# How and why are these three graphs visually related? $y=ex^2 \sin\left(\frac{1}{x}\right)$, $y=ex^2$, $y=-ex^2$

While graphing equations I come across, or find interesting, I found the relationship between graphs $$y=ex^2 \sin\left(\frac{1}{x}\right) \qquad y=ex^2 \qquad y=-ex^2$$

https://www.desmos.com/calculator/nhf6tphhbb

How are these three graphs related, and is there some pattern to their points of intersection?

• It's nothing deep, it's just that the factor $\sin(1/x)$ oscillates between $-1$ and $+1$. Nov 5 '19 at 16:45

As $$-1\le\sin{(\frac{1}{x})}\le1$$, we have $$-ex^2\le ex^2\sin{(\frac{1}{x})}\le ex^2$$.
So, the functions $$ex^2$$ and $$-ex^2$$ serve as upper bounds for $$ex^2\sin(\frac{1}{x})$$