How does a function's differences in terms contribute to it's equation?

I know calculus, so feel free to factor that into your explanation, I will understand.

Question

How does a function's differences in terms contribute to it's equation?

I know calculus, so feel free to factor that into your explanation, I will understand.

anonymous · Answer

What exactly do you mean by this? What kind of equation and how does its terms differ?

anonymous · Answer

It's difference in it's terms increase in increments of 1. I should add that I know the equation to it's differences, which is y=x-0.5. I have tried finding the anti-derivative,  but it never lines up when i plot it along with the points given. Attached is a hastily drawn table of the points.|dw:1360557097694:dw| 
my apologies, i wasn't very clear with my initial question.