The attached 4-page paper is the start of an article that might appear in a journal such as the Mathematics Teacher -- the audience being mathematics teachers who might use some of the ideas for instruction.

It is a start; incomplete, unclear, maybe in error; maybe glossing over significant points and stressing some obvious or trivial points.

Your assignment:

Sign on as a co-author. Rewrite and complete the article. This means you must come to grips with whatever points are to be essential,
what to add, what to delete, and what to edit. The "different" approaches to this topic are really in the graphs in the xb, xc, or xa planes.
You might want to examine a bunch of these before trying to re-write.


Some Different Ways to Examine

ax^2 = bx = c = 0


James W. Wilson and LaShonda Davis
University of Georgia

It has now become a rather standard exercise, with availble technology, to construct graphs to consider the equation

ax^2 + bx + c = 0

and to overlay several graphs of

y = ax^2 + bx + c

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

ax^2 + bx + c = 0

can be followed. For example, if we set

y = x^2 + bx + 1


for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.


We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis ( the point
(0,1) with this equation). For b < -2 the parabola will intersect the x-axis 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 x-axis 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 x-axis -- the original equation has no real roots. Similarly
for b = 2 the parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersets the x-axis twice to show two
negative real roots for each b.

Now consider the locus of the vertices of the set of parabolas graphed from

y = x^2 + bx + 1 .

As you can see from the graph below, the locus of the vertices of the set of parabolas is itself a parabola. Upon calculating this value for
each of the graphs pictured below by using the first derivative of the functions, the vertices are:

All of these points are on the same parabola which is the one that concaves downward. Since the two x-intercepts or the places that satisfy the problem when it equals zero of this parabola are x=1 and x=-1, it is easy to determine the factors of the parabola to be (x-1) and (x+1). In addition, we know that because it has a downward turn that the roots are multiplied by a -1. Therefore, the function of the locus of the vertices of the set of parabolas is:

y = -1*(x-1)*(x+1) = -x^2 +1


Graphs in the xb plane.


Consider again the equation

x^2 = bx + 1 = 0

Now graph this relation in the xb plane. We get the following graph.


It is a hyperbola with asymptotes of x = 0 and y = -x.

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 x-axis. 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


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 this case, we find that the curve is also a hyperbola with asymptotes again of x=0 and y=-x.
It seems that for any value of b we want where we let b vary, we can find one negative root and one positive root.

Graphs in the xc plane.

In the following example the equation

x^2 + 5x + c = 0


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 orignal equation at that value of c. In the graph, the graph of c = 1 is shown. The equation

x^2 + 5x + 1 = 0

will have two negative roots -- approximately -0.2 and -4.8.




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.


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