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

We can discuss the "movement" of a parabola as a is changed. The parabola always passes through the same point on the y-axis (the point (0,1) with this equation.). For a < 0 the parabola will intersect the x-axis in two points yeilding one positive root and one negative root. For a = o, the graph is a straight line, crossing the x-axis at the single point (-1,0). For a > 0, the parabola does not intersect the x-axis--thus the original equation has no real roots.

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

Show that the locus is the line y = .5x +1.

Graphs in the xa plane

For instance, let us look at graphs in the xa - plane where the y-axis acts as the a-axis.

This differs from the xy- plane in that roots are found by finding the intersection of the curve and the parallel line a = n for varying values of n.

Let us use this method to find the roots of the equation: . Therefore we will need to graph the line a = -3 and overlay this onto our curve.

We will graph in the xy - plane and compare our solutions. Since the parabola crosses the x-axis at x= -.434259, this verifies that x = -.434259 is, in fact, a root of the polynomial.

Let us look at the graph of for differing values of b. 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. The parabola alwayspasses through the same point on the y-axis ( the point (0,1) with this equation). Forb < -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.

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 = 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.

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

Let us look at the graph of the equation of for differing values of c.

Now consider the locus of the vertices of the set of parabolas graphed from . Show that the locus is the line .

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.

Though the ax, bx, cx planes are not typically discussed in a high school mathematics classroom, this exploration might be an interesting way to introduce the idea of various planes to students with previous knowledge of quadratic functions.

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