# How to find the equation of a graph of a rational function from a set of points?

For example, the data points are:

(1,1)

(2,1/2)

(3,1/3)

(4,1/4)

(5,1/5)

How do I find the equation from those points? Do I look at the common ratio of the y-values or something?

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You might note a mapping of the for $x \mapsto 1/x$, right? – Pedro Tamaroff May 30 '12 at 2:38
If it is stated that the datapoints are necesarily that of a rational function of the form $$y=\frac{a}{x+b}+c$$ you can always find $a$, $b$, and $c$ with $3$ entries, by producing a system of equations of 3 variables and 3 equations. – Pedro Tamaroff May 30 '12 at 2:40

Since you're just asking for a curve fitting those points, note that each given point $(x,y)$ has the form $\left(x,\frac{1}{x}\right) \Rightarrow y=\frac{1}{x}$. For this particular problem, the problem-solving skill you're using is called pattern recognition.
For other problems that ask you to find the curve given a set number of points, you generally need $n+1$ points to determine an equation with degree $n$. If a pattern isn't immediately evident, then you generally need computer software to find the best fit equation.