**By:
James W. Wilson and Keith Schulte
University of Georgia**

Last modified on **June 30, 2003**.

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 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 for tthis equation, always passes through the same point on the y-axis
( the point (0,1 . 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
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

.

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 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. When we let b = 3, we have two roots and both are negative. The roots are approximately -.382 and -2.618. In the following graph, the red line represents b=3. We can see that it intersects in two places and that both are negative.

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, if we let b=3, we would get the
following graph. Once again, the
red line represents b=3. With c= -1, we see that our new equation,

has a different orientation. When b=3, the new equation, returns one positive root at about .303 and one negative root at about -3.003.

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.

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