My comment yesterday about the annoyingly imprecise use of the phrase

“inflection point” by innumerate scribblers generated a good quantity of

e-mail, some querying, some quibbling.

In my dual capacity as Ombudsman and Math Guy, I am now going to lay down

the law about just when you may use “inflection point” as a metaphor.

Listen up!

First, cast your eye upon this. It’s a curve with an

inflection point at the origin (i.e. dead center of the picture).

Just as defined in Frost’s CURVE TRACING, the inflection point is one at

which the tangent to the curve cuts the curve, like this.

(Note it is necessary that the tangent to the curve AT THAT POINT cut the

curve AT THAT POINT. The tangent at any old point might cut the curve at

SOME DIFFERENT POINT — like this. That doesn’t make the

tangent point a point of inflection.)

Looking at the thing non-mathematically, what’s actually happening here at

this inflection point?

Well, if you follow the curve from left to right along the diagram, what is

happening is DECELERATION changing to ACCELERATION at the inflection point.

So a proper use of “inflection point” as a metaphor is when you have some

phenomenon — the U.S. economy, say — that has been decelerating for a

while, then begins to accelerate. That’s an inflection point — the point

when deceleration stops and acceleration starts.

(Note that the deceleration need not go down all the way to zero — it

doesn’t in this diagram, though it might in another one.)

Suppose an ACCELERATION switches to a DECELERATION. Would that be an

inflection point? Sure would: like this.

Everybody got that? Good. Now let’s see no more sloppy usages of

“inflection point,” please, not in my bailiwick at any rate.