No Longer XY


I was a little confused by this, as everyone knows their is only one coordinate plane the XY plane but my Grandma told not so. She said their is an endless possibilities of planes one has just to be willing to explore them and see what kinds of graphs one will get. Apparently, the Xb plane is her favorite as it has some curious possibilities within it. I asked her to explain what these different planes can do and she gave me an example. She said to consider the equation 1x2 + bx + 1 = 0 and to graph the equation in the xb plane. I was confused on how to do this as I had never been asked to do such a thing. She patted my hand and told me that I would learn one day when I was a true mathematician, but until then she explained that in this particular situation we already know that y=0, so our unknown are b and x. So in order to set up the xb plane, we need to solve for b in order to have our input be x and our output be b.

With this in mind I began my furious algebra work and found that when one solves for b we have:

b = (-x2 - 1)/2

So using my graphing calculator I was able to see what the graph would look like in the xb plane.

I showed Grandma and she did not seem to impressed, she told me that maybe one day I would be able to create such a fine design like that of her flower garden. This set me one a mission to figure out how to manipulate this function in the xb plan.

Before I left she explained that what this did for us. This graph helps us to see what types of solutions (real or complex) the quadratic will have. This first one tells us that the original equation will have 2 negative real roots when b>2, one real negative root when b=2, no real roots when -2 < b < 2, one real positive root when b =-2, and two real possitive roots when b<-2.

Using this new bit of information I knew what I wanted to find to try to impress my Grandma. I wanted to know if it was possible to find the equation of the locus of the maximums and minimums of each of the two curves created by the quadratic. After that I will try to recreate my Grandma's flower garden.

First thing is first I would have to do a little exploration with these types of functions. I began by changing the a coefficeint but that began to stretch and shrink the curves, which was not something I wanted to change, so I decided I would have to leave the coefficient a as 1.

With no changes to the coefficient a that means that the coeffienct c was next, so I began to manipulate c and some amazing things began to happen.




Part II: The manipulation of C