James W. Wilson and Jeff Lalaian

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

Show that the locus is the parabola

Generalize.

Consider again the equation

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

If we take any particular values of b, say b = 3, 2, 1, -1,
-2 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.

so we can see that when c is negative we have two roots. In fact, we can see that when c is negative one root will be positive, one negative. Let's plug in multiple values to see this:

As c approaches zero we see that we get a line that is asymptotic to the hyperbola.

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.

Let's let c also equal values 2, 6, and 7 to see this:

Let's further investigate.

** Graphs in the x,a plane**.

Let's begin by looking at the equation

Let's see how this will look along with a values -1, .25, 1:

From this we can see that when a>.25 there are no solutions. When a=.25 there is exactly one solution. when a< .25 there are two solutions.

So we have seen the differences when in the (x,b), (x,c), and (x,a) planes, a very interesting study.

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