values of a or b ≥ 1, any change in "a" while "b" is constant will result in the sides of the bell drooping along the y-axis and any change in "b" while "a" is constant will result in the sides of the bell drooping along the x-axis.  Furthermore, when the values of a or b < 1, any chnage in "a" while "b" is constant will result in the sides of the bell drooping along the x-axis and any chnage in the "b" while "a" is constant will result inthe sides of the bell drooping along the y-axis. 

My next step is to determine what the affect on the shape of the graph is when a or b < 0. Below is the graph where I have kept b = 1 and changed a = -1.  The result is that instead of the sides along the x-axis drooping downward to zero, they are drooping upward toward zero.  The same the concept, but opposite, is found when b = -1 and a = 1. Below are the two graphs: 

To make sure that as either "a" or "b" decreases in value, their respecting sides will droop upwards their respecting axis, I did a quick check where a = 1 and y = -2 to find that this wasn't the case like I had originally expected.  As the values of "a" or "b" decreases below zero while the other is constant, the resulting sides don't droop upward toward their respecting axis, but continue to droop downward like before, only the droop begins from below zero. Below is the graph: 

Furthermore, I wasn't sure whether or not fraction values of "a" and "b" when a and b < 0, changed any part of the graph besides the droopiness of the respecting sides.  So below I have included several graphs where a = 1 and b = -1/2, -1/6 where I found no difference in the trend that I have before determined. 

My next step was to test my theory of whether or not my original bell shape when a = 1 and b = 1 would be the exact same thing, only extended along the negative z-axis (meaning it would be flipped upside down) when a = -1 and b = -1.  Below is the resulting graph: 

I now now what effect the negative sign has the graph of the original function.  My last step is to determine what the resulting graph looks like when the values of "a" and "b" are not equal to each other AND neither equals 1.  Below I have graphed a = 3 and b = 4 and found a mixture of results.  In the graph, it shows that both the sides of the bell along the x-axis and y-axis are drooping toward zero only varrying in degree of droopiness depending on the choosen values of "a" and "b".   

To make sure that this trend continues (which it does) , I have included some graphs where the values of "a" and "b" are greater than one and not euqal to each other: