EMAT 6680 Assignment 3
Some Different Ways to Examine
by
James W. Wilson and Kanita DuCloux
University of Georgia
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. 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
The graph opens downward because of the negative
coefficient. It passes through points all of the vertices of other
graphs of .
Graphs in the xb plane.
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. 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. I have also graphed the case c = -3. There is always a solution.
Graphs in the xc plane.
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 equations
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. In the example when c = 4, we get
both negative for 0 < c < 6.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.