This is the writeup of Assignment #3 
Brian R. Lawler

EMAT 6680 
12/8/00

and to overlay several graphs of
for different values of a, b, or c as the other two are held constant. From these graphs discussion of the patterns for the roots of
can be followed. For example, if we set for b = 3, 2, 1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained. (Click on the graph itself to download a file for exploration. Do this on any graph on the page.)
We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the yaxis (the point (0,1) with this equation). For b < 2 the parabola will intersect the xaxis in two points with positive x values (i.e. the original equation will have two real roots, both positive). For b = 2, the parabola is tangent to the xaxis and so the original equation has one real and positive root at the point of tangency. For 2 < b < 2, the parabola does not intersect the xaxis  the original equation has no real roots. Similarly for b = 2 the parabola is tangent to the xaxis (one real negative root) and for b > 2, the parabola intersects the xaxis twice to show two negative real roots for each b.
Now consider the locus of the vertices of the set of parabolas graphed from .
See that the locus is the parabola .
To prove that is the locus of the vertices of the parabola, consider the set of vertices of all quadratics of the form .
By rewriting in vertex form 
, 

the general form of the vertices can be seen to be 
. 

Substituting 3 values for b will identify 3 particular points. The unique quadratic through those three points can identified with linear algebra. 
Hence, passes through all vertices.
To generalize this hypothesis, observe similar results in quadratics with other a and c coefficients. For example, in the case below, a = 0.7 and c = 3.
See that the locus of this family of quadratics can be defined in general by .
Again, by rewriting the general quadratic in vertex form 
, 

the general form of the vertices can be seen to be 

Substituting 3 values for b will identify 3 particular points. The unique quadratic through those three points can be verified to be with linear algebra.
Consider again the equation
Now graph this relation in the xb plane. We get the following graph.
If we take any particular value of b, say b = 5, and overlay this equation on the graph we add a line parallel to the xaxis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.
For each value of b we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when b > 2, one negative real root when b = 2, no real roots for 2 < b < 2, one positive real root when b = 2, and two positive real roots when b < 2.
Consider the case when c =  1 rather than + 1. (In the example below, I had to use the y variable to mimic the b. Otherwise, you will see everything works the same  this is why I left the original equation in green.)
Again, for each value of b we select, we get a horizontal line. It is clear on a single graph that we get two real roots of the original equation for all values of b . One root will always be positive, and the other negative.
It is possible to consider many values of c all at one time. First, move the c to the other side of the functions. Then, observe a contour plot or a 3D graph to observe the effects of changing c as x and b are free.
is considered. If the equation is graphed in the xc plane, it is easy to see that the curve will be a parabola. For each value of c considered, its graph will be a line crossing the parabola in 0, 1, or 2 points  the intersections being at the roots of the original equation at that value of c. In the graph, the graph of c = 1 is shown. The equation
will have two negative roots  approximately 0.2 and 4.8. (Again,note that in the example below, I had to use the y variable to mimic the c. Otherwise, you will see everything works the same.)
There is one value of c where the equation will have only 1 real root  at c = 6.25. For c > 6.25 the equation will have no real roots and for c < 6.25 the equation will have two roots, both negative for 0 < c < 6.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.
Interestingly, varying this value of b while graphing in the xc plane will yield the same behavior as was seen in the initial problem of graphing in the xy plane. We will leave it to the reader to explore and explain this phenomenon.
Also, how about graphing in the xa plane? Happy mathematics!
Comments? Questions? email either of us at: 
Dr. Jim Wilson

jwilson@coe.uga.edu  
Brian Lawler

blawler@coe.uga.edu 
Last revised: December 28, 2000 