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). 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 twice to show two negative real roots for each b.
Now consider the locus of the vertices of the set of parabola graphed from
,So the vertex has the form 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 = 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 as you can see in the following graphs.
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 original 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 in the following graphs.
Likewise the case of b, we can clearly get no real roots for c > 25/4, only one real root when c = 25/4, and two real roots for c < 25/4.
