Some Different Ways to Examine

by

James W. Wilson and Vilma Mesa
University of Georgia

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. From these graphs discussion of the patterns for the roots 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). 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 -1 < 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.

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

Completing the square of the expression given, we obtain:


And then

This expression is of the form:. But this expression let us know that the parabola has vertex at (h,k). That means that for our equation, any parabola will have vertex at:

Observe that the h and k are related; we can see k as dependent from h in this ordered pair in a quadratic way; with a suitable change of variable, we obtain:

Or in a more standard way, .
Observe that this parabola is symmetric with respect to the Y axis. A possible conjecture, derived from the way in which this relationship was founded is that if the independent term of the original equation is given by c, then the locus of the vertices of the parabola will be given by . The following group of graphs illustrate this conjecture.

Here c =3;

Here c = -2 and the locus of vertices is given by.
We can prove this analytically. Observe that in the process of transforming into , c remains unaltered. Let's complete the square for :

What happens if we introduce a coefficient for? Let's see some examples, for c fixed and different values for a and b in:






So it seems plausible to think that there is a relationship between a and the amplitud of the parabola that represents the locus of the vertices of. A natural conjecture may be that the equation for the locus be. Let's check with some graphs:

and the locus of the vertices is given by .
Another example:

Here .
We can verify this analytically:

So, calling , we got the conjectured relationship for the locus of the vertices of the parabola:
.
Observe that this is the general equation for the locus of the vertices of all the parabolas , when b varies.

Another look at the parameters

Consider now a relationship between b and c; let's say c =b:


What would be the locus of the vertices for ?


Lets see another example; lets set c = :
Then we obtain:

The locus seems to be a parabola again. Let's consider the first case, c=b. From our previous work we determined that are the coordinates for the vertices of the parabolas; using the change of parameter we obtain . But , and the equation for the locus of vertices becomes: or . Let's verify this in our previous example:

What if the relationship between b and c is quadratic? Lets say. Let's take a look at the parabolas generated by :


Here b = 0, 1, 2, 3, 4, -1, -2, -3, -4

Observe that again we can see a quadratic relationship for the coordinates of the vertices of the parabolas. But now we are getting a special family of parabolas; while in the linear case, the locus was a parabola that was not symmmetric with respect to the Y axis, the locus for these new family of parabolas, seem to be. What would happen if the relationship is still quadratic, but with a constant? Let's say c = What would be tha family of parabolas? Do you expect this?:

Then what would be the family if ? Was this your prediction?

The aim of this paper was to explore the general expression for a quadratic function giving special attention to the behaviour of the parameter b, the coefficient of the linear term, and to relate the behavious with the other coefficients, an in particular with the independent term, c. This guidelines mey help to pursue further eplorations, in the same vein, like for example, what happens if the relationship between b amd c, is cubic, or logarithmic, or exponential. Analyzing the functions form a different perspective, indeed will help us to consolidate our perception of this interesting mathematical objects.

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