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).
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 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 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.