Regression using Higher Order Polynomial feature

$1$ point possible $($graded$)$

Assume we have $n$ data points in the training set $\{(x^{(t)},y^{(t)}){\}}_t=1,$dots,$n$ where $(x^{(t)},y^{(t)})$ is the $t-$th training

example:

A biochemist is considering the depicted data and we're helping them.

We want to find a non$-$linear regression function $f$ that predicts $y$ from $x,$ given by

$f(x$;$,{}_0)=*{}^{}(x)+{}_0$

where $(x)$ is the polynomial feature vector that includes all and only the monomials of degree at most $k($in

this case, since $x$ has dimension $1,$ this means $$ has $k+1$ components; the degree$-0$ component is

redundant with the bias term ${}_0,$ but that doesn't matter for this problem$).$ What degree $k$ would you

recommend the biochemist use? Note that this is a soft, not$-$completely$-$mathematical question, much like the

question, 'Does Louisiana look more like a boot or a mitten?$\text{'}-$ there's consensus here among folks who know

the terms involved, even though it's a soft question. Common sense and human experience are important in

$ML$ engineering, so this question is fair game.