James W. Wilson and Robert H. Childres III

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.

Let's begin by examining effects of **a** on the equation by setting **b** = **c** = 1. This results in
the following equation

If we take **a** = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

We can see how the line from **a** = 0 "curves" as **a** is changed. Regardless of the value of **a**,
the graph always crosses the y-axis at the point (0, 1). We can also determine that for **a** = 1 or **a** > 1, the
graph has no real roots. For **a** = 0, it has one real root at -1 (the x-intercept of the line). When **a** < 0,
the graph has two real roots. Further scrutiny would show that when 0 < **a** < 1/4, the graph has two real
roots, and at **a** = 1/4 the graph is tangent to the x-axis at -2.

Consider again the equation

Now graph this relation in the **xa** plane. We get the following graph.

If we take any particular value of **a**, say **a** = -2 (the green line), **a** = 1 (the blue line),
and **a** = 1/4 (the black line), and overlay these equations on the graph we add a line parallel to the
x-axis. If it intersects the curve in the **xa** plane the intersection points correspond to the roots of the
original equation for that value of **a**. We have the following graph.

For each value of **a** we select, we get a horizontal line. It is clear on a single graph that when **a** > 1/4
there are no real roots. When **a** = 1/4, there is one real root, namley x = -2. While it is not immediately
obvious from the graph all values of **a** < 1/4 have two real roots; except for **a** = 0, which has one real
root at x = -1. For **a** on the interval (0, 1/4), there are two negative real roots. For **a** < 0, there
is one positive real root and one negative real root.

We continue by examining the effects of

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

Generalize.

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 (the blue line), 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. Which results in the green line added to the graph.

Let's continue by examining effects of **c** on the equation by setting **a** = **b** = 1. This results in
the following equation

If we take **c** = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

We can see how the parabola from **c** = 0 "moves" as **c** is changed. The graph always crosses
the y-axis at the point (0, **c**). We can also determine that for **c** = 1 or **c** > 1, the
graph has no real roots. For **c** = 0 or **c** < 0, the graph has two real roots. Further scrutiny would
show that when **c** = 1/4 the graph is tangent to the x-axis at -1/2, thus it has only one real root.

Consider again the equation

Now graph this relation in the **xc** plane. We get the following graph.

If we take any particular value of **c**, say **c** = -2 (the green line), **c** = 1 (the blue line),
and **c** = 1/4 (the black line), and overlay these equations on the graph we add a line parallel to the
x-axis. If it intersects the curve in the **xc** plane the intersection points correspond to the roots of the
original equation for that value of **c**. We have the following graph.

For each value of **c** we select, we get a horizontal line. It is clear on a single graph that when **c** > 1/4
there are no real roots. When **c** = 1/4, there is one real root, namley x = -1/2. For **c** on the interval [0, 1/4]
there are two non-negative real roots. For **c** < o, there are two negative real roots.

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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