EMAT 6680

Assignment # 3

Some Different Ways to Examine


James W. Wilson and Carol K. Sikes
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.

Consider that the solution set for the equation

for these same values of b can be found by determining where the graph of

crosses the x-axis for the given value of b.

We can discuss the characterisitcs of the solution set by observing 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 values of 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 positive root at the point of tangency. For -2 < b < 2, the parabola does not intersect the x-axis, and therefore 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 at the values for the two negative real roots for each b.

Now consider the locus of the vertices of the set of parabolas graphed from


Consider that the vertex is the point (h, k) in the general form of the parabola

and that for the given equation a = 1, so we have the equation

which becomes

Considering that k represents y, h represents x, we can substitute the point (0, 1), which is always on the graph of the original parabola, regardless of the value of b. This gives us the equation

and substituing y for k and x for h and then solving for y gives us the equation for the locus of the vertices of the parabolas graphed from the original equation.

Graphs in the xa plane.

Now we will consider the following equation

When this equation is graphed in the xa plane, we have the following graph.

If we consider the graph of a specific value of a, and overlay this onto the original graph this adds a line parallel to the x-axis. If this line intersects the curve in the xa plane, then the points of intersection correspond to the roots of the original equation for the given value of a. The following graph shows the graph of the original equation in the xa plane and the graph of a = 1 and a = -1.

We can see that when a = 1, there are no roots for the original equation. When a = -1, there are two real roots for the equation, one positive and one negative. If we take the maximum value of the function

we find the largest value of a that will result in a graph of a parabola of the equation

with the parabola intersecting the x-axis in at least one point. The maximum value of a for which this occurs is a = .25. This means that the equation above will have no real solutions for values of a > .25. This can be observed by looking at the graphs of the equation for a parabola shown above for values of a = -3, -2, -1, 0, 1, 2, 3.


Now we consider the equation

where the value of b = -1 instead of 1, and we can characterize the roots of this equation for various values of a by graphing the relation

in the xa plane as shown below.

From the graph we can see that there are again no real solutions to the original equation for values of a > .25.

Graphs in the xb plane.

Now we will consider again the equation

and then graph this relation in the xb plane. This results in the following graph.

Similarly to the graph in the xa plane, if we take a 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.

Because the graph of each value of b is a horizontal line, we can determine the characteristics of the roots of the original equation for each possible value of b by observing the intersection of the graph of the horizontal line with the graph of the relation in the xb plane. 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 as shown by the red graph.

Again each value of b that is selected generates a horizontal line. It is clear for this graph that there is one negative and one positive real root for all values of b.

Graphs in the xc plane.

We have already demonstrated that graphs in the xa and xb planes can be used to show limiting values for the coefficients a and b that will result in equations that have real solutions. Now we will consider graphs in the xc plane. For this investigation 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 horizontal line that intersects 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 graph illustrates the fact that the equation

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

We can see that the vertex of the parabola is the one value of c where the equation will have only 1 real root, and this occurs as 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 equal to zero when c = 0, and one negative and one positive when c < 0.

Further explorations can be done using graphs in the xa, xb, and xc plane to determine more about the nature of the roots of quadratic equations. Students might want to explore some of the following questions: