The Department of Mathematics Education
The Department of Mathematics Education
It has now become a rather standard exercise, with availble technology,
to construct graphs to of 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 a discussion of the patterns for the roots of
can be developed. 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 implying 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
.
Show that the locus is the parabola
The diagram below shows the graph of
overlaid on the graph of
It is clear to see that the vertices of the first group of parabolas
is the graph of the second parabola.
To show this more clearly, we will solve algebraically to find the intersection
point(s) of the two parabolas.
Generalize. So in general, the two parabolas will intersect when x = 0 (the point 0,c) and when x = -b/2, which is the x value of all the vertices of the first parabola.
The general pattern for the above parabola which is the locus of the vertices is defined by the equation
To get the points of intersection, we now solve the general equations.
So the two general parabolas will intersect when x = 0 ( at point 0,c) and when x = -b/2a ,which is the x value of the vertex of each individual parabola for a,b,and c.
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.
When c = -1,
Here are the equations of the family of graphs for b = -3,-2,-1,0,1,2,3
Here is the picture showing the graphs overlaid.
For all values of b, the equation has two real roots (one positive and
one negative) and will always pass through the same point on the y-axis
(0,-1).
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.
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.
