I think the premise of the question is false. When asked how the graph of $y=x^6-2x^2$ looks like, one does not reach into the catalog of "known graphs of functions". Instead, one parses the formula and identifies two summands: $x^6$ and $-2x^2$. Three observations are made then:
- Both summands are even powers, therefore the function is even
- For small values of $x$ ($x\approx 0$) the first summand is much smaller than the second
- When $x$ is large, the first summand is much greater than the second.
The above is enough to picture the graph, which is flat at $x=0$, moves down symmetrically in both directions as upside-down parabola, but then goes upward (also symmetrically) roughly like a parabola but steeper. If one wishes for the picture to be on accurate scale, then looking for intercepts $y=0$ and critical points $y'=0$ is the next logical step.
So, instead of a catalog of graphs one should know, the question might be about the skills one needs to develop for this purpose. But I don't see how to put this latter question in SE format: it looks too broad for it. Qualitative understanding of functions cannot be acquired separately from other skills, such as the ability to see that $10\cdot 10-1.78\approx 100$ without reaching for a calculator.