WU Number 3

David W. Stinson

The following 4-page paper is the start of an article that might appear in a journal such as the Mathematics Teacher -- the audience being mathematics teachers who might use some of the ideas for instruction.

It is a start; incomplete, unclear, maybe in error; maybe glossing over significant points and stressing some obvious or trivial points.

Your assignment:

Sign on as a co-author. Rewrite and complete the article. This means you must come to grips with whatever points are to be essential, what to add, what to delete, and what to edit. The "different" approaches to this topic are really in the graphs in the xb, xc, or xa planes. You might want to examine a bunch of these before trying to re-write.

Some Different Ways to Examine

James W. Wilson, Ph. D. and David W. Stinson

University of Georgia, Athens 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 y = x^2 + bx +1 for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

To view Quick Time movie click here.

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 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

y = x^2 +bx+1

.

Show that the locus is the parabola

y = -x^2 +1

 

Generalize.

 

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 = 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.

 

 

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 positi>

 

Transfer interrupted!

> Consider the case when c = - 1 rather than + 1.

 

 

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.

 

 

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.

Return to Homepage