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
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 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.
Now consider the locus of the vertices of the set of parabolas graphed from
Show that the locus is the parabola . Click here in order to see how we find this equation.
Consider again the equation
Now graph this relation in the xb plane. We get the following
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.
The following figure illustrates several graphs of for c an even integer from -2 to 10. The specific values of c were chosen so that we might compare the roots of these graphs with the equation graphed in the xc plane. As we can see, the change in c amounts to a vertical shift of the graph.
Notice that when c=0, the roots of the equation are x=-5 and x=0. Furthermore, we can find the vertex of this graph fairly easily. We can use the axis of symmetry to locate the x value of the vertex at x=-2.5. The y value of the vertex can then be found using substitution to be y=6.25. Play around with the graph by changing the linear coefficient. What effect does this have on the graph? Compare the graphs and . What effect did using the opposite linear coefficient have on the graph?
Now let us consider the equation . 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.
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. Does this value ring a bell? 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.
Graphs in the xa plane.
Now consider the equation graphed below for integer values of a from -3 to 3. Notice that the graphs with negative leading coefficients are concave down and those with positive leading coefficients are concave up.
The purple graph below shows the equation in the xa plane.
The red graph is the graph of a=2 illustrating that the equation has has two negative real roots. From this graph we regognize that for a>6.25, the graph has no real roots. For all a<0, the graph has one negative and one positive root. Furthermore, as a approaches zero, the graph has one small negative root and one large positive root. What happens when a=0?
There is more to be discovered regarding the patterns of roots of the equation . One could take this exploration further by considering what will happen to the graph in the xa plane if we change the linear coefficient once again. One could also try to find an equation for the locus of points determined by the vertices of the set of parabolas graphed by . If there is an equation to be found, could it be generalized for changing values of the linear coefficient?