What is the proper adjective/adverb for a power function?

Question

I have a function where space grows as a power of time: $x= at^2$. In my report, I've been using the adjective 'exponential' or adverb 'exponentially' to describe the expansion with time. However, looking at the textbook definition, it appears that an 'exponential' expansion would look like $x = 2a^t$. If 'exponential' isn't appropriate, then what is the proper adjective/adverb for my expansion?

Answer 1

For exponent $2$, you say quadratic, for $3$ and following, cubic, quartic, quintic... For general exponent, power function or power law.

[In French you can use potentielle, but this can cause polysemic ambiguities.]

The adverb quadratically can be freely used; I would abstain from "cubicly", "quarticly"...

Also avoid the confusion with quadric, which describes a quadratic surface in 3D space.

Answer 2

The growth of $x^n$ is called linear ($n=1$), quadratic ($n=2$), cubic ($n=3$), quartic ($n=4$), quintic ($n=5$)... You can find here a few names after that, but I don't expect anyone to say "octavic/octic" with a straight face. In general if you don't know $n$ it's just called "polynomial growth".