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The following investigations will utilize Graphing Calculator 3.2 software for illustrations:

Let's explore patterns of roots and vertices in various graphs of the equation:

First, look at different values of b, i.e. b = -3,-2,-1,0,1,2,3:

Do you notice a common point for each parabola?

Yes, regardless of the value of b in this equation, the parabola will always intersect the y-axis at (0,1).

Roots

Now, the roots of a parabola are found where y = 0, i.e. where the parabola intersects the x-axis. When b = 2, the parabola has one real root which is negative. When b = -2, the parabola has one real root which is positive. When b > 2, the parabola will have two real roots with both being negative. When b < -2, the parabola has two real roots with both being positive. So, when does the parabola have no real roots? If you look at the above graph, you should notice when b < 2 and b > -2 that the parabolas never intersect the x-axis.

Vertices

Do you notice any pattern when you look at the vertices of the above parabolas?

Let's mark each vertices of the graph with a point and remove the "lines".

These points seem to form their own parabola. Let's consider the following equation as the representation of this parabola:

Now, let's combine this proposed equation and the points of the vertices above:

Hence, as the value of b varies, our original equation will generate a new parabola from the vertex points. The original parabolas will have one, two, or no real roots, whereas the new parabola has two real roots, namely x = -1, x = 1.

Now, let's investigate graphs in the xb plane. Consider our original equation with y = 0:

Here's how this equation looks in the xb plane:

Now, observe the additional graph of b = 3:

The interesting part of the interaction of these two equations is their relationship with the real roots of our original equation. In other words, where b = 3 intersects with x^2 + bx + 1 = 0 is the exact x value of the real roots of our original equation where b = 3. This investigation, also, confirms what we discussed in the first section regarding real roots.

Let's consider the following formula and add it to the graph above:

Observe that where b = 3 and 2x + b = 0 intersect, we find the x value of the vertex of the quadratic formula such that x = -b/2. We know the formula x = -b/2a which cooresponds to our investigation here.

Let's consider a graph in the xc plane and xa plane, respectively:

In conclusion, the investigation of the pattern of roots can be expanded to more polynomial equations. Similar procedures should be followed as found above.

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