Finding formula for rational point ($x$,$y$) on the y = $x^2$

Question

How to find formula for rational point ($x$,$y$) on the $y$ = $x^2$ in term of rational parameter $t$. And also how would I write in form of ($x$,$y$) =($f$($t$),$g$($t$))

Answer 1

Yes what you are doing seems correct.

Take $x=t$ where $t \in \Bbb Q$. Then $y=x^2=t^2 \Rightarrow y \in \Bbb Q$.

Also it is not true that '$x^2$ is rational $\Rightarrow x$ is rational'.(Take $x^2=2$, then $x=\pm \sqrt 2 \notin \Bbb Q$)

Hence $\{(t,t^2):t \in \Bbb Q\}$ is the set of all rational points on the parabola $y=x^2$.