The attached 4-page paper is the start of an article that might
appear in a journal such as the Mathematics Teacher
-- the audience being mathematics teachers who might use some
of the ideas for instruction.
It is a start; incomplete, unclear, maybe in error; maybe glossing over significant points and stressing some obvious or trivial points.
Sign on as a co-author. Rewrite and complete the article. This means you must come to grips with whatever points are to be essential, what to add, what to delete, and what to edit. The "different" approaches to this topic are really in the graphs in the xb, xc, or xa planes. You might want to examine a bunch of these before trying to re-write.
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.
We begin our discssion with a look at the efffects of a on the equation
To do this we must allow b and c to remain constant. Lets examine a for the values of -3(purple), -2(red), -1(blue), 0(grey), 1(green), 2(aqua), and 3(yellow), while b and c are 1.
Graph in the xa Plane
The graph in the xa plane look as followed;
If we take any particular value of a, such as a = -3(blue) and 1(green), overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xa plane the intersection of points correspond to the roots of the original equation. We will overlay a = -3(blue) and 1(green) onto our original equation.
For a = -3(blue), we have two real roots and for a=1(green), we have no real roots. Notice for a = 0, one real root is revealed.
We continue our discussion by examining b for the equation
We will look at b for -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
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.
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 and c = -5 is shown. The equation
will have no real roots at c =1 and two real roots at c = -5.
There is one value of c where the equation will have only 1 real root -- at c = 0.25. For c > 0.25 the equation will have no real roots and for c < 025 the equation will have two roots, both negative for 0 < c < 0.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.
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