Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?
What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being:
{ (x,y) : y=sin(x) is true }
We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power.
Does that help?
x^2 + y^2 = k
and can't even begin to understand how you could graph something like that.
It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.
But the gist of it I got. And I admit, I'm unusual.
This is literally the entire point of graphing inequalities.
He later realized that functions are a type of relation, (which is made explicit with the notation "y = f(x)") and that the graph is the set of points that satisfy that relation. Of course if you realize that you can also generalize to g(x) = f(y), f(x,y) = 0 etc. Which he seems to have done.
It also seems he hadn't completely realized that a graph is merely a collection of points, not a nice line-like object. This last is more explicit in non-continuous functions, but you don't usually see those in high school.
Well, I find whole thing rather controversial, so to say. Depends on what do one mean by "collection" and by "point", eg. if we take point literally: (x, y) \in RxR, then we have to assume that "collection" is uncountable (which seems counterintuitive, esp. from the school student's pov). OTOH, one could treat "point" as representation of that point, eg. pencil mark, so that point is no longer single element, but rather an uncountable subset containing the point of interest, along with its neighbourhood => our "collection" becomes countable, so we could think about it as bunch of indexed points (much more intuitive, isn't it?), shifting our attention towards mapping's representation from the mapping itself.
tl;dr Treating graph as a collection of points is no more correct/better, compared to treating it as a continous line. Different problems require different approaches and different "angles" of view. Barely an insight from the author - the most of the mathematics is about abstractions and pattern matching.
The whole paragraph does not make sense to me. For example, how is a collection uncountable somehow unintuitive? Why should a point not be a single element, but a subset or a pencil mark? Are you confusing the theoretical graph with the graph actually drawn (which has finite width)?
- enumerable (countable) sets are more intuitive to reason about
- no matter, how do you comprehend the function, you'll end up drawing it by means of composing continuous chunks
> Are you confusing the theoretical graph with the graph actually drawn (which has finite width)? Probably. From my point of view, graph is the something that is drawn/plotted, whilst relation/mapping being something that you've called "theoretical graph". So I'd question, how treating graph as a bunch of points is superior concept, giving that one could easily slip into substituting actual mapping's points by their graphical representation, for the sake of simplicity, thus possibly hiding mapping's behaviour from own mind.
PS Nevertheless, I do now understand, that the whole thing author has meant to say, was: "Given x = f(y), it is not only some explicit line on a plane, but also a mapping f: (x, y) \in RxR (which also could be drawn, btw)".
The whole subthread is more of a dialectical excercise, with the definition of "graph" not being synchronized among participants :)
From there he generalizes the idea to drawing the set of points that makes a statement true, without necessarily having a formal function defined.