**Investigation 1**.

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.

Let's go over the following basic concepts about the quadratic functions

How about c?

To see the graph movie, click** here!**

and we can see the information about the coefficient b from below graphs

From these graphs discussion of the patterns for the roots of

can be followed. For example, if we set

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). 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) and for b > 2, the parabola intersets the x-axis twice to show two negative real roots for each b.

show that the locus is the parabola

Generalize.

**Investigation
2**.

**Graphs in the xb plane.**

Consider again the equation

Now graph this relatio in the xb plane. We get the following graph.

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.

Consider the case when c = - 1 rather than + 1. In that case, it always has roots.

Graph other values of **c **on the same
axex.

we can see it totally by clicking following **here !**

**Investigation
3**.

**Add the graph of 2x + b = 0 to the picture and discuss its
relation to the quadratic formula.
**

If the sum is

and **here**
is its graph.

and we can know it has always a root from its graph.

Algebraically, it's trivial!

From the quadratic formula, it's the following....

Whatever the real number b is, the equation has at least a root.