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, -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 always passes through the same point on the y-axis (the point (0,1) with this equation). Viewing the graphs of parabolas allows us an easy way to see the roots (or solutions) of each parabolic equation, which may also be found by factoirng, using the quadratic formula, or completing the square. However, the graph is a quick reference for the roots. The x-axis is used as the indicator of how many and what type of solutions a parabola has. If the parabola intersects the x-axis in two places, the parabola or equation of that parabola has two real roots. When the parabola is tangent to the x-axis, then there is one real solution, and if the parabola does not intersect the x-axis, then are are two non-real (or imaginary) roots.

For the particular values of b shown above, when 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 intersects 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

The picture is shown below and is also a parabola itself.

Using the general form of a parabola , with (h,k) as the vertex, we'll use (0,1) for (h,k) being that it is the vertex of the parabola formed by the locus of the above parabolas. Since the parabola is upside down a=-1, resulting in the following:

Therefore, is the equation of the parabola formed by the locus of the vertices of the parabolas mentioned and pictured above.

**Graphs in the xb plane.**

Consider again the equation

Now graph this relation in the xb plane. We get the following graph.

This is the graph of a hyperbola. 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. 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.

**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.

Analyzing graphs of this nature in conjunction with evaluating the roots (or solutions) by hand or calculator would be especially meaningful for students to see the connection between the graph and its solution. Here is proof that a picture is worth a thousand words when a student can visually verify his calculated result.