EMAT 6680 Assignment 3

Last modified on August 20, 1998.

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

by

James W. Wilson and Steven F. Imler
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:

By using the general equation to determine a, for the focus (0,a), 4a = -1. Negative 1 being the coefficient of x^2. A = -1/4; adding 1 results in a = 3/4. The focus = (0,.75) and the directrix is y = 1.25 since the vertex is equidistant from the focus and the directrix.

Graphs in the xa plane.

Consider the equation

ax^2 +5x+1 = 0

This relation will be graphed in the xa plane:

Note that the x and y axes are horizontal and vertical asymptotes. The vertical blue line is the vertical asymptote. If we take any particular value of a, say a = 5, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xa plane the intersection point corresponds to the root of the original equation for that value of a. We have the following graph. Note that the value of the root is given:

For each value of a we select, we get a horizontal line. It is clear on a single graph that we get one negative real root of the original equation when a >0 and one positive real root when a < 0.

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 graphs. Note that the values of the roots are given (where y = b):

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. Note the blue graph is the same as above:

while the green graph is c = -1 instead of +1. The roots of the equation with b = 5 are given:

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.

Send e-mail to sfimler@arches.uga.edu