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
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
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 = 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.
Consider the cases when c=5, 3, 1,-1, 3, 5.