Mathematics
Education
EMAT 6680,
Professor Wilson
Exploration 3, Quadratic Equations by Ursula Kirk
Consider the following quadratic equation
Graph this relation in the xb plane.
Consider
equation
, we will analyze the effects of b on the roots.
The first step will be to solve the equation for b; 
What is the mathematics?
Our
curve equation graphs as a hyperbola with a vertical asymptote at 
and a diagonal asymptote at 
Now, we
can graph the plus we
will take a particular value for b, here b=3.
The two graphs overlay as shown below.
What is the
mathematics?
The horizontal purple
line and the hyperbola have two points where they meet. These two points are
the roots of
.
When 
the
solutions are imaginary; when 
, there is one negative solution; when 
, there is one positive solution; when 
, there will be two real solutions.
Now we can analyze the
equation when 
of 
and 
At is our original hyperbola from exercise 1,
here on red.
At is our new hyperbola, here on blue
What is the
mathematics?
In this case, we see a
hyperbola with asymptotes are 
and
Our new hyperbola on
blue crosses the x-axis at two points 
and 
Next, we will plot equation 
when 
What is the
mathematics?
Here we have created a
family of graphs. As long as
,
it seems that we will always have two points where the hyperbolas cross the
x-axis, these points are one positive and the other one negative. The
asymptotes are at and.