How to sketch a function such as r(t) = (t^2)i + (t^3)j

Question

How can I sketch this function by hand? I am not even sure what I should be expecting,.

Answer 1

Note that the pairs $x\mathbf{i} + y\mathbf{j}$ satisfy $x=y^{2/3}$ and that $y$ ranges throughout $(-\infty,\infty)$.

Answer 2

The vector is $$\mathbf{r}(t)=x\mathbf{\hat{i}}+y\mathbf{\hat{j}}=t^2\mathbf{\hat{i}}+t^3\mathbf{\hat{j}}\\ \implies x=t^2,y=t^3\\ \implies t=\sqrt{x},y=t^3,\text{ which holds }\forall x\geq0\\ \implies y=x\sqrt{x}$$ This gives us the graph as the graph of the function $\boxed{y=x\sqrt{x}}$ on $\mathbb{R}^2_{x\geq0,y\geq0}$, i.e., the coordinates of any point on the graph is $(x,x\sqrt{x})$.