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 b2.
Hence, the number of real roots will be
á 0 if b2 < 4ac
á 1 if b2 = 4ac
á 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
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.
