Section 1
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 check the behaviour of the roots of each of this parabola using the discriminant test. Click HERE to read more about the discriminant test.
here, we will investigate the roots of the parabola
where a =1 and c =1.
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.
Section 2
Now consider the locus of the vertices of the set of parabolas graphed from
We want to show that the locus is the parabola
For every parabola of the form , the
vertex is found at
, 
For this example ,, where a =1, c=1,
the vertex is at
Read more about the Vertex of a parabola here.
We see that all the vertex of the parabola of the form
lies on
Hence, the locus of the parabola is
Consider again the equation
When we graph this relation in the xb plane we get the following
graph.
In actaul fact, we are graphing the equation,
From the above equation, we see that the domain of b is all real numbers except at x=0. The asymptote are given by b=-x and b=0. 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. That is,
= 
=
=
=
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.
Section 4
Consider the case when c = - 1 rather than + 1. We obtained the
following
Re-arranging the equation, we have
The asymtotes are clearly given by x=0 and b=-x.
If we take any particular values of b, say b=-5 and overlay this equation on the graph, we add aline paralle to the x-axis. it intersects the parabola in thexb plane at 2 points for every value of x. Using the disciminant rule, its disciminant > 0 for all values of b. hence it has 2 distinct real solutions for all values of x.
Section 5
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. In this example,
For
to have one distinct root, 25 - 4c =0 i.e. c=6.25
There is one value of c where the equation will have only 1 real root -- at c = 6.25.
For
to have no real solution,
For
to have 2 real solutions,
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.