**It has now become a rather standard exercise,
with available technology, to construct graphs to investigate
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.**

**To start our investigation of "how
changing our b-value" effects our graph, let's first take
a look at the equation, ......(where our
b = 0)**

**Let's take a look at the graph by itself
to get a good look at it and use it as our basis....**

**For example, if we set:**

**for b = -3, -2, -1, 0, 1, 2, 3, and overlay
the graphs, the following picture is obtained.**

**Let's 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).
We can see that when b>0 the vertex of the graph is on the
left-hand side of the y-axis, and when b<0 then the vertex
of the parabola is on the right-hand side of the y-axis. We also
see that 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:**