 # EMAT 6680 Assignment 3

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.

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 Melissa Bauers 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.

We begin our discssion with a look at the efffects of a on the equation To do this we must allow b and c to remain constant. Lets examine a for the values of -3(purple), -2(red), -1(blue), 0(grey), 1(green), 2(aqua), and 3(yellow), while b and c are 1. Graph in the xa Plane The graph in the xa plane look as followed; If we take any particular value of a, such as a = -3(blue) and 1(green), 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 of points correspond to the roots of the original equation. We will overlay a = -3(blue) and 1(green) onto our original equation. For a = -3(blue), we have two real roots and for a=1(green), we have no real roots. Notice for a = 0, one real root is revealed.

We continue our discussion by examining b for the equation We will look at b for -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 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 positive real roots when b < -2.

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 and c = -5 is shown. The equation will have no real roots at c =1 and two real roots at c = -5.

There is one value of c where the equation will have only 1 real root -- at c = 0.25. For c > 0.25 the equation will have no real roots and for c < 025 the equation will have two roots, both negative for 0 < c < 0.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.

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