A point of inflection of a graph is where the concavity of that graph changes. On particle- or object-motion graphs, these points are where the acceleration reverses. A point of inflection is a zero of the double derivative (something that is not always easily-found). Take the derivative of the derivative of the original function, set it equal to zero, and solve for X.

Example:
If the original equation is given as:
F(X) = X^4 - 4X^3 + 4

Then the double derivative is given as:
F``(X) = 12X^2 - 24X

Simplified, set equal to , and factored:
12X^2 - 24X = 0
  X^2 - 2X  = 0        (divide both sides by 12)
  X(X - 2)  = 0        (factor)
So F``(X) = 0 when X = 0 or 2.
By plugging 0 and 2 into the original F(X) equation, we find that the points of inflection are (0, 4) and (2, -12). In other words, when X = 0, the concavity flips around and curves the graph, making it concave-down, and when X = 2, the concavity reverses once more, making it concave-up.

Try it for yourself: Plug X^4 - 4X^3 + 4 into your graphing calculater and see how much better you can understand the curves in a graph from the points of inflection.