James W. Wilson and Kimberly Burrell

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. The equation is a quadratic equation, which
generates a U-shaped graph known as a parabola. From these graphs
discussion of the patterns for the roots of**

**can be followed.**

**The ****Fundamental
Theorem of Algebra**** states that any
polynomial function of degree one or greater has at least one
root. A corollary states that the number of roots will be equal
to the power of the variable of the leading coefficient. A
quadratic function will have two roots, but they may not be
distinct. The root might be double. If the function crosses over
the x-axis, then the roots will be real roots. If the function is
tangent to the x-axis, then the roots will be the same. When the
function does not intersect the x-axis, the roots are complex
imaginary roots. Viewing the graph is a visual way to see the two
roots which can also be found by factoring, using the quadratic
formula, or completing the square.**

**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).****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

.

**The locus of the vertices of the
parabolas form another parabola that is reflected over the line y
= 1. Since it is opening in a downward direction, the coefficient
of squared term is less than 0.**

**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, which is a hyperbola.
**

**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 real
positive 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 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.**