The Department of Mathematics Education

EMT 668 Assignment 3

Last modified on September 15, 1997.
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" parts 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 Karline Feller
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 value; that is, 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 root at the point of tangency. It is positive. For -2 < b < 2, the parabola does not intersect the x-axis. That means the original equation has no real roots. Similarly for b = 2 the parabola is tangent to the x-axis and has one real negative root. For b > 2, the parabola intersets the x-axis twice to show two negative real roots for each b.

The xb plane

Consider again the equation

Now x is one variable and b is the second variable with y fixed at zero. 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.
This seems easier to see when graphed in the xb plane.

Let's see what happens when we graph

Notice that the intersections of this equation are coincident with those from the previous two equations.

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

Now we have hyperbolas opening in the opposite direction.

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 will be at the roots of the original equation at that value of c. Below 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.

Again for each value of c selected, there is a horizontal line and it will be easy to see how many roots there are and whether or not they are real. Let's see what happens if this graph is overlaid with
one in which b is replaced with -b.

That merely shifts the graph to the left. What if the c is replaced with a -c?

That's simply inverts the graph. It seems that one can easily read the roots of the equation by graphing each desired value for c. This is similar to the situation in the xb plane.

The xa plane

Here is a graph that is undefined at 0. Just as with the xb and xc planes we can graph in values for a, say a=2 and a = -4 to get the roots.

If a>4 there are no real roots; when a=4 there is one negative real root. If 0 < a <4, there are two real negative roots; for a < 0, there are two roots, one positive, one negative. So we see that whatever value of a is used we can easily find the roots from using a graph in the xa plane.

Conclusion: Graphs in the xb,xc, and xa plane are helpful in finding all the roots of an equation quickly and easily when a, b, and c are varied.

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