It has now become a rather standard exercise,
with available technology, to construct graphs to consider the
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.
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.
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
For b > 2, the parabola intersects 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
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 <
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.