Explorations in the xy plane

By: Sydney Roberts

 


 

In mathematics, the equation  is typically referred to as the quadratic equation and can take on many different forms. In this equation, x is a variable while a, b, and c are parameters. For the purpose of the below exploration, we are going to let a=1 and b = y. Now, b is also a variable. Therefore, we can graph the equation  for different values of c in the xy plane. Below shows just that where c ranges from negative four to positive four.

Description: graph.png

 

Notice that all of these equations have asymptotes at x = 0 and y = -x. Also, each of these equations either has 0, 1, or 2 real roots.  However, what if we overlay the equation y = d for different values of d, then we can see that this horizontal line will also intersect any of these hyperbolas at 0, 1, or 2 different values of x.

How can we tell how many times the horizontal line will intersect any given hyperbola?

Well, consider again the quadratic equation . Typically, we can determine the number of roots this equation will have by looking at the discriminant: . The number of real roots will be

    0 if the discriminant is less than zero

    1 if the discriminant is zero

    2 if the discriminant is greater than zero

Another way of looking at this is setting the discriminant equal to 0, and solving for b2. Hence, the number of real roots will be

    0 if b2 < 4ac

    1 if b2 = 4ac

    2 if b2 > 4ac

Remember we rearranged the quadratic equation for this exploration so that it is of the form  for different values of c. So in this situation, the discriminant is . Therefore, we can determine the number of intersection points by comparing y2 to 4c, or simply comparing y to .Hence, the number of intersection points will be

    0 if y < |

    1 if y =

    2 if y >||

Lets look at this graphically for a specific hyperbola. Consider the graph of .

Description: graph.png

Here,  =  = . Hence, if we set y equal to anything between negative root 8 and positive root 8, this horizontal line will not intersect the equation in the xy plane. Consider, for example, the line y = 1.

Description: graph.png

Also, if we set y equal to , it will intersect the hyperbola at one place as seen below.

Description: graph.png

And if we set y equal to a value greater than , it will intersect the graph in two places. Therefore, consider the line y = 5.

Description: graph.png

 


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