WRITE-UP 3

by

Matt Sorrells

Some Different Ways to Examine

by

James W. Wilson and Matthew P. Sorrells

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.

Looking at the same graphs as above, one can determine the locus of the vertices of the set of parabolas graphed from:

The locus of the vertices is the line or curve that intersects each of the graphs at its respective vertex. The vertex can be found by taking the derivative of the graphs, and then by either setting the derivatives equal to zero and solving for x or by graphing the derivative and finding the point of intersection with the derivative and its respective graph. This intersection will also be a point of tangency.

The locus of the vertices for the above equation is the parabola

The original graph along with the locus of the vertices is below:

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 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. The two graphs along with the graph of b=5 are below.

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 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 as can be seen below when we graph the two equations,

and c=1

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 at 0 when c = 0 and one negative and one positive when c < 0.