## Some Different Ways to Examine

### by James W. Wilson and Julia M. Neal 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.

Notice that:

*The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation).

*Statements can be made pertaining to the number and kind of roots.

1. 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).

2. 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.

3. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots.

4. 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

.

We can show that the locus of the vertices is a parabola.

Click here to open an animation that shows the parabolic path of the vertices.

For this case, the locus of the vertices follows the equation:

We can generalize this statement for all quadratic equations of the form

The vertices will follow the path with an equation of

### 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 graph.

For each value of b we select we get a horizontal line. We can make several conclusions from the graphs.

Notice that:

When b > 2, we get two negative real roots of the original equation.

When b = 2, we get one negative real root.

For -2 < b < 2, there are no real roots.

When b = -2, there is one positive real root when b = -2.

When b<-2 there are two positive real roots.

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

Notice that in the xb plane, a negative value for c changes the axis of the hyperbola.

It is interesting that the graph approach asymtotes of the b axis and the line x = -b.

### 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.