How to graph the parametric without calculator: $x=\sin t$ , $y=t^2$

Question

Question

How to graph the parametric without calculator: $x=\sin t$ , $y=t^2$

I first make t by itself by doing the following: $\sqrt{y}=t$ and plugging into the other function to get $x=\sin(\sqrt{y})$

But then I realize that I have no clue how to graph that by hand, any tips on doing so?

I believe that I have to get rid of the trigonometric function but I don't know how.

Answer 1

I think you have answered your own question. The two parametric equations can be written in Cartesian form as

$$x = \pm \sin \left( \sqrt{y} \right).$$

These are two (not one!) functions of $y$. Notice that they are symmetric with respect to $x=0$. So it suffices to plot $x=\sin \left( \sqrt{y} \right)$ and reflect it about the $y-$ axis.

To plot $x=\sin \left( \sqrt{y} \right)$, you can use any standard calculus technique: find roots, first and second derivatives, maxima, minima, inflection points, and so on... The only difference is that your independent variable is now $y$, not $x$