It has now become a rather standard exercise, with availble technology, to construct graphs to observe the roots of the quadratic equation:
by overlaying 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 observed. For example, if we set
for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following pictures are obtained:
We can discuss the "movement" of a parabola as b is changed as follows:
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 (sometimes called a double root).
For -2 < b < 2, the parabola does not intersect the x-axis (i.e. the original equation has no real roots).
For b = 2 the parabola is tangent to the x-axis (one real negative root).
For b > 2, the parabola intersets the x-axis twice (two real roots, both negative).
Summary:
Two real roots when:
b < -2 (both positive).
b > 2 (both negative).
One real root when:
b = -2 (positive root).
b = 2 (negative root).
No real roots when -2 < b < 2.
Now consider the locus of the vertices of the set of parabolas graphed from
First look at the graphs of the locus of the vertices alongside the graphs of this equation with b = -3, -2, -1, 0, 1, 2, 3.
Note that the locus of the vertices is shown in black. At this point, students should investigate this further in order to develop some conjectures regarding the locus of the vertices. Some guiding questions include:
Is the locus always a parabola?
Does the parabola always open downward?
Is the locus always symmetrical to the y-axis?
Notice that the locus of vertices crosses the x-axis at x = 1 and x = -1. Thus the roots of the locus are x = 1 and x = -1. So (x - 1) and (x + 1) are factors of the polynomial describing the locus. Thus the equation for the locus of vertices is y = a(x - 1)(x + 1). We can solve for "a" by substituting x = 0 and y = 1 (from the point (0,1)). Trivially a = -1. Thus the equation for the locus of vertices is given by:
So the answers to the three questions listed above is, "yes."
Consider again the equation
The graph of this relation in the xb plane is shown below:
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. The intersection points of the horizontal line and the curve correspond to the roots of the original equation for that value of b. We obtain the following graph:
For each value of b we select, we get a horizontal line. From this graph it clear to see that there are:
Consider the case when c = - 1 rather than c = 1.
Note that when c = -1, there are always two real roots for any value of b. One root is negative, while the other is positive. This would be an excellent extension for students to explore (i.e. Why does this happen?).
So let's investigate the equation while keeping b constant and varying c. In order to do this, we will graph the equation
in the xc plane. Once 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 -4.79 and -0.21).
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.
Final Comments:
Graphing the quadratic equations in the xb- or the xc- planes will yield the same information concerning the roots of the equation as graphing in the xy- plane. This method may be easier for some students to see and recognize the relationships between the roots and the coefficients (a, b, and c) of the quadratic equations.