How does one create a polynomial equation based on graphic features? So as many of you know, graphs of polynomial equations with degree greater than 2 have what are known (to me) as local minima and local maxima (the point(s) on its graph where the derivative of the function is zero, the point to which output values converge inside a certain range of input values). Based on the coordinates of these points, and other information taken from the graph, is it possible to create an equation that describes it? For example, if I asked, "Create the equation of a cubic function with positive-positive end behavior that has a local maximum at (3,5) and a local minimum at (6,2), with the graph passing through the origin. Is there a systematic way to do this? Also, if there is not enough information, I would like to know, and why.
Thanks!

Taking the points with the maximum and minimum, which are sufficient to define the cubic, we can just write the cubic as $y=ax^3+bx^2+cx+d$ and plug in the data we have.
$$5=a3^3+b3^2+c3+d\\2=a6^3+b6^2+c6+d\\0=3a3^2+2b3+c\\0=3a6^2+3b6+c$$
This is four equations in four unknowns which can be solved by the usual techniques to yield $a,b,c,d$.
