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.