The x Coefficient of a
Quadratic Function

By

Alex Moore

For
this investigation we will consider the function f(x)=(x^2)+bx+n where b and n
are allowed to vary. We start with
the case n=1. Consider the
equation (x^2)+bx+1=0. Here is the
graph of this equation in the x,b plane, that is, the graph of the relationship
between x and b.

What
is the significance of this graph?
For each fix values of b we have at most two corresponding values for
x. For example, if b=3 then we
have the new graph

and
we see the that the intersection of these graphs occur at the corresponding x
values. What are these x
values? These are the two roots of
our quadratic function! To see what these x values are explicitly we use the
quadratic formula:

What
if b=1? Then we obtain the graph shown below in which no values of x correspond
to b=1.

Why
are there no roots to our function?
This is because our discriminant is (b^2)-4=1-4=-3 and -3 has no real
roots. To find what values
of b correspond to roots we calculate

(b^2)-4>=0

b^2>=4

-2<=b<=2.

Therefore,
for values of b smaller than -2 and larger than 2 we have 2 real roots and for
b=2 and b=-2 we have one real root.
For values of b between 2 and -2 we have no real roots.

What
would happen to the number of roots to our function if we vary n? For example,
what if n=-1? Then for any value
of b the function will have two real roots!

Looking
back at the discriminant, if n>0 then for b>(4n)^(1/2) and
b<-(4n)^(1/2) we have two real roots, one real root if equality holds, and
no real roots if
-(4n)^(1/2)<b<(4n)^(1/2).
If n<0, however, we have that (b^2)-4n>0 for all b and so there
will always be two real roots.
What happens if n=0? We get the following graph:

It
appears as though we get only one real root but this is not the case. If n=0 we have 2 real roots! If n=0 then we have (x^2)+bx=x(x+b)=0
and so x=0 or x=-b. If b=0 then we
have 0 as a repeated root. Clearly
the number of roots of our function f(x)=(x^2)+bx+n=0 vary with n and b and
there relationship is most seen by the discriminant, that is, the quantity
(b^2)-4n. Below is the graph of
values for n=2,1,0,-1, and -2.