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 0, 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.