Disclaimer: I am aware that the policy of the site is not to give complete solutions to homeworks. But this does not look like a homework, and my answer outlines a general procedure that could be used to solve many similar problems. Indeed, it would have been pointless to solve a different problem just for the sake of not giving the full answer, because generalization is straightforward. Furthermore, I did not provide the numerical solution to the problem.
What you are trying to achieve is called a "fit": finding a function that best approximates your data points. There are many tools that can do it for you, but more on that later.
The first thing you need when you want to fit some data points is a model. A model is a function $f(x;a,b,c,...)$, where $x$ is the independent variable and $a$, $b$, $c$... are called parameters. Your goal is to find the values of the parameters such that the function passes near your data points.
In some cases you already know what model to use. It may be given by the physical situation. For example, if the X data represent time and Y data represent the position of a free falling object, the model you want to use is the law of uniform acceleration:
$$f(x;a,b,c) = ax^2+bx+c$$
By performing a fit of your data, you will obtain the value of the acceleration $a$, the velocity $b$ and the initial position $c$.
Since you don't mention any given model in your question, I assume that you are not aware of any particular physics law that would describe your data. You only have the requirement of the two asymptotes.
So we need to chose a model that fulfills the requirement, and the model needs to have few parameters. This is really important. You definitely want a model with less parameters than the number of available data points, but the less, the better. The best model is the one that is able to reasonably reproduce your data points with the smallest number of parameters. There are ways to make this statement more rigorous, but I will not cover it here.
A simple model that presents the two required asymptotes is a power law with negative exponent:
$$f(x;a,b) = ax^{-b}$$
Now comes the tool part. There are many powerful tools to perform a fit. Some examples are Matlab, Mathematica, R, and even Excel. I prefer to use python and this is what I will show in this answer.
IMO, the easiest way to begin with python is to use Google Colaboratory. You don't have to download anything and can run python directly from the browser.
Once you are in Colab, the first thing you need to do is importing the necessary libraries:
import numpy as np # math functions and arrays
import matplotlib.pyplot as plt # plotting functions
from scipy.optimize import curve_fit # fit function
Then you can define the arrays of data points
X = np.array([1,10,100,2e6])
Y = np.array([600,400,240,4])
And the model function
def f(x,a,b):
return a*x**b
Now you can finally call the fitting function and let python do its magic
p1,_ = curve_fit(f,X,Y)
$\texttt{p1}$ now contains the best values of $a$ and $b$ found by python.
You can plot the result, in this case a logarithmic scale may lead to a better visualization.
x = np.logspace(0,7)
plt.plot(x,f(x,*p1))
plt.scatter(X,Y)
plt.xscale("log")
plt.yscale("log")
plt.grid()
What to do next
If you were able to follow the steps until here, congratulations, you have done a fit. But this is only the beginning. You may have noticed that the first three points are perfectly aligned. They are truly following a power law then.
The last point is a bit off, tough. This may be because the chosen model was not adequate to describe the data, or the lower position of the last point may be due to experimental error. The only way to know is to estimate the uncertainties of your measures and run a statistical test on your fit, such as a Chi-squared test (easier to implement, but not always reliable) or a Bayesian test (more difficult, but usually preferred)
You can also give the uncertainties to the function curve_fit, to obtain a better result. To learn more about the many ways to use the function curve_fit visit the scipy documentation
A formula for linear models
A linear model is a model that depends linearly on the parameters, like
$$f(x;\alpha,\beta) = \alpha + \beta x$$
A fit with this model can be found by using the formula for the simple linear regression.
Let the measurements of the independent variable be $X = (x_1,x_2,..., x_n)$, and the measurements of the dependent variable $Y= (y_1,y_2,..., y_n)$. The averages are defined as:
$$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i; \ \ \bar{y} = \frac{1}{n}\sum_{i=1}^n y_i$$
The best estimate of the parameters $\alpha$ and $\beta$ will be
$$\hat{\beta} = \frac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n (x_i-\bar{x})^2}$$
$$\hat{\alpha} = \bar{y}-\hat{\beta}\bar{x}$$
The power law is not a linear model, but this formula can be used nonetheless, because the logarithm of the power law is $\log(f(x;a,b)) = \log(ax^{-b}) =\log(a) -b\log(x)$. Thus, if we define $\alpha = \log(a)$, $\beta = -\log(b)$, $x' = \log(x)$ and $f' = \log(f)$, then $f'(x', \alpha, \beta) = \alpha + \beta x'$ is a linear model.
This means that you can take the logarithm of your data X and Y, plug them into the formula to obtain $\hat{\alpha}$ and $\hat{\beta}$, from which you can find $a$ and $b$.
