Chapter 5.2 — Exercise 5.2 — Cartesian System

Four Regions, Four Sign Patterns

Abscissa and Ordinate: Two Names for the Two Numbers

A point's x-coordinate has its own separate name, the abscissa; its y-coordinate likewise has its own name, the ordinate. For (4, −8), the abscissa is 4 and the ordinate is −8. For (0, 0), both are 0. These aren't different quantities from "x-coordinate" and "y-coordinate" — just older, more formal names for exactly the same two numbers, and they still turn up constantly in textbook and problem-statement wording even when "x-coordinate" would communicate the same thing just as clearly. The two names are also a useful memory aid for what each number actually measures: the abscissa is how far a point sits from the y-axis (a horizontal distance), while the ordinate is how far it sits from the x-axis (a vertical distance) — each coordinate is named for the axis it measures distance away from, not the axis it happens to sit closest to.

Points That Land Exactly on an Axis

A point doesn't have to sit inside one of the four quadrants at all — it can land exactly on the boundary between two of them. (0, 8) has abscissa 0, so it sits directly on the y-axis rather than off to one side of it. (3, 0) has ordinate 0, so it sits directly on the x-axis instead. Whenever the y-coordinate is 0, the point lies somewhere on the x-axis; whenever the x-coordinate is 0, it lies somewhere on the y-axis — and the one point where both coordinates are 0 at once, the origin, is the single point lying on both axes simultaneously. This is worth stating as a genuine two-way rule, not just a one-directional observation: a zero ordinate doesn't merely happen to coincide with the x-axis, it's the exact defining condition for sitting on it — every point on the x-axis has ordinate 0, and every point with ordinate 0 is on the x-axis, with no exceptions running in either direction.

• (−5, −8): neither coordinate is 0 — does not lie on either axis.
• (0, 13): x-coordinate is 0 — lies on the y-axis.
• (−2, 0): y-coordinate is 0 — lies on the x-axis.
• (0, 0): both coordinates are 0 — lies on both axes, the origin.

What Plotting a Whole Family of Points Reveals

Plotting (1, 0), (3, 0), (−2, 0), (−5, 0), (0, 0), (5, 0), and (−6, 0) together produces seven points scattered along a single straight line — the x-axis itself. The pattern is the ordinate: every single one of these seven points has y = 0. Plotting the mirrored set — (0, 1), (0, 3), (0, −2), (0, −5), (0, 0), (0, 5), (0, −6) — produces the same result rotated 90°, all seven landing on the y-axis, because this time every point has x = 0.

The general rule both examples point to: any point with ordinate 0 lies somewhere on the x-axis, and any point with abscissa 0 lies somewhere on the y-axis, regardless of how large or small the other coordinate is. It's worth noticing what this rule is really claiming: it isn't just that these particular seven or fourteen points happen to line up — it's that literally every possible point with ordinate 0, an infinite set with no gaps, sits on that one line, and a finite handful of examples can only ever illustrate a rule like that, never fully verify it by exhaustive checking.

True or False: Where the Definitions Get Tested

• "The horizontal line is the y-axis." False — the horizontal line is the x-axis; the vertical one is the y-axis.
• "The point lying on both axes is called the origin." True.
• "(2, −3) lies in the third quadrant." False — a positive x and negative y is the fourth quadrant; (2, −3) actually lies there.
• "(−5, −8) lies in the fourth quadrant." False — both coordinates negative is the third quadrant, where (−5, −8) actually lies.

The two false statements share the same underlying mistake: naming a quadrant without re-checking both signs against the table above. A single wrong sign on just one coordinate moves a point to an entirely different quadrant altogether, never merely to a nearby or adjacent one — there's no such thing as "close" between quadrants, only a shared axis boundary or a diagonally opposite region. Q1 and Q3 share only a single point, the origin, and are otherwise as far apart as two quadrants can be; Q1 and Q4 at least share an entire edge, the positive x-axis, which is why swapping just the y-coordinate's sign moves a point to its immediate neighbour rather than to the far side of the plane.

Reading Coordinates Straight Off a Graph

Once points are already plotted and labelled on a grid, the reverse skill matters just as much: reading a coordinate back off the picture. Given a labelled graph, a point's ordinate is read off by tracing a line straight across to the y-axis, and its abscissa by tracing a line straight down or up to the x-axis — the same two distances that defined the point in the first place, just measured off an existing drawing instead of computed from a written pair of numbers. On a graph where several points are already labelled with letters, this becomes a matter of locating the right letter first, then reading the two distances off carefully — the skill is entirely in careful reading at that stage, not in any further calculation, since the axes themselves have already done the measuring.

Turning Coordinates Into an Actual Drawing

Every rule established here — which quadrant a sign pattern points to, what abscissa and ordinate mean, when a point sits on an axis instead of inside a quadrant — becomes hands-on in Exercise 5.3, where actual points get plotted, joined into shapes, and used to compute real areas. Every one of those later constructions still leans on the same four facts fixed here first: which quadrant a sign combination points to, what "on an axis" specifically requires, and which of the two coordinates is the abscissa versus the ordinate whenever a problem's wording insists on the formal names rather than the more casual x and y.