Quadratic growth

In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta notation,

f

(

x

)

=

Θ

(

x

2

)

{\displaystyle f(x)=\Theta (x^{2})}

. This can be defined both continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural number variable).

Examples

Examples of quadratic growth include:

Any quadratic polynomial.

Certain integer sequences such as the triangular numbers. The

n

{\displaystyle n}

th triangular number has value

n

(

n

+

1

)

/

2

{\displaystyle n(n+1)/2}

, approximately

n

2

/

2

{\displaystyle n^{2}/2}

.

For a real function of a real variable, quadratic growth is equivalent to the second derivative being constant (i.e., the third derivative being zero), and thus functions with quadratic growth are exactly the quadratic polynomials, as these are the kernel of the third derivative operator

D

3

{\displaystyle D^{3}}

. Similarly, for a sequence (a real function of an integer or natural number variable), quadratic growth is equivalent to the second finite difference being constant (the third finite difference being zero), and thus a sequence with quadratic growth is also a quadratic polynomial. Indeed, an integer-valued sequence with quadratic growth is a polynomial in the zeroth, first, and second binomial coefficient with integer values. The coefficients can be determined by taking the Taylor polynomial (if continuous) or Newton polynomial (if discrete).