interpolating function with inclined asymptote

Question

I would like to find an interpolating function that is a curve with an inclined asymptote. I already tried with the parabolic and with the homographic (y= (ax+b)/(cx+d)). Unfortunately they are both phisically inconsistent with the phenomenon that I am analysing. I need a function like the c) in the image attached, which is a curve that has an inclined asymptote. Can you help me ?

Answer 1

I recommend looking at functions of the form polynomial/polynomial, where the top polynomial's degree is 1 larger than the bottom polynomial's degree. You'll also want to make sure the bottom polynomial never equals 0 when x is nonnegative. One family of examples (that go through the origin, as seems to be desired from your pictures) is $$ y = \frac{ax^2+bx}{x+c} \tag{1} $$ where $a,b,c$ are positive. (The inclined asymptote will then be the graph of $y=ax+b$, for that matter.)

If you don't insist that the graph go through the origin, then you can expand this family to $$ y = \frac{ax^2+bx+d}{x+c}. \tag{2} $$ Another way of writing this same family is $$ y = ex+f+\frac g{x+c} $$ (where the inclined asymptote is now $ex+f$). Whether $g$ is negative or positive controls whether the graph lies below or above the asymptote (equivalently, whether the curve is concave or convex). Both shapes are possible with different choices of the parameters—and the same is true for the forms (1) and (2).