University of Georgia

It has now become a rather standard exercise, with the available 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 now discuss the "movement" of a parabola as **b** is
varied while keeping both **a** and **c** constant.

1.All these parabola's have one point in common,(0,1)(the point where each parabola crosses the y-axis).

2.Forb<-2, the parabola intersects the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive).

3.Forb=-2, the parabola is tangent to the x-axis and so the original equation has one real(positive) root at the point of tangency.

4.For-2<b<2, the parabola does not intersect the x-axis -- the original equation hasnoreal roots.

5.Forb=2, the parabola is tangent to the x-axis and so the original equation has one real(negative) root at the point of tangency.

6.Forb>2, the parabola intersects the x-axis in two points with negative x values (i.e. the original equation will have two real roots, both negative).

Let us now take a different approach to investigating the equation

Consider a version of the equation above

We will now graph this equation 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 above we add a line parallel to the

x-axis. If this line intersects the curve in the xb plane, the intersection
points correspond to the root(s) of the original

equation for that value of b. The graph with **b = 3** overlayed looks
like the following.

For each value of b we select, we get a horizontal line. It is clear from this 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.

Viewing the quadratic in the **xb-plane** allows one to see more clearly
that there are no real roots on the interval **-2<b<2**. The graph
below illustrates this point by overlaying the equations **b=1** and
**b=-1** onto the **xb-plane** which already shows the graph of the
quadratic.

From this graph one can see that since the graphs of b=-1 and b=1 do not cross the graph of the quadratic that there are no real roots associated with the quadratic with b=-1 and b=1.

Now consider the case when c = - 1 rather than c = 1.

Overlaying graphs of this equation in the **xy-plane** for b={-3,
-2, -1, 0, 1, 2, 3} gives you the following.

Now graphing the quadratic in the **xb-plane** yields

In contrast to the graph in the **xy-plane**, this graph makes it
easy to see that for any value of **b** you get two real roots, one positive
and one negative.

We will now move on to the case when **c** is varied and **a**
and **b** are held constant. We will use the following equation in this
example:

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 below, the graph of c = 1 is overlayed on the previous graph.

From the two previous graphs one can see, there is one value of **c**
where the equation will have only 1 real root -- at **c=6.25**. Also,
for **c > 6.25** the equation will have no real roots and for **c
< 6.25** the equation will have two roots. Both roots will be negative
for **0 < c < 6.25**, one root will be negative and one 0 when
**c = 0** and one root will be negative and one positive when **c <
0**.

This equation,, will have two negative roots -- approximately -0.2 and -4.8.

The graphs in the **xa-plane** are nearly as easy to read as those
in the **xb** and **xc**. We have not explored the **xa-plane**
here; however, we leave it to the reader to do this exploration if his/her
curiousity is running wild.

We have just demostrated some different ways of looking at the quadratic

for different values of **a**, **b** and **c** and determining
what type(s) of roots the equation will yield. Viewing the quadratic in
the **xb** or **xc-plane**, may make determining roots easier for
some people. If you are comfortable with determining roots using the **xy-plane**
that is great but you should still understand these other methods and vice-versa.
It is worth understanding how many different methods work even if you prefer
one over the other(s).