EMAT 6680 Assignment 3

Last modified on September 10, 2001.
The attached 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 and James M. Meneguzzo
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


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 positive real roots when b < -2.

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

For the case where c is negative, it appears that we get one positive root and one negative root for all values of b. As the values for b get very large, the positive root approaches 0 and the negative root approaches negative infinity. As the values for b get very small, the negative root approaches 0 and the positive root approaches positive infinity.

Graphs in the xa plane.

In the following example, the equation:

is considered. This graph produces a hyperbola. As we consider different values for y and graph them, these will be lines intersecting the hyperbola and these intersection points will be the roots for those values of a in the original equation. Let us look at the graph for different values of y:

From the graph, it appears that when a has negative values, this produces one positive and one negative root for the original equation. When a =o, their is one root at -1. For positive values of a only one root is produced and it will be negative.

Graphs in the xc plane.

In the following example the equation

is considered. This graph will produce a parabola. Let us consider different values for y which will be horizontal lines crossing our parabola. The intersection points between these lines and the parabola will be the roots for those values of c in our original equation. Let us look at such a graph:

For negative values of c, we again produce one negative and one positive root, similar in nature to our graph in the xa plane. However, we differ her in this plane for c=o, where we produce two roots instead of one, we still get the -1 root and add to it the root of 0. As the c values start to become positive, we produce two negative roots. There is a maximum positive value for c, where we get just one negative root of -0.5and any greater values of c produce no more roots. This value for c is 0.25.

Send e-mail to jwilson@coe.uga.edu
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