The Department of Mathematics Education

EMT 668 Assignment 3

Some Different Ways to Examine


James W. Wilson and Angie Head
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 consider the graphs of

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

From this, 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). This point is known as the y-intercept. It is found by setting x=0 and solving for y in all of the above functions. By looking at the discriminant,

one can tell how many roots or x-intercepts exist. If the discriminant is greater than 0, there will be two x-intercepts; equal to 0, one; and less than 0; there will be no x-intercepts. In our above graph, 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 positive real root at the point of tangency. For -2 < b < 2, the parabola does not intersect the x-axis -- thus 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

The locus is the following parabola:

We can generalize this to be true because the vertex of this parabola is the point (0,1), which is also the point in which the set of parabolas

all pass through.

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. 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 in the above equation. We get the following graph.

It is clear from this graph for any value of b that we pick we will get two roots, one positive and one negative. It is also clear that zero will never be one of the roots.

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 following graph, the graph of c = 1 is shown. The equation

will have two negative roots -- approximately -0.2 and -4.8.

By looking at the graph, one can determine how many roots and the type of roots the above equation will have according to whether c =0, c >0, or c<0. 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.

What would the above graph look like if we let a=-1 instead of 1 in the above equation? Let's see.

As one would expect, our findings are just reversed. Now for c<0 there are two positive roots, for c>0 there is one negative and one positive root, and for c=0 we have two real roots 0 and 5.

Click here to go to Angela Head's EMT 668 Page