Formulating the exponential equation given a certain pair of points

Question

I have the following pairs of points:

(0 , 100) ; (0.81 , 41) ; (1.38 , 20) ; (1.75 , 9) ; (2 , 4)

How can I determine the equation of the curve passing through this points?

Thanks very much!

UPDATE

What I'm trying to achieve is to get the function representative of the height decrease depicted in the picture.

Answer 1

There are infinitely many curves passing through these points. You have to be more specific.

If you want the minimal degree polynomial that passes through them you can use Lagrange interpolation polynomial.

With your example it would be a polynomial of degree 4. But as I said if you allow for higher degrees you would get infinitely many results.

Answer 2

You can use lots of representations, depending on your requirements for continuity at each point.

Piecewise linear segments will "work", but you could also go with higher order piecewise interpolations. You could use Bezier or some other splining technique.

You could assume a single polynomial of order 4 and calculate the coefficients so it passed through each point.

There are lots of ways to do it.

Answer 3

If your data is x_i,y_i (i=1..n) you could fit a line through x_i,log(y_i); if that line is log(y_i) ~ a*x_i + b, then x->exp(b)*exp(a*x) may well fit the original data.

Answer 4

I found this website which seems to be precisely what I need. I just have to input my data pairs and choose the appropriate degree (in my case 4). It then compute the correlation factor and the equation.

http://www.arachnoid.com/polysolve/index.html