x=x(t)=m/cos(t), y=y(t)=n(tan(t) - t),

The simplest model would be a segmented regression model.

However, the assumption of linearity does not seem ideal particularily for x < 150. It seems negative values for x are not possible and for x->0 follows y->Inf. Thus, the left part should be some kind of 1/x or exp(-x). The right part is probably a saturation function, but here the difference to a straight line does not seem relevant. If you need a model to explain features of these data, you would need to come up with some theory and build an appropriate model from that theory.

If you need it just to get a simple description, you can use segmented regression. If you want to fit a smoother curve without sharp corner, you can use a loess curve or a spline model.

Here's an application of your question. The current computation would be best done in R with using a robust M -estimator. For full details see these references:

Article Robust regression-based analysis of drug-nucleic acid binding

https://b-ok.cc/book/5825760/ed313f

https://b-ok.cc/book/466042/d5d065

Intro to whole topic: is attached

Best wishes, David Booth

A function can have at most two oblique asymptotes, but only certain kinds of functions are expected to have an oblique asymptote at all. For instance, polynomials of degree 2 or higher do not have asymptotes of any kind. (The degree of a polynomial is the highest exponent on any term. For example, 10x^3 – 3x^4 + 3x – 12 has degree 4.)

As a quick application of this rule, you can say for sure without any work that there are no oblique asymptotes for the quadratic function f(x) = x^2 + 3x – 10, because it’s a polynomial of degree 2.

On the other hand, some kinds of rational functions do have oblique asymptotes.

The reciprocal function f(x)=1/x more sutable in this case.

Here is one interesting example: a function given in a parametric form

x=x(t)=m/cos(t), y=y(t)=n(tan(t) - t),

has two oblique asymptotes.

Thank you Jochen Wilhelm, David Eugene Booth, Karen A. Darbinyan , and Anton Vrdoljak for your comprehensive answers.

Dear Anton Vrdoljak. I tried to draw the equation you proposed. I changed its shape to

y(x) = n(tan(arccos(m/x)-arccos(m/x)))

to be able to draw it.

Please find attached the drawn shape for equation

y(x) = 5(tan(arccos(5/x)-arccos(5/x))). As it is observed in the attached figure, there is just one oblique asymptote.

x=x(t)=m/cos(t), y=y(t)=n(tan(t) - t), has two oblique asymptotes.

Polynomial Division to Find Oblique Asymptotes. The idea is that when you do polynomial division on a rational function that has one higher degree on top than on the bottom, the result always has the form mx + b + remainder term. Then the oblique asymptote is the linear part, y = mx + b. https://magoosh.com/hs/ap-calculus/2017/oblique-asymptotes/

Thank you Chinaza Godswill Awuchi for your answer.

I changed the equation to this form in order to draw it

y(x) = n(tan(arccos(m/x)-arccos(m/x))). I was wondering to know whether I made any mistake in converting the equation so I do not see two oblique asymptotes.

Also about the rational function you mentioned, I tried it before. I tried a rational function with a squared polynomial in the numerator and a first degree polynomial in the denominator. It gave me an oblique asymptote and a vertical asymptote. And, I could not find any rational function with two oblique asymptotes that match my data.

Mohamadreza Haghani

I suggest you try tablecurve 2d. It is an excellent application to find functions that fit your data. I promise you will not regret.

1.Is the function rational?

2. Is the degree on the top is one more than the degree on the bottom.

If the answers to 1&2 are yes use polynomial division.

To Mohamadreza Haghani: I am able to draw a function in a given parametric form with GeoGebra - leading provider of dynamic mathematics software, supporting STEM education and innovations in teaching and learning worldwide; it is an open source multiplatform software, available in many languages for Windows, Linux, Mac OS, Android, iPad and it has an easy-to-use interface. Try it, and you will be able to see that there are two oblique asymptotes...

Thank you Jorge Ortiz Pinilla, Pradeep Paraman, Anton Vrdoljak. I appreciate your answers.

Mohamadreza -

It would be nice, I think, to know, if possible, what application is being considered here. That might help visualize the problem. Trying to find some function that might appear to fit this set of data, without regard to subject matter, might take you down the wrong, arbitrary road.

Also, I note heteroscedasticity seems apparent. Perhaps you should treat this no differently than any other case where you try to use the best regression model that you can. Attention could be paid to a graphical residual analysis and a cross-validation of some kind.

Perhaps you had a reason for starting this discussion as you did. Perhaps you could share that reasoning.

Cheers - Jim

x=x(t)=m/cos(t), y=y(t)=n(tan(t) - t),