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).
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
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
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,
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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