It has now become a rather standard exercise, with availble technology,
to construct graphs to consider the equation
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.
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
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, the green concave-down one. Since the two zeroes of this parabola are x=1 and x=-1, we can find the factors of the parabola to be (x-1) and (x+1). In addition, we must account for the downward turn of the graph by multiplying it by -1. Therefore, our function of the locus of the vertices of the set of parabolas is:
which is further translated as:
Upon inspection, you will find that each of the above stated vertex points is on this parabola.
Consider again the equation
Now graph this relation in the xb plane. We get the following graph.
Notice it is a hyperbola with asymptotes of x=0 and y=-x. It behaves as a graph of y=-1/x does.
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 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 this case, we find the curve to be that of a hyperbola with asymptotes again of x=0 and y=-x. But this time, the curve acts like y=1/x. Now we can see that for each value of b we select, we can find one negative root and one positive root all along the curve.
In the following example the equation
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
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.