James W. Wilson and Richard Moushegian

University of Georgia

__General__:

It has now become a rather standard exercise, with available 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, we can observe the patterns for the roots of

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; hence, 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; hence, the
original equation has no real roots. Note that for b=0, the equation
of the parabola has no linear term in x, and the graph of the
parabola in centered on the y-intercept of (0,+1).

Similarly for b = 2, the parabola is tangent
to the x-axis (one real __negative__ root) and for b > 2,
the parabola intersects 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

where the curves with __positive__ b-values
are shown in **red** , __negative__ b-values are shown in **green**, and
the __zero__ b-value is shown in **blue**.

The locus is a parabola which is shown as a
**light-blue** curve with the following equation:

In general, the set of parabolas can be shifted
along the y-axis by varying the constant, and the parabola passing
through the vertices will also shift. This shifting can be shown
with a different __y-intercept as -3 __with the associated
graphs.

__Graphs in the xb-Plane__:

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 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, therefore, have the following graph:

For each value of b we select, we get a horizontal line. Judging from the above graph, it is clear on a single graph what the roots are based on how the horizontal line intersects the hyperbola:

(1) If b > 2, there are two

negativereal roots for the original equation.(2) If b = 2, there is one

negativereal root.(3) If -2 < b < 2, there are no real roots.

(4) If b = -2, there is one

positivereal root.(5) If b < -2, there are two

positivereal roots.

Suppose the conjecture is made that, half-way
between the two points on either side of the b = 5 line, is the
value **(-b/2a)** as part of the quadratic formula. Then the
midpoint on the b = 5 line, in general, would be (-b/2a, 5). The
**slope** of the line through the origin will be **-2a/b**.
Hence, the **equation **of the line will be **y = (-2a/b)x**
as illustrated by the following **purple** line:

Also consider the case when c = - 1 besides c = + 1. Then we get a hyperbola that takes up the space on the other side of the asymptotes of the original hyperbola. Again the intersection points will be the roots of the equation.

__Graphs in the xc-Plane__:

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 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.

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.