This might seem like a weirdly elementary question - and it is certainly meant to be a fundamental one, rather than a practical one (that is you are not supposed to just plug it into a graphing software to verify it). How do I know that a graph on the cartesian plane is indeed the correct one?
For instance, consider the equation: $y = x^3 + 5$. And consider a picture which purports to be its graphical representation (this one is generated by a computer so it happens to be correct): cube plot
I think for equations beyond straights lines and conic sections, the only way to verify if the graph is indeed correct is by checking it for all possible points, which isn't actually feasible. Sure, one could propose that by taking the derivative, you can check the function is increasing/decreasing etc., and similar claims can be made in regard to its concavity by using the second derivative. But is that information sufficient to determine what the representation of a graph would be, visually?