The Department of Mathematics Education

J. Wilson, EMAT 6680


Jeffrey T. Daniel, EMAT 6680

Some Different Ways to Examine


It has now become a rather standard exercise, with availble technology, to construct graphs to solve 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, roots of

can be estimated or actually found and other patterns or methods for solving such roots may be boldly explored.


For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

Click here for a live demonstration of how the above parabola changes as b = (-10,10).

It becomes obvious which functions have real roots and which ones do not, but furthermore 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.

After briefly analyzing the locus of the vertices of the set of parabolas graphed from


we have numerous conceptual and applied options when we introduce a new function to the mix.


being introduced to the family of curves above.


Click here and view the interaction as b is allowed to vary.

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. We have the following graph.

For each value of b we select, we get a horizontal line. Click here to see an active demonstration of this.

What do you notice happens to the line with certain values of b?

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.

One may view or create similar graphs in graphing calculator 3.1 by these functions of x and y

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.

In closing, these explorations can be widely adaptable to different levels of students and grade levels. This may be a good opportunity to introduce asymptotes and limits at a conceptual level for Algebra 2 and Pre Calculus students.

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