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 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 (one real negative root) and for b >
2, the parabola intersets the x-axis twice to show two
negative real roots for each b.
Looking at the same graphs as above, one can determine the locus of the vertices of the set of parabolas graphed from:
The locus of the vertices is the line or curve that intersects each of the graphs at its respective vertex. The vertex can be found by taking the derivative of the graphs, and then by either setting the derivatives equal to zero and solving for x or by graphing the derivative and finding the point of intersection with the derivative and its respective graph. This intersection will also be a point of tangency.
The locus of the vertices for the above equation is the parabola
The original graph along with the locus of the vertices is below:
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. The two graphs along with the graph of b=5 are below.
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 as can be seen below when we graph the two equations,
and
c=1
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 at 0 when c = 0 and one negative and one positive when c <
0.