Some Different Ways to Examine

by

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.

Graphs in the xb plane.

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.

Graphs in the xc plane.

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.

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