The
coeficient of the x in has an effect on
the axis of symmetry of a parabola and on the roots (x-intercepts)
of the parabola.

As
b varies, the locus of the vertices of the parabolas form another
parabola with equation . Click here
to see an animation of this.

The
vertex of a parabola is given by the point .

Investigation 2: We
will now keep a = 1 and c = 1 in the equation .
We essentially have the equation and
can now sketch the equation (We
sketch this equation and not the equation just
because Graphing Calculator draws graphs in the xy plane)

From the graph above we learn that:

The
graph drawn in the xb-plane has a discontinuity at x = 0.

The
two parts of the graph have vertices at x = 1 and x = -1. The
significance of this will be explained in investigation 3.

Investigation 3: Now
we will place two specific parabolas and the hyperbola
on the same set of axes and see what we learn.

From the graphs above we learn that:

The
vertices of the hyperbolas correspond with the places where and touch the x-axis.

For
b values inbetween -2 and 2 the parabola has no roots. For b values
2 and -2 the parabola has only one root. For b values greater
than 2 or smaller than -2, the parabola has two roots. To see
this vividly we can look at the following graphs.

The
graph of has no discontinuity because of
the fact that the parabola always has real roots. The blue line
y = b in the animation above confirms
this fact.

Investigation 5:
Let us investigate a cubic function and its roots.

From the graph above we learn that:

The
same relationship between the three graphs exists as in the case
of the parabola.

When
b = -2.5, the cubic graph has exactly two roots.

When
b < -2.5, the cubic graph has three roots.

When
b > -2.5, the cubic graph has one root . Click
on the graph above to see an animation of this fact.