**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.