Let $f(x,y)$ be an affine curve over a finite field $\mathbb{F}_q$. Assume that it has some rational points over $\mathbb{F}_{q^r}$, i.e. it has some $\mathbb{F}_{q^r}$-rational points for some integer $r$. Is there any method to check which of these $\mathbb{F}_{q^r}$-rational points are lying on $\mathbb{F}_q$?
In other words, let's say $f(x,y)$ has n $\mathbb{F}_{q^r}$-rational points, how many of these $n$ points can be lying on $\mathbb{F}_q$?
Thank you very much in advance!