The Department of Mathematics Education

J. Wilson, EMAT 6680

EMAT 6680 Assignment 3

Some Different Ways to Examine


James W. Wilson and Amy Benson
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. From these graphs discussion of the patterns for the roots of

can be followed. 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) and 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


Show that the locus is the parabola

It is important for students to visualize the path of the locus and to generate the equation of the locus from this visualization and from the coordinates of the plotted points.

Here we see the graph of the set of the parabolas overlayed witht the graph of the locus.

Graphs in the xb plane.

Consider again the equation

Now graph this relation in the xb plane. We get the following graph.

If we take any particular value of b, say b = 3, 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. Notice that this follows the graphical approach to solving a system of equations. In this case, we have the two equations

b = 5 and

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 ( the intersection of the horizontal line and the hyperbola) 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.

The value of this exercise being that it provides students the opportunity to find the roots of a parabola using an alternative method. It also emphasizes the idea that a and c are constant with only the value of b changing. It is also valuable for students to be able to determine the equation to be graphed in the xb plane.

Consider the case when c = - 1 rather than + 1, shown here in red.

We may again find the roots of the original equation by locating the intersection of the hyperbola and horizontal line. In this instance with c = -1, notice that for all values of b there will be one negative real root and one positive real root.

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. Students should graph c = -x^2 -5x to see this parabola. This equation will reinforce that c is now treated as y in the xc plane. 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 (the vertex of the parabola). 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.

Return to Amy's homepage