Assignment #3Nicole MostellerEMAT 6680

Some Different Ways to Examine The Roots of a Parabola by James W. Wilson and Nicole Mosteller 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 ofa,b, orcas 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 forc = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained. We can discuss the "movement" of a parabola ascis changed, and with this movement, we can discuss the changes in the roots for the general equation . Forc < 0, our equation will have two real roots - one positive and one negative. Forc = 0, our equation also has two real roots - one positive and one the origin. For0 < c < 1, our equation will have two positive real roots. Forc = 1, our equation is tangent to the x-axix at the its vertex, and our equation has only one root at this point of tangency,(1,0). And, forc >1, our equation will have no roots.

This method has proven successful when investigating roots of parabolas, but even with access to technology this method can be quite time consuming, and we are still plagued with a bit of guess-work. With the assistance of technology, our investigation into the effect of a, b, and c on the roots of parabolas can be simplified by investigating just the root equation but in thex-a plane, thex-b plane, or thex-c plane. Below shows an investigation into these various planes.

In thex-a plane, let's investigate the roots of the parabola given by the quadratic equation . In thex-a plane, thea-coordinateshows the different values for the leading coefficient for a quadratic, and thex-coordinategives the root at each of the values ofachosen. For example, seeFigure 1below.From this picture, we can see not only the number of roots for each case ofFigure 1:a, we can also find the actual value of the roots by observing the intersection of the roots equation with a equal to some value. Whena = -3, we see two roots - one at-1and the other at1/3. Also from this graph we can make the same generalizations as we did from a multi-graph representation. Fora < 0, there are two real roots - one positive and one negative. Fora = 0, there is only one real root.* (*Remember when a = 0, our equation becomes linear.) For0 < a < 1, there are two real roots - both positive. Fora = 1, there is only one real root where the parabola is tangent to the x-axis. Fora > 1, there are no real roots.

In thex-b plane, let's investigate the roots of the parabola given by the quadratic equation . In thex-b plane, theb-coordinateshows the different values for the coefficient of the linear term for a quadratic, and thex-coordinategives the root at each of the values ofbchosen. For example, seeFigure 2below.Figure 2:From this graph, again we can see both the number of roots for each case ofbas well as the values of each of the roots. Whenb = 5, we see two negative, real roots -both irrational- near-5and near0.* (*The decimal values are -4.79129 and -0.208712.) Also from this graph we can make the same generalizations as we did from a multi-graph representation. Forb < -2, there are two real roots - both positive. Forb = -2, there is only one real root. This positvie root is were the parabola is tangent to thex-axis. For-2 < b < 2, there are no real roots. Forb = 2, there is only one real root where the parabola is tangent to the x axis again. Forb >2, there are two real roots - both negative.

The original multi-graph representation was to observe the roots for the equation . In thex-c plane, let's investigate the roots of the parabola given by the quadratic equation . We should expect the same results for roots as we anticipated from the original set of graphs. Remember, in thex-c plane, thec-coordinateshows the different values for the constant for a quadratic, and thex-coordinategives the root at each of the values ofcchosen.Comparing our results forFigure 3:c = -3, notice that we have two roots - one root at-1and the other at3. In addition to finding the roots at c = -3, we can now find the roots at any given value ofc. Notice that can make the same generalizations as we did from a multi-graph representation. Forc < 0, our equation will have two real roots - one positive and one negative. Forc = 0, our equation also has two real roots - one positive and one the origin. For0 < c < 1, our equation will have two positive real roots. Forc = 1, our equation is tangent to the x-axix at the its vertex, and our equation has only one root at this point of tangency,(1,0). And, forc >1, our equation will have no roots.

Using the multi-graph representation for parabolas certainly give our students an idea of the effect of the values of a, b, and c in the equation for parabolas Finding different representations, like the presentations seen in thex-a plane,x-b plane, andx-c planeserve to enrich our students' mathematics experience as well as to cement the idea of the roots of parabolas. If time allows, an extension worth noting is the investigation of thex-a plane,x-b plane, andx-c planeand the interpretation of the locus of the vertices of the parabolas from these graphs.Return to Nicole's Page