It has now become a rather standard exercise,
with 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. The equation given is a quadratic equation
(2nd degree) and its graph is that of 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 and a corollary to that states that the number of roots will be equal to the power of the variable of the leading coefficient. Hence, a quadratic equation will always have two distinct roots. If the function crosses over the x-axis, then the two roots will be real rational roots. If the parabola is tangent to the x-axis, then the two roots will not be different but the same. When the parabola does not intersect the x-axis, the two root are complex imaginary roots. Viewing the graph is a visual way to see two roots which may also be found by factoring, using the quadratic formula, or completing the square.

Now consider

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).

Now consider the locus of the vertices of the set of parabolas graphed from

The locus of the vertices of the parabolas forms another parabola which is reflected over the y = 1 line. Since it is going in a downward direction, the coefficient of is less than 0.

From transformations of quadratics and using standard form of , a = -1 when then graph is reflected, h is 0

since there is no horizontal shift and k = 1 since there is a vertical shift one unit up. Therefore, the equation of the parabola formed by the locus of the vertices of the given quadratics is .

Consider again the equation

Now graph this relation in the xb plane. We
get the following graph, which is the graph of 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 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 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.

Graphing Using Technology

Even though it is imperative that students
learn to solve quadratic equations by hand when working on units
involving quadratic equations, using visuals on a graphing calculator
or computer program brings to life what we mean by roots, zeros,
and solutions. Technology utilized in this manner will be much
more efficient and meaningful than graphing with pen and paper.

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