Some Different Ways to Examine

James W. Wilson and Rita Meyers
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.

When looking at this graph we can visually see what type of roots our equations have...per prior knowledge we know that if the graph of a parabolic equation intersects the x-axis in two places then the equation has two real roots.  If the parabola is tangent to the x-axis than we only have one real root.  Finally, if the graph of the parabola does not intersect the x-axis at all , then we have two imaginary, or non-real, roots.

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

By looking at the following graph we can see that the locus is a parabola itself

You can acquire the equation for the locus by using the general form of a parabola, with a = -1 since the locus is and upside down parabola.  Doing the algebra you get...

Graphs in the xb plane.

Consider again the equation


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

Here we see that the equation leads us to 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.

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.

Being able to see this graphs and equations visually can be a tremendous help to is manipulating them before their very eyes.  This can help the students to comprehend what is being taught and pull a better connection between the equation and the graph.

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