The Department of Mathematics Education
Write Up #2
Observations of Asymptotic Behavior Associated with Adding nxy Term to a Quadratic
Starting with a quadratic function of the form y=ax2+bx+c, a term nxy is added to this function. The original quadratic is shown in Figure 1 with a=2, b=3 and c=-1.
Figure 1: Original quadratic equation.
When adding the term nxy to this function, y=ax2+bx+c+nxy, two obvious features of the resultant function is that the locus of points on this new function always pass through the roots of the original quadratic and through the point (0, -1) regardless of the value of n. This is demonstrated in the animation, Figure 2.
The cause of the behavior of always passing through the roots of the quadratic is readily apparent by examining values of x for y=0. Since this causes the added term nxy to also be equal to zero, regardless of the value of n, we are left with the original quadratic for y=0. Of course, the solution to this is the roots of the original quadratic. The behavior of passing through the point (0, -1) regardless of the value of n can be seen just as easily by noting that the term nxy is again zero for x=0, are the second and first degree terms. This leaves the value “c”, or the point (0,c). In our example, c=-1.
One additional very apparent feature is that the two function coincide when n=0 for obvious reasons; n=0 is a third way to set the nxy term to zero, but this holds for all x and y values. In other words, n=0 leaves us with the original quadratic.
Beyond some of these immediately apparent features, graphing the new function shows interesting behavior near the roots of the original function. The remainder of this paper explores this behavior.
In the example y=2x2+3x-1+nxy, the graph bears a marked resemblance to y=+1/x, as n approaches +∞, noting of course that the result is still forced to go through the roots and (0,c) as previously mentioned. But there are a couple of critical transition points that occur in the neighborhood of x=-0.56 and x=3.6 where the “1/x” type graph abruptly changes from opening toward quadrants I / III to opening toward quadrants II / IV. This is demonstrated in Figure 3 for the neighborhood of x=3.6..
This behavior becomes apparent by rearranging y=ax2+bx+c+nxy as follows:
Assume the roots of the quadratic equation are r1 and r2. Then this equation can be rewritten as:
Note that the original quadratic is in the numerator of this result. It is now apparent what is occurring at the points x=-0.56 and x=3.6. Factoring the quadratic results in roots at r1=0.281… and at r2=-0.562…When n =1/r1=3.56 or n=1/r2=-0.56, terms cancel and the results becomes:
Figure 4 and Figure 5 dynamically shows the new functions behavior in the neighborhoods of the critical n values. Note that on either side of the critical value, the linear portion of the graph agrees with the above description with a slop of –a/n and a y intercept of –(a·r1)/n = –a·r1·r2 = c. Note this is already know to be a necessary result from earlier in our discussion when we noted that this graph must always include the point (0,c). The two slopes are the just -a·r1 and a·r2.
Figure 4: Transition point at root r1.
Figure 5: Transition point at root r2.
So now that the linear portion of the resulting function is understood, what about the abrupt change in quadrants of the resulting function? This is apparent from noting the as x approaches1/n in the denominator (x approaches a root), the denominator approaches zero from one direction, say positive. As x passes the root, the sign of the denominator abruptly flip negative, hence the abrupt quadrant change of the 1/x type result. Note also that the vertical line at the root values on the x axis is undefined, strictly speaking, since at this point, y=0/0.
To conclude, the addition of nxy results in 1/x type graphs for values of n approaching infinity. The behavior of the graph can be understood from considering the roots of the original quadratic function. One final note, if the quadratic is adjusted to not have any roots, the abrupt transition behavior is absent from the animation as one might expect (see Figure 6).