What is Expo in RC airplanes

It seems to me through the many videos that I watched that many people don't really understand what the "Expo" (Exponential) function does on an RC transmitter. Most people believe that the expo will smoothen/dampen out the movements of the airplain and make it not so sensitive, which is not true, in fact, partially, the expo will MAKE the airplane more sensitive and twitchy. As I am studying engineering, I see exponential curves almost everyday, so I will try my best to explain how they work.

First of all, let's start by showing the difference between an exponential and a linear function. An exponential function is on the form y=e^x (^ means "to the power") and if we draw x vs y on a curve, here is how it will look like.
Opposed to exponential functions, there are linear functions whose graph should look like a line. Here is an example of how they look.
You should naturally be wondering now, what the hell does this have to do with RC flying. I will explain that right now. Imagine that your control sticks are the X axis of the graph, when you move your stick to a certain position, your transmitter translates that into the y value of the curve, and that's the value that's actually sent to the receiver and thus to the servos.
Now let's see how each graph affects the airplane. The linear function, which is the default one, if you tilt (move) your control stick 1 degree (think of it as 1 mm), the servo moves 1 degree in that same direction (think of it also as 1 mm). Let's say you "really" want to dampen your airplane movements. This you can do by going to your airplane's curves and just making then look like a line that is not very steep, a little closer to the horizontal, this means that now, if you tilt your stick 1 degree, the servo will move only 0.5 degrees, which is "damping/smoothing" the movements. But this has a disadvantage, let's say that when you tilt you stick to the max it goes 10 degrees. Now your servo will go 5 degrees. When in fact, your servo can physically go up to 10 degrees. Thus you are using some maneuvrability, and actually some capability to recover from a possible crash for example.
The expo now comes in handy, let's see how the expo function behaves. In the expo function, the amount that the servo moves does not depend only on how much did you tilt your stick, but also on where it is. So for example if you center your stick to zero and then tilt it 1 degree, the servo will move 0.5 degrees. Now, take your stick and tilt it 9 degrees and consider that your new starting position. If you tilt your stick 1 more degree, the servo will react to that with a movement of 2 degrees now. I know it might be a little hard to grasp, just sit and look at the curves and think about it. What the expo function now does for you is that it makes movements near the zero position of the stick move the servos less, while movements near the extremes of the stick move the servo more. This is very useful because small movements near the center are usually used when you are hovering a helicopter and just making small adjustments to keep it in place, or flying and air plane on idle and also making small adjustments to keep it streigt or make a wide bank or something like that. At the same time, movements near the extremes of the stick are used when you are making stunts or trying to recover from a possible crash. In those situations, sensitivity doesn't really matter since you are already making big movements and jamming the stick left and right all the way.

So, to sum it up, expo doesn't decrease the twitchiness or the plane or dampen the movements, it does that only near the center of the stick, but compensates for this by making movements near the end of the stick more twitchy and sensitive to give you the full range of your control surfaces/servos. A linear function is just an almost direct map between your control sticks movement and the servo movement, which means, if you move your control stick 1 degree the servo will move 1 degree not depending on whether the stick moves near the center or near the extremes. (Please note that it doesn't necessarily mean that your servo will move 1 degree for each 1 degree tilt of the stick, the servo could move 2 or 3 degrees for each 1 degree from the stick... this was just an example to demonstrate. But you can change the amount the servo moves for each "x" movement of the stick in your curves or there was another place where you could set the throws or something like that. I just don't remember the name in the menus, you would specify how much % of the servo movement should correspond to your 100% stick movement, and the computer calculates everything else for you. Note, you have 2 settings for this, one is called Normal-Rates, and the other is called Dual-Rates, so basically you can configure 2 sets of curves, one for each.)

I hope this little article was helpful for somebody out there, please let me know if you like it.