A linear changing difference is a difference that increases or decreases in greater amounts each time. For example, if $f(1) = 1$, $f(2) = 4$, $f(3) = 9$, $f(4) = 16$, then the differences are $3$, $5$ and $7$, respectively, and changing linearly, in this case with the linear rule, $2x + 1$. What kind of function has this kind of change?

The original function is quadratic.

Take a quadratic function and list out some successive differences. For example, let's take the differences from the earlier hint: $3$, $5$, and $7$. What is the difference between these differences and how do they change? If they don't change, what is the term for that?

The differences of the differences are constant.