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.
Now consider the locus of the vertices of the set of parabolas
graphed from
.The locus is the parabola
The graph of the locus is shown below.
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.
It becomes clear on a single graph that we always get two roots of the original equation, for all values of b, one negative root and one positive root.
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.
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.
Consider the case where c=-1,
This graph is also 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.
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 -6.25<c<0, one negative and one 0 when c=0, and one negative and one positive when c>0.
Graphs in the xa plane.
In the following example the equation
is considered. If the equation is graphed in the xa plane, the graph is an unusual curve.
For each value of a considered, its graph will be a line crossing the curve in 0, 1, or 2 points. There are two values of a where the equation will have only one real negative root -- at a=0.25 and at a=0. For values of a>0.25 there are no real roots, and for 0<a<0.25 the equation will have two real roots, both negative for 0<a<0.25, and one negative and one positive when a<0.
Consider the case when c=-1.
If the equation is graphed in the xa plane, the graph is an unusual curve.
For each value of a considered, its graph will be a line crossing the curve in 0, 1, or 2 points. There are two values of a where the equation will have only one real positive root -- at a=-0.25 and at a=0. For values of a<-0.25 there are no real roots, and for -0.25<a<0 the equation will have two real roots, both positive for -0.25<a<0, and one negative and one positive when a>0.