How do you determine if a curve in $\mathbb R^2$ is bounded?

Question

Say you have a curve $D\colon y=x^2+1$. If I'm not wrong, $D$ is closed as it belongs to $\mathbb{R}^2.$

Is the curve bounded? If so, how do I determine if $D$ is bounded? And if it isn't, how can I prove that?

Answer 1

The curve can be parametrized as $$ D = \{\, (t,t^2+1) \mid t\in\mathbb R\,\}. $$ Hence, the norm squares of the points on $D$ are $$ \|(t,t^2+1)\|^2 = t^2 + (t^2+1)^2. $$ But $t^2+(t^2+1)^2$ is unbounded when $t\to\infty$, hence $D$ is unbounded.