How do I graph the following curve without using a calculator $y^2=x\sin x$?

Question

I have to graph the following curve by hand without taking the derivative of it. If you could help in any way that would be lovely. Thanks in advance!

Answer 1

Let's first understand $y=x\sin x$. It's bound between the lines $y=\pm x$, which form two "cones" touching at the origin. The curve touches each line infinitely often, where $\pi/2|x$. So it looks a bit like $\sin x$, but with larger oscillations as we move away from $x=0$ in either direction. It's an even function rather than an odd one, but the roots - multiples of $\pi$ are the same. You should be able to sketch it.

Now, what about $y^2=x\sin x$? This time, there are three key changes. Firstly, the bounds are now $\pm\sqrt{x}$ instead, so the oscillations don't grow as much. Secondly, the regions where $x\sin x<0$ are deleted, so there are infinitely many width-$\pi$ gaps. Finally, $y=\pm\sqrt{x\sin x}$, so if you work out what happens with $y\ge0$, it's mirrored below the $x$-axis.