Explorations of Quadratic Functions
By: Laura Lowe
It
has now become a rather standard exercise, with available 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 yaxis (the point (0,1) with this equation). For b < 2 the parabola will intersect
the xaxis 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 xaxis 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 xaxis  the original equation has no real roots. Similarly for b = 2 the parabola is
tangent to the xaxis (one real negative root) and for b > 2, the parabola
intersects the xaxis twice to show two negative real roots for each b.
Now consider the locus of the vertices of the set of parabolas
graphed from .
First
letŐs consider the set of vertices of the set of parabolas of the form .


From
the above graph we can see that the vertices of the graphs of the form are (1.5, 1.25), (1, 0), (0.5, 0.75), (0, 1), (0.5, 0.75), (1, 0), and
(1.5, 1.25). The locus of these
points has the equation as seen below.


Now
suppose we start with the equation and we let b =
3, 2, 1, 0, 1, 2, 3, and overlay the graphs, the following picture is
obtained.


From
the above graph we can see that the vertices of the graphs of the form are (0.75, .125), (0.5, 0.5), (0.25, 0.875), (0, 1), (0.25, 0.875), (0.5,
0.5), and (0.75, .125). The locus
of these points has the equation as seen below.


In
general, the locus of the vertices of the equations of , is the equation .
Click
here
to see a movie when a varies and c = 1.