It has become a rather standard exercise, with available technology, to construct graphs to consider the equation

for different values of a, b, or c as the others are held constant. From these graphs we can examine the patterns for the roots of this equation.

For example, if we set

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

We can discuss the "movement" of the parabola as b is changed. We see that the parabola always passes through the same point on the y-axis. For this equation the point is (0,1). For b < -2, the parabola intersects the x-axis in two points with positive x values implying that the original equation will have two real positive roots. For b > 2, the parabola also intersects the x-axis twice but in this case there are two real negative roots for each b. For b = -2 and b = 2, the parabola is tangent to the x-axis and so the original equation has one real positive and one real negative root, respectively. For -2 < b > 2, the parabola does not intersect the x-axis and the original equation has no real roots.

If we now consider the locus of the vertices of the set of parabolas graphed from

We can show that the locus is the parabola

since the graph is concave down, symmetric about the y-axis, and has maximum value of 1.

**Graphs in the xb plane.**

Consider again the equation

where

If we graph this relation in the xb plane, we get the following graph.

We can represent any particular value of b by a line parallel to the x-axis. Where the line intersects the curve will correspond to the roots of the original equation for that value of b.

For example, if b = 3 we have the following graph

We get two real negative roots for the original equation.

Since for each value of b we select we get a horizontal line, we can determine the roots of the original equation from the graph. I t is clear that we get two negative real roota 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.

We see that in this case the equation will always have two real roots, one positive and one negative.

**Graphs in the xc plane**

In this example the equation

is considered.

Where,

If this 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. These points of intersection represent the roots of the original equation at that value of c.

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

has two negative roots, approximately -0.2 and -4.8.

There is one value of c where the equation will have only one 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. For

0 < c > 6.25 both roots will be negative, one negative and one 0 when c = 0 , and one negative and one positive when c < 0.

**Graphs in the xa plane**

Consider the equation

where

If we graph this relation in the xa plane, we get the following graph.

For a = -1, the equation

has two real roots, one negative and one positive.

For a > 0.24, the original equation has no real roots. For 0.24 < a > 0, there is only one root and it is negative. For a < 0, there are two roota, one negative and one positive.

This investigation provides another way for investigating the roots of quadratic functions. Instead of using the xy plane we can use the xa, xb, or xc plane depending on which coefficient we wish to vary. This approach gives a better perspective of the patterns for the roots of the equation as you vary either a, b, or c as the other two are held constant.