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I tried asking something similar to this before, but I guess I didn't explain it very well. Hopefully this will be more clear.

I need to get an equation which describes a curve with certain points.

For example, I have a list of 5-10 different points along the x-axis where a curve either touches or crosses the x-axis.

Now, how can I get an equation to describe any curve (at least one of the possible curves) which satisfies this?

I'm only concerned with x values within the range of the given values. So, if the curve interacts with the x-axis at other points outside this range, that would be ok too, it doesn't matter....

So to put it another way,

if y = 0; and 0 < x < 100; also x = (list of given points eg. 0,4,17,19,56,67,100)

It doesn't matter what the other x values are when y ≠ 0. And it doesn't matter what other x values might be outside the given range (in this case when x > 100).

Is there any website or free software which can take the coordinates and spit out an equation? Or is there a method you can explain, which I can use to write my own program for doing so?

To be more clear, I'm given a list of points all where y=0, How can I get an equation such that, someone else could then look at the equation (without seeing the given points), and figure out what those given points were (each place it touches the x-axis, within the range I specify)?

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I think you should check the Wikipedia article on Curve Fitting – Sovos Dec 30 '12 at 3:43

If $x_1,\ldots x_n$ are the points in the x-axis where the graph touches or intersects then $f(x)=\prod_{i=1}^n (x-x_i)$ is one possibility.

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