Exploring
Quadratics
By
Tonya
DeGeorge
In
this investigation, we will be looking at quadratics again, but in a different
way. We will be looking at the
following equation:
This
graph will show the relationship in the (x,
b) plane.
By
adding a horizontal line in the (x, b)
plane, it the intersection of the two graphs corresponds to the roots of the
equation for a particular value of b.
For example, suppose b = 2, we get
the following graph:
If
we were to solve the equation by
hand, we would find:
This
corresponds to the graph above because the two graphs intersect at the point
(1,0) where x = 1. So how can we generalize this for
different values of b? There are a number of cases to
consider:
Case (i): b > 2
Case (ii): b = 2
Case (iii): 2 < b < 2
Case (iv): b = 2
Case (v): b < 2
LetÕs
look at these different regions.
WeÕve already seen what happens when b
= 2. We found that there
exists one double root: x = 1. For Case (i), where b > 2, lets
choose b = 5. We find the following:
This
implies that the equation has 2 negative roots. LetÕs check this by looking at the equation:
Therefore, we see that there
exists 2 negative real roots.
Below
are summaries for the next three cases:
For
Case (iii),
where 2 < b < 2, letÕs choose
the value b = 1. We find:
There are two imaginary
roots (hence, the line does not intersect the graph).
For
Case (iv),
where b = 2, we get:
There is one repeated root
at x = 1 (where the graph intersects
at one point).
For
Case (v),
where b < 2, letÕs choose b = 4. We find:
There exists two positive
real roots.
In
conclusion, we find:
Values of b

Types of
roots

b > 2 
Two negative real roots 
b = 2 
One double root (x = 1) 
2 <
b < 2 
Imaginary roots 
b = 2 
One double root (x = 1) 
b < 2 
Two positive real roots 
Now
suppose we look at the value of c and
change it to 1 and compare what happens to the graph when this change is done.
As
we can see, when we change the value of c
to 1, the graph changes into a hyperbola. What happens when we use other values of c?
LetÕs use the following values:
c = 2, 1, 0, 1, 2 (all on
the same graph):
From
here, we can see that the value of c changes
the graph in three different ways.
It could be a hyperbola (two different types) or a line. The three cases seem to be:
Case 1: when c < 0: hyperbola (asymptotic to x = 0 and y = x)
Case 2: when c = 0: a line (y
= x)
Case 3: when c > 0: a hyperbola (asymptotic to x = 0 and y = x)
*Note that the hyperbola is restricted to different
areas of the plane depending on the value of c.