**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.

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

á 0 if the discriminant is less than zero

á 1 if the discriminant is zero

á 2 if the discriminant is greater than
zero

Another way of
looking at this is setting the discriminant equal to 0, and solving for b^{2.}
Hence, the number of real roots will be

á 0 if b^{2} < 4ac

á 1 if b^{2} = 4ac

á 2 if b^{2} > 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 y^{2} to 4c, or simply comparing y
to .Hence, the number of intersection points
will be

á 0 if y < |

á 1 if y =

á 2 if y >||

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

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.

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

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.