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James W. Wilson and Shannon M. Umberger

University of Georgia

It has now become a rather standard exercise, with available technology, to construct graphs in order to consider the equation

and to overlay several graphs of

for different values of a, b, or c as the other two variables 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 then overlay the graphs, the following figure is obtained:

We can discuss the "movement" of a parabola as b
changes. Notice that the parabola always passes through the same
point on the y-axis ( the point (0,1), using the above equation).
For b < -2, the parabola intersects the x-axis in two points
which have positive x values (i.e., the original equation has
two positive real roots). For b = -2, the parabola is tangent
to the positive x-axis, and so the original equation has one positive
real root (at the point of tangency). For -2 < b < 2, the
parabola does not intersect the x-axis; therefore, the original
equation has no real roots (i.e., the roots are imaginary). Similarly,
for b = 2, the parabola is tangent to the negative x-axis (so
the original equation has one negative real root), and for b >
2, the parabola intersects the x-axis twice (so the original equation
has two negative real roots).

Now consider the locus of the vertices of the set of parabolas
graphed from

We can show that the locus of the vertices is the parabola which has the equation

The figure below shows the original graph plus the graph of the locus of the vertices. What can you generalize?

There are other ways to examine the roots of a quadratic equation by examining the graph. One way is to graph in the xa plane instead of in the xy plane. This simply means that we will substitute y for a into a quadratic equation and then graph.

Let us consider the equation

Now we will graph this relation in the xa plane. To do this, we need to graph the equation

We obtain the following graph:

If we take any particular value of a, and overlay the equation y = a on the graph, we add a line parallel to the x-axis. If the line intersects the curve in the xb plane, the intersection points correspond to the roots of the original equation for that value of a.

Take a look at the following figure. For a = 2, the equation y = 2 is graphed in green. For a = 0.25, the equation y = 0.25 is graphed in blue. And for a = -2, the equation y = -2 is graphed in purple.

We can see on this single coordinate plane that we get no real roots of the original equation when a > 0.25, one negative real root a = 0.25, two negative real roots when 0 < a < 0.25, and one positive real root and one negative real root when a < 0.

Now, instead of graphing in the xy plane, we will graph in the xb plane. This simply means we will substitute y for b into a quadratic equation and then graph.

Consider again the equation

Now graph this relation in the xb plane. To do this, we need to graph the equation

We obtain the following graph:

If we take any particular value of b, and overlay the equation y = b on the graph, we add a line parallel to the x-axis. If the line intersects the curve in the xb plane, the intersection points correspond to the roots of the original equation for that value of b.

Take a look at the following figure. For c = -3 and c = 3, the equations y = 3 and y = -3 are graphed in green. For c = -2 and c = 2, the equations y = -2 and y = 2 are graphed in blue. And for c = -1 and c = 1, the equations y = 1 and y = -1 are graphed in purple.

It is clear on this single coordinate plane that we get two
negative real roots of the original equation when b > 2, one
negative real root when b = 2, no real roots when -2 < b <
2, one positive real root when b = -2, and two positive real roots
when b < -2.

Now, consider the case when c = - 1 rather than + 1. Therefore,
the equation

graphed (in turquoise) below in the xb plane, looks like this:

What can you deduce about the roots of the equation

for different values of b by examining the above graph?

Finally, instead of graphing in the xy plane, we will graph in the xc plane. This simply means we will substitute y for c into a quadratic equation and then graph.

In the following example, consider the equation

To graph this relation in the xc plane, we need to graph the equation

We obtain the following graph:

If the equation is graphed in the xc plane, it is easy to see that the curve will be a parabola. If we take any particular value of c, and overlay the equation y = c on the graph, we add a line parallel to the x-axis. If the line intersects the curve in the xc plane, the intersection points correspond to the roots of the original equation for that value of c.

In the following figure, for c = 1, the graph of y = 1 is shown in green. For c = 6.25, the graph of y = 6.25 is shown in blue. For c = -1, the graph of y = -1 is shown in turquoise. And for c = 0, the graph of y = 0 is graph in purple (partially hidden by the x-axis).

We can observe that there is only one value of c where the equation will have one real root: at c = 6.25. For c > 6.25 , the graph equation will have no real roots. For c < 6.25 the equation will have two roots; more specifically, both roots are 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 Dr. Jim Wilson at **jwilson@coe.uga.edu**

or to Shannon Umberger at **umbie51698@aol.com**