Some Different Ways to Examine by

James W. Wilson and Mark C. Cowart

It has now become a rather standard exercise, with available 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, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is

obtained:   The value of b can be matched by looking at the colors. We can discuss the "movement" of a parabola as b is changed. Note the following observations:

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

2) For b=2 or -2, the parabola is tangent to the x- axis.

3) For negative b values the parabola shifted right. For positive b values it shifted to the left.

4) For b>2 or b<-2, the parabola will intersect the x-axis at two distinct points that are real roots.

5) The locus of points of the vertices of each parabola is a new parabola whose equation is . It might be helpful to examine a graph this equation in order to see the vertices being connected.

A generalization drawn from statement 5 above is that for equations of the form , the locus parabola passing through the vertices of the set of their graphs will be of the form .

Graphs in xb plane: (Note: Instead of using , let b = y in order to graph better.)

Consider again the equation where the b value has been replaced with y. The graph of this equation in the xy plane is: (This graph of this relation also examines the xb plane.)

If any particular value of y is taken, such as y=3, and overlaid on the above equation, the graph would add a line parallel to the x-axis. If it intersects the curve in the xy plane the intersection point corresponds to the roots of the original equation for that value of b. Notice the following graph: For each value of y selected, we get a horizontal line. The following generalizations can be found:

1) On a single graph there are two negative real roots of the original equation when y>2.

2) There are no real roots for -2<y<2.

3) One positive real root when y=-2.

4) There are two positive real roots when y<-2.

Similar explorations can be done for the xc plane as well. Make your own generalizations.