yes
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The second difference of a quadratic equation being one indicates the second derivative at that point is positive. What you do from there depends on what property or transformation you're looking for with respect to the equation.
These are called the second differences. If they are all the same (non-zero) then the original sequence is a quadratic.
No, it is not.
Yes.
A quadratic function is a second degree polynomial, that is, is involves something raised to the power of 2, also know as squaring. Quadratus is Latin for square. Hence Quadratic.