## Some Different Ways to Examine

### by James W. Wilson and BJ Jackson 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 of the graphs in the xy plane.

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.

When overlaying the inverse, the inverse intersects the graphs at the minimums or the verticies of the other parabolas. This intersection is coordinate of the vertex.

### Graphs in the xb plane.

Consider again the equation

Now graph this relation in the xb plane. To graph in the xb plane, we change the y-axis into a b-axis. To make this work in a graphing calculator one can use n = by. We get the following graph.

Click here for a movie that shows what happens to the graph when n varies from -10 to 10.

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.

Click here for a movie that shows how the number of solutions vary as n varies.

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

Here again we get a hyperbola that would have the same asymptotes as when c = 1. The difference is that there are always two roots with one of the roots being positive and one of the roots being negative.. There is never an instance when c=-1 that has no solutions.

### 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.

Click here for a movie of the graphs of the parabolas for n = -10 to 10.

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.

Click here for a movie that illustractes the solutions for n=-10 to 10.

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

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