Parabolas and the xb plane
In this exploration we will focus on looking at graphs of quadratics in a less than typical manner. Rather than examing the traditional f(x) = ax2 + bx + c = 0 equation we will let the value b be a function of x. For purposes of graphing such an expression we will rewrite ax2 + bx + c = 0 as b = -(ax2 +c) / x .
If we were to let a = 0 We would no longer have a quadratic to explore, and our graph would be simply be b = c/x . We will investigate for non zero values of a, as well as non zero values of c ( c = 0 gives a line through the origin with slope -1).
To begin let's consider a=1 , c=1 and c=10.
(Follow this link to interact with the changing values of c)
It appears that as the value of c increases the two branches of the hyperbola retreat from one another. Likewise as the value of c becomes smaller, the branches become closer to one another, whence we get c = 0 and we obtain a straight line. As c becomes negative we see from the following pictures that the branches of the hyperbola open in different directions.
here we have values c = -1, and c = -10 respectively
Since we have represented b as an explicit function of x above, if we substitute this explicit expression into our original quadratic, ax2 + (-(ax2 +c) / x )x + c = 0, it is easy to see that we have an identity. Thus from a graph in the xb plane to obtain a root of the original quadratic we need only find the intersection(s) of the hyperbola in the xb plane with a horizontal line. The intersection piont has coordinates (x,b) , and as previously mentioned b is a function of x which satisfies ax2 + (-(ax2 +c) / x )x + c.