to Examine

James W. Wilson

Helene Chidsey

Lou Ann Lovin

In order to determine the pattern for the roots of

we may consider the equation

for different values of a, b, or c as the other two are held constant.
If we overlay several of these graphs, we can see how the graph changes
as the chosen coefficient varies.

For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, we can explore
the relationship between the roots of the equation and b. Due to the possible
confusion caused by six graphs on one axes, we chose to separate the graphs
into -3 <= b <= 0 and 0 < = b <= 3. The following are graphs
for -3 <= b <= 0.

We can discuss the "movement" of a parabola as b changes.

equation).

positive root at the point of tangency.

roots.

The following are graphs for 0 <= b <= 3.

To summarize, we see from both graphs that, the equation has

b <-2 (both roots are positive) or

b > 2 (both roots are negative)

b = -2 (root is postive) or

b = 2 (root is negative)

Now consider the equation

in the **xb plane**. We obtain the following graph.

If we take any particular value of b, say b = 3, and overlay this equation
on the graph, i.e., adding a line parallel to the x-axis, we find that b=3
intersects the curve in the xb plane. These points of intersection correspond
to the roots of the original equation for b=3. See the following graph.

For each value of b we obtain a horizontal line. Thus, 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. In other words, from this one graph students can glean the
same information, they obtained from the two previous graphs, with less
work and less visual confusion.

Now let's consider an equation in which we will vary c instead of b. For
example,

If the equation is graphed in the **xc plane**, 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 are the roots of
the orignal equation for a particular value of c. In the picture below,
the graph of c = 1 is included.

Thus, 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.

For further investigation, we could vary the coefficient a.

In summary, we wish to emphasize that using this type of graph (the xb or
the xc plane, versus the xy plane), we can get the same amount of information
pertaining to the relationship of the roots to the changing coefficent with
less work and less visual confusion!