Second Differences and Quadratics

Take any quadratic and plug in equally spaced x-values, like 0, 1, 2, 3, 4. The outputs won't grow steadily, but the first differences, the gaps between consecutive outputs, will grow steadily, in a straight line pattern. That happens because the first differences of a quadratic form a linear sequence. Take the differences of those differences, the second differences, and they stop changing entirely. They land on one constant number, and that number is always 2a, twice the leading coefficient. The square numbers 1, 4, 9, 16, 25 show this directly: first differences 3, 5, 7, 9, then second differences 2, 2, 2. Since squaring is x², a = 1, and 2a = 2, exactly what shows up.

This constant second difference is the discrete version of a parabola's constant curvature. A graph that bends the same amount everywhere is exactly what average rate of change turning into a straight line looks like, step by step, rather than smoothly. Where linear functions have constant first differences (constant slope), quadratics push that constancy one level deeper, into the second differences. Cubics push it deeper still: their first and second differences change, but third differences settle into one constant nonzero value. In general, a degree-n polynomial's n-th differences are constant. This gives a quick test for degree from a table of equally spaced data: keep differencing until the row goes flat, and count how many rounds it took.