I was messing around in Desmos and wanted to create a chart to show the x values, function values, first differences, and second difference of some quadratics. The graph I was using is here. However, I had to create explicit formulas for the first differences and found some really strange stuff going on.
It's common knowledge that the second difference of a quadratic is just the $a$ value times $2$, assuming that the function is in the form $f(x) = ax^2 + bx + c$. If I recall correctly this is because that is the slope of the derivative of the function.
However, I found that the first difference of a quadratic at $x$ is $2ax+a+b$, where the first difference is the change in y over the next step of one unit in the x axis.
This is definitely not the derivative, but for all of the values I tested (I used Desmos's little slider thingies) it seemed to work. However, I couldn't find this formula anywhere on the internet.
I haven't proved anything, so have I just had happy mistakes and it doesn't actually work? If it does work, why does it work? Additionally, an explanation that explains the difference between the 'first difference' and the derivative would be great.
Let $f(x) = ax^2 + bx + c$ be a quadratic equation. The first difference of this quadratic can be expressed as: $$f(x+1) - f(x) = a(x+1)^2 + b(x+1) + c - (ax^2 + bx + c) = 2ax + a + b$$