What inflection point?

June 5, 2013

It’s incredible how often I’ve heard people say that a company “hit the inflection point of an exponential growth curve and took off!” I’m writing this post so that I can give people a URL instead of arguing with them in a loud bar until foam comes out of my mouth.

Dude, an exponential growth curve does NOT have an inflection point! It simply doesn’t. And if a growth curve has an inflection point (in which case it cannot be exponential), you certainly don’t want to hit it.

If what I just said wasn’t readily obvious, please continue reading.

An inflection point is a point at which the second derivative of a curve changes sign. That is, it goes from a bowl to a dome, or vice versa. An exponential growth curve looks like this:

Image

Do you see an inflection point? Me neither. So stop the shenanigans! The only time you are allowed to use “growth curve” and “inflection point” in one sentence is when you’re referring to a market saturation or S-shaped adoption curve, which looks like this:

Image

But reaching the inflection point of this curve isn’t a good thing, is it?

Now what I suspect most people are trying to say when they bring up the inflection point shenanigans is an area of the curve where the growth curve appears to start accelerating (sometimes called the knee of the curve, as shown here).

exponential4

Ummmmm…. I have more bad news for you guys: An exponential growth curve does NOT have a knee either! It’s all a matter of scale you used to plot the curve. You can make it look like the actual takeoff happened anywhere along the x-axis by manipulating the scale on the y-axis. Depending on your definition, an exponential growth curve is “always hockey-sticking” or “never hockey-sticking”.

In short, no point on an exponential growth curve is special! Your eyes are tricking you!

PS. Ok Let me answer couple of questions I got so that I can put this thing behind me and sleep well tonight 🙂

Q: Is the point where the derivative of the curve crosses above 1 special?

A: No! The x-axis measures time while the y-axis shows revenue ($) or units sold. Apples and oranges. There is nothing special about the derivative reaching a value of 1 because it depends on the units you chose. Switch units ($1/$1000/1cent for the y-axis and 1 1hour/1day/1month for the x-axis) and you get vastly different points for where the value of the derivative crosses 1.

Q: How about the point where the curve achieves a 45 degree slope?

A: Same as above. Absolutely not! You can make ANY point on the curve have a 45 degree slope by changing the scale on the y-axis!

Q: But visually, there is a point where the curve suddenly slopes up.

A: Yes, I also have a photo on Facebook that shows me holding the moon in my hands. Not real. That takeoff point on the curve is purely a function of the scale used to plot the curve. Choose $1 or $1000 as unit for the y-axis and it gives you completely different “knees”. Check this out for an awesome visualization: http://www.abarry.org/knee.htm.

Also read this: http://www.kazabyte.com/2011/12/we-dont-understand-exponential-functions.html

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