Equation by Graph

Question

Given a random Graph,is there any known way to find an equation for it ? If I create a random graph is there a way that i can find an equation that totally describes my random graph?

Answer 1

Let's suppose that the "random" graph you drew represents some function $y = f(x)$. In other words, above each point of the $x$-axis, there is at most one point on your graph.

Then there are a few ways to get an equation that gives a very good approximation of $f$, at least. If the function $f$ is continuous (i.e. you drew it without lifting your pen from the paper), then you can find a polynomial function that's everywhere very close to $f$. By "very close" I mean as close as you want, actually. This is is guaranteed by something called the Weierstrass Approximation Theorem.

If you actually want to compute the equation of an approximating polynomial (rather than just being assured that it exists), then I recommend a piece of software called chebfun. If you look on the chebfun web site here, you'll see some of the wiggly random-looking graphs for which it can find equations.