How do we know that graphs have the shapes they do?

Question

This is a pretty simple question I think, but I don't know how to answer it.

For example, we all know that a parabola such as y=x^2 looks something like this: parabola

But my question is, how do we know that the graph actually looks like this? We can't actually plot out an infinite amount of points to see that the graph follows a smooth pattern, but instead we can only individually plot points and we just connect them. Why do we just assume that the points can be connected?

If anyone needs any clarification about my question, feel free to ask.

Answer 1

In the particular case for the parabola, it is a conic section, so you can obtain its shape by imagining the intersection between a cone and a plane.

Answer 2

Continuity and smoothness aren't really enough by themselves to be certain that we've drawn a given graph correctly: maybe in the interval $(17.123487634827631, 17.123487634827632)$ there's a huge spike! We wouldn't see this just by plotting a bunch of points unless by sheer dumb luck we picked one in that interval, so how can we rule it out?

To rule this out, we analyze the function. For example, we can prove that on positive reals, the function $f(x)=x^2$ is increasing; this rules out such a spike, because "half" the spike would have to be decreasing (think about it). Similarly we can figure out how fast $f$ can ever increase, in any interval (that is, we can find the maximum of the derivative of $f$), and so forth. Perhaps the most useful fact about $f$ in terms of graphing it is that $f$ is convex: if we let $L$ be the line connecting $(a, f(a))$ to $(b, f(b))$ (for $a<b$), the graph of $f$ on $(a, b)$ lies below the line $L$. This gives a lot of information about what the graph looks like! And this fact can be proved from the basic properties of the real numbers.

More complicated properties of graphs (e.g. being "steeper" over here than over there) can be precisely defined and verified using calculus. In the end, these facts can give us a very complete picture of what the graph of $f$ looks like.