James W. Wilson and Angel R. Abney

University of Georgia

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

To show that the locus of vertices is a parabola, we'll use the vertex formula. Let , where a = 1.

Therefore, we can see that the locus of vertices is the parabola:

See graph.

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 = 3, 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.

b

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. So we have the
equation

See the red curve on the graph:

b

Notice that for each value of b, which corresponds to a horizontal
on the graph, we get two real roots of the equation above. Note
also, for all values of b, we get one positive and one negative
*x*-intercept for the function

This is clear to see graphically, but it is also easy to show analitically. The discriminant,

is always greater than zero, which implies that the function
has two distinct, real roots for all values of b. In fact, we
can say that *f* will have two distinct, real roots for all
values of b if a>0, and c<0.

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 below, the
horizontal line at c = 1 is shown.

Thus, the equation

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

Notice, there is one value of c where the equation will have only 1 real root of multiplicity two -- 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 distinct, real roots: both negative for 0 < c < 6.25, one negative and one equal to zero when c = 0, and one negative and one positive when c < 0.

In the following example the equation

is considered. If the equation is graphed in the xa plane, it is easy to see that the curve is a rational function with a horizontal asymptote at y=0 and a vertical asymptote at x=0. For each value of a 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 a. In the graph below, the horizontal line at a= 2 is shown.

Thus, the equation

has two negative roots -- approximately x = -2.28 and x = -.22.

Notice, there are two values of a where the equation will have only 1 real root -- at a = 6.25 (root of multiplicity 2) and a = 0 (root of multiplicity 1). Note, if a = 0, the equation is no longer a parabola, but a line. For a > 6.25, the equation will have no real roots and for 0 < a < 6.25 the equation will have two distinct, real roots, both negative. For a < 0, the equation will have two distinct, real roots, one negative and one positive.

Again, this is easy to show analytically. The discriminant,

is only positive when a < 6.25.

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