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 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.
Click here to see the "movement."
Notice that:
*The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation).
*Statements can be made pertaining to the number and kind of roots.
1. 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).
2. 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.
3. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots.
4. 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
.We can show that the locus of the vertices is a parabola.
Click here
to open an animation that shows the parabolic path of the vertices.
For this case, the locus of the vertices follows the equation:
We can generalize this statement for all quadratic equations of the form
The vertices will follow the path with an equation of
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
= 5, 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. We can make several conclusions from the graphs.
Notice that:
When b > 2, we get two negative real roots of the original equation.
When b = 2, we get one negative real root.
For -2 < b < 2, there are no real roots.
When b = -2, there is one positive real root when b = -2.
When b<-2 there are two positive real roots.
Notice that in the xb plane, a negative value for c changes the axis of the hyperbola.
It is interesting that the graph approach asymtotes of the b axis and the line x = -b.
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.
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