### James W. Wilson and Mike Cotton

Investigation 1.

It has now become a rather standard exercise, with available 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 (or where the graph touches the x-axis) 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.

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). The graph's behavior on the x-axis has three different trends, which translate into zero, one, or two roots for the equation.

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 intersects the x-axis twice to show two negative real roots for each b.

2. 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. Similarly for b = 2 the parabola is tangent to the x-axis (one real negative root) and

3. For -2 < b < 2, the parabola does not intersect the x-axis -- the original equation has no real roots.

There will never be more than two roots because the equation's highest term is squared. The number of roots an equation has is less than or equal to the degree to which it is raised. A squared equation is a parabola, which can only cross the x-axis a maximum of 2 times due to its shape.

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

.

As mentioned in the last assignment, the x-coordinate of the vertices of the parabolas is

,

and the y-coordinate is

.

Solving both equations for b, we get the equations

b = -2ax

and

.

These two equations are equal because the variable b is the same in each equation. So now we get the equation

.

Solving this equation for y, the resulting expression is

.

This is the equation for the vertices of all parabolas shown above. Please note the following graph.

Investigation 2.

Graphs in the xb plane.

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. This horizontal line will intersect the graph twice, once, or not at all, giving us two roots, one root, or no roots.

1. When b > 2 (see previous figure) we get two negative real roots, and when b < -2 we get two positive real roots. In this case each horizontal line crosses the graphed curve at two places.

2. When b = 2 we get one real negative root, and when b = -2 we get one real positive root. In this case each horizontal line crosses the curve at one point.

3. When -2 < b < 2 we do not get any real roots.

Now consider the case when c = - 1(red curves) rather than + 1(green curves).

In this case there are two real roots for every value of b. Unlike the green curve, there is no horizontal line that doesn't cross the red curve in two places.

The cases where c = 1 and c = -1 can be explored further by using the quadratic formula where a = 1. In the case where c = 1 we get the equation

and in the case where c = -1 we get the equation

.

In the first case (c = 1) there are no real solutions for -2 < b< 2, but in the second case (c = -1) every value for b has a solution.

Investigation 3.

Graphs in the xc plane.

In the following example the equationis 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 original equation at that value of c. In the graph, the graph of c = 1 is shown. The equation

will have two negative roots -- approximately -0.21 and -4.79.

For each value of c we select, we get a horizontal line. This horizontal line will intersect the parabola twice, once, or not at all, giving us two roots, one root, or no roots.

1. When c < 6.25 (see previous figure) we get two real 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. In this case the horizontal line crosses the parabola at two places.

2. When b = 6.25 we get one real negative root. In this case the horizontal line crosses the curve at one point.

3. When c > 6.25 we do not get any real roots. In this case the horizontal line doesn't intersect the parabola.

The case where a = 1 & b = 5 can be explored further by using the quadratic formula. In this case we get the equation

In this case there are no real solutions for c > 6.25,but for c less than or equal to 6.25 there are many solutions.

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