When going from velocity to position..

Hi, I finally sat down and got to understand what the workings are when you derive position -> velocity -> acceleration -> jerk etc.

I made some graphs on a paper, since I work better when I get a visual representation, and the thing is, I can't get it to make sense in the link between position and velocity.

I have a movement that goes through: 1 m/s, 2 m/s, 3 m/s and 4 m/s plottet on the position vs time graph. So it's an expontentially growing curve.

When plotting on the velocity vs time graph, it's a growing curve with 'x' (inclination of 1), since for each second passing the object is moving 1 meter extra per second compared to the previous second (acceleration).

So looking at it visually, it makes sense to me at this point. But when I go to derive the 'x' function it becomes '1/2x^2', naturally, and that function doesn't fulfill my position vs. time graph.

The values at the position vs. time plot would be 1/2, 2, 4.5, 8, when I'd want it to be 1, 2, 3, 4 to fit my graphical representation in the position vs. time plot.

So that's why I'm a bit confused and would love a pointer to what I'm not seeing. So if anyone could help me understand.

I am wondering if the position vs. time graph is wrong compared to the velocity vs. time one, that's the only explanation I can think of, and it might be that my position function can't be described by a simple function?

I'm going to keep thinking about this, and I might have found the answer by the time I get an answer here, but well, if not then I have this to look forward to :)