# How do you explain cubic growth of a function

When reading scientific papers I have seen people explain the growth of a variable linearly, exponentially. However how would one say for a variable which grows in quadratic fashion, or cubic?

I don't remember reading something like: "y grows cubically with x"

Can I say, `y` grows cubically with `x`?

What would be the proper way to explain cubic or quadratic growth of a variable as a function of another one?

How would you explain the given lines:

• Mathematicians tend to use increase rather than grow in such contexts. For any such "to the power n" relationship you can say Y grows exponentially without mentioning the other factor. For n=2 (or n=3), you can say Y increases as the square (or cube) of X. For larger exponents, it's Y increases as the nth power of X. I think, but there are plenty of "real" mathematicians here who should know better than me. Commented Aug 16, 2015 at 20:17
• Also, the 4th power is quartic, not quadratic. Commented Aug 16, 2015 at 20:22
• @Hot Licks: Dunno if there's a highly specialised mathematical definition, but so far as I'm concerned a value increases exponentially if the relevant exponent is greater than 1 (in consequence of which each "increase" must be greater than the preceding one, which means the rate of increase will continually go up). Commented Aug 16, 2015 at 20:25
• Kristol: 'cubically' is fine as an analog to 'quadratically'. This is a question better asked on math.SE or cs.SE as they might have their own preferred specific jargon for this. Commented Aug 16, 2015 at 21:12
• @FF z = y x y, z = 6y x y x y etc are called power functions not exponential functions (though power, exponent and index overlap in meaning). The base y varies, the exponent is constant. In an exponential function, the exponent is the independent variable. Commented Aug 16, 2015 at 21:43

Growth as a quadratic, cubic, quartic, or other fixed-power multiple is referred to generally as polynomial: formulas such as x2 or x3+2x+7 or even x99+99x98+98x97+... are all polynomial. (These formulas can be characterized as "when you double your input X, your output will be 4 times (or 9 times, or 64 times) as big.")

Exponential growth occurs when the variable itself appears as the power, e.g. in a formula such as 2x, for example in a situation where "the population at time t is approximately equal to 1.1t." (These formulas can be characterized as "when you add 1 to your input X, your output will be doubled / tripled / multiplied by some other factor.")

• +1 this is the correct mathematical answer. For a sufficiently large x the largest exponent in a polynomial dominates all others, so one usually refers to the "degree" of the polynomial (the largest exponent). In a cubic (3rd degree) polynomial (with a positive coefficient) the function is proportional to the cube of the input (x) for sufficiently large x. Related to programming Big "O" notation rob-bell.net/2009/06/a-beginners-guide-to-big-o-notation
– amdn
Commented Aug 16, 2015 at 21:46

The mathematical term for a function of a variable that changes with a fixed power (actually the sum of fixed multiples of fixed non-negative powers) is "polynomial." The multipliers are called "coefficients; the powers, "exponents." ("Fixed" here means constant, i.e, the values of the coefficients and the exponents don't change with the value of the variable.

The highest power, called the "degree" or "order" of the function, gives the name of the polynomial:

0 constant
1 linear
3 cubic
4 quartic
5 quintic

After exponent=5, the names are simply the "n-th power."

I'm not sure what you mean by "explain" the graph. A polynomial function operates by iterated multiplication on the value of the variable, the number of iterations given by its degree. Because multiplying a number between 0 and 1 by itself results in a product that's smaller than the number, the higher the degree of the polynomial, the slower its growth between 0 and 1.

In your graphical example, your polynomials have only one term each, and all their coefficients are 1. These are called "monic monomials." Their values are zero starting at 0 because multiplying 0 by itself any number of times results in 0. The values increase toward one as the variable approaches 1, reaching one at 1. This because multiplying 1 any number of times by itself just results in 1.

I think the only practical phrase is "Y increases as the cube of x" (or "as the fourth power" as the case may be). But it is rare that it is important to the actual argument whether the increase depends on the third power or the fourth, and if it is the audience will presumably study the graph, so exponentially is the usual term no matter what the exponent.

It appears from the question that you dislike this because you think exponentially refers to 'y = x squared' only. That is not just over-general but incorrect: the term for that is geometrically.

• Or if there is a proportionality constant, y is proportional to x cubed etc. But 'exponentially' does not refer to 'y = x to the power n' but to 'y = a to the power x'. Commented Aug 16, 2015 at 21:46
• you think exponentially refers to 'y = x squared' only might be a bit misleading Commented Aug 17, 2015 at 8:32