Everyone knows that a function is a machine: enter x, exit y. What almost never gets shown is the moment when two numbers become a place — and it's that moment, repeated, that makes the graph exist. Here x lives on a line and y lives on another, each with only one dimension; the graph point is the meeting of the two, and the curve is the trace the meetings leave behind.
Do: drag input x and watch the two numbers walking each on their own line. Turn on leave trail and move again — the curve wasn't there, it is constructed. sweep all inputs makes the machine work alone.
1. They are two separate lines, and each is a one-dimensional world. On the bottom line there is only position: a number, and nothing more. On the left one, the same. None of the two, alone, has shape — there is nowhere to curve.
2. The graph point is not on either of the two lines. It is where the rise coming from x crosses the traverse coming from y. That's why it needs two numbers to exist: one from each line. It's sheet 3 of C1 — "the pair becomes a point".
3. The graph is a trail, not a drawing. Turn on leave trail and sweep the inputs. Each position of x produces an encounter, the encounter is marked, and the curve appears. No one drew the curve: it is the collection of all encounters. "Do this with all the inputs. The trail that the points leave is the graph."
4. The machine can refuse. In 1 / x, zero doesn't enter: there is no
output for it. That's the domain — the inputs the machine accepts — and in the
drawing it appears as a place where there was never a meeting.
The other two tools of the family speak of the relationship between the two axes and
assume that the plane already exists. The desenrolamento.html shows where the
first coordinate comes from; the taxa.html shows how the second changes when
the first moves. This one shows what these two numbers are, and why there are two —
and it is the only one that constructs the stage where the others play.
Local instrument, self-contained: no network, no external library. First of the family;
the sisters are desenrolamento.html and taxa.html.