How to determine root multiplicity from ONLY the graph? If you were given the graph of a function, without the function's equation, is there a way to determine exact multiplicity (not just parity) of the roots of the function?
 A: Well, naslundx is correct that we can determine what the equation of a polynomial is if we had a perfectly drawn graph. (We can then use that equation to figure out the multiplicity of the root.) 
But if by "determine from the graph" you mean, just by looking at a hand-drawn figure, then no, there is no way. draks makes a good comment that illustrates why; I have better than 20-20 vision, but just by glancing at the graphs of $x^{2012}$ and $x^{2014}$, I could not determine which function was being graphed (unless the scale was huge). In fact, if the scale were small, I probably couldn't even tell the difference between $x^6$ and $x^{10}$ (or $(x+4)^5$ and $(x+4)^7$ etc.). As a result, I could not determine the multiplicity of the root from the graph alone, without the equation.
A: If we know the function $f$ is a polynomial, and that we can take arbitrarily many exact points along the graph, we can.
From the plot we can pick $n$ points $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ and using a Vandermonde matrix we can solve for all the coefficients, assuming $\deg f = n$.
From there we can 'easily' factorize (since we know the roots from the plot) to find the multiplicity of all roots.
Of course, this all depends on being able to find more or less exact values for the coordinates of the points.
A: Yes. Kind of. You can distinguish at least between multiple and simple roots.
The thing is that if a polynomial has a root $a$ of multiplicity $n$, then its derivative has that same root with multiplicity $n-1$. In particular, if (and I think only if) $n>1$ (it's a multiple root), then the derivative is zero at $a$. So, multiple roots are zeros where the graph is flat. A root is simple iff the polynomial "cuts" the $x$-axis.
As an example that should already be familiar to you, a quadratic has a multiple root if and only if it "kisses" $x$-axis.
Also, I believe the first two derivatives of a curve are zero at a point if and only if the curve is flat at that point, but not a local extremum (ie, like $x^3$ at $0$). So you can even tell the difference between roots of multiplicity $2$ and $>2$.
