Assignment # 3

by Shannon Umberger

## Some Different Ways to Examine ax2 + bx + c = 0

### by 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

ax2 + bx + c = 0,

and to overlay several graphs of

y = ax2 + bx + c

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

ax2 + bx + c = 0

can be followed. For example, if we set

y = x2 + bx + 1

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

y = x2 + bx + 1.

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

y = -x2 + 1.

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

### Graphs in the xa plane

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

ax2 + x + 1 = 0.

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

yx2 + x + 1 = 0.

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.

### Graphs in the xb plane

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

x2 + bx + 1 = 0.

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

x2 + yx + 1 = 0.

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

x2 + yx - 1 = 0

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

What can you deduce about the roots of the equation

x2 + bx - 1 = 0

for different values of b by examining the above graph?

### Graphs in the xc plane

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

x2 + 5x + c = 0.

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

x2 + 5x + y = 0.

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