The Department of Mathematics Education

J. Wilson, EMAT 6680

EMAT 6680 Assignment 3

Some Different Ways to Examine


James W. Wilson and A. Kursat Erbas
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 picture in Figure 1 is obtained.

Figure 1

Now, 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 is not surprising because when we substitute x=0 in the equation , we may get y=c whatever the coefficients a and b are. For example in the above case, since c=+1, the quadratic equation satisfies the point (0,1) . In other words, the graphs (parabolas) of always passes through the point (0,1) as the coefficient b changes.

As we may observe from Figure 1:

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)

For b > 2, the parabola intersects 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 vertex of the above parabola is located at

If we look at the relation between x and y intercepts of the vertex, we may see that

So, we may expect that the vertices of each parabola for different values of b lie on the graph of the parabola (See Figure 2)

Figure 2

If we think about the general case, a vertex of a parabola is located at



Graphs in the xb plane.

Consider again the equation

Obviously, the root(s) and the graph of this equation depends on the values of 'b'. So, in this situation, there is two variable in the above equation: 'x' and 'b'. In particular, we may express this equation in the following form:

Now graphing this relation in the xb plane. We get the following graph in Figure 3.

Figure 3

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. For example, for b=3, we may observe that the roots of the equation is approximately x = -2,6 and x = -0.38 (See Figure 4)

Figure 4


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.

Now, let's consider the case when c = - 1 rather than c = + 1. (See Figure 5)


Figure 5 (c=+1, c=-1)

As we may observe from the graph, for every value of b, the horizantal line intersect the graph at two points for c=-1 although it intersects the graph at two points if b>2 and at a single point if b=2 for c=+1. In other words, for ever value of 'b', the equation

has two real roots. This shows that when the signs of the coefficients 'a' and 'b' are opposite then the quadratic equation has two real roots whatever the coefficient 'b' is.

This can be explained in the following manner: The dicriminant of the above equation is

As we see, it is always greater than zero (0). So, the quadratic equation has two real roots. But, if we look at the discriminant in the case that c=1, it is pozitif when b>2, negatif when b< and equal to zero(0) when b=2. So, the equation has real roots when b is greater than or equal to 2 in the case that c=1 (Calculate the discriminant and see this)

Graphs in the xc plane.

Above we have investigated the quadratic equation

by fixing the coefficients 'a' and 'c' for various values of the coefficient 'b'. Now, let's try to see what happens if we fix 'a' and 'b' but vary 'c'.

Let's consider the following example to carry out aim

As what we did in the varying 'b' case, we can write the above equation in the following way

If this equation is graphed in the xc plane, it is easy to see that the curve will be a parabola (See Figure 6). 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.

Figure 6


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 (See Figure 7).

Figure 7

As we have investigated, the coefficients b, c of the quadratic equation

plays an important role for the roots of the equation. In the same manner one can investigate the behavior of the equation in xa-plane (i.e. keep 'b' and 'c' fix while varying 'a'.

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