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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;

 

Graph_1.JPG

 

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.

 

Graph_3.JPG

 

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

 

Graph_4.JPG

 

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

Table_2.JPG Graph_5.JPG

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.

 

 

 

 

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