Lesson · Lines, Functions and Graphs · The coordinate plane

The Coordinate Plane

Two number lines at right angles turn a pair of numbers into a point, and an equation in x and y into a picture. The order inside the pair is not a formality — it is the whole content of the idea.

Open the interactive lesson Practice

(x, y) x across first, then y up (3, 5) ≠ (5, 3) the pair is ORDERED QI (+,+) QII (−,+) QIII (−,−) QIV (+,−) on an axis → in no quadrant at all

The plane in four lines

Core ideas

What this lesson establishes

Linear Equations and Inequalities solved equations in one unknown, and an answer there was a number on a line. An equation in two unknowns has pairs for answers, and pairs need two dimensions to be drawn.

A point is an ordered pair

(3, 5) and (5, 3) are different points. The first number always measures horizontal position and the second always vertical, and no context ever overrides that.

A graph is a solution set, drawn

The graph of 2x + 3y = 12 is the set of every pair that makes the equation true. A point is on the graph exactly when substituting its coordinates gives a true statement — that is the only test.

The axes belong to no quadrant

A quadrant is an open region. (0, −3) and (−5, 0) sit on an axis, so the answer to "which quadrant?" is none, not the nearest one.

The material

Naming a point

Two number lines, one convention about order, and four regions.

Definition · Coordinate plane

Two perpendicular number lines — the x-axis horizontal, the y-axis vertical — crossing at the origin (0, 0). Every point of the plane is named by exactly one pair of numbers, and every pair names exactly one point.

That last sentence is doing real work. The naming is a perfect matching in both directions, which is why a question about points can be answered with algebra and a question about algebra can be answered with a picture. The rest of this course lives on that exchange.

Definition · Coordinates

In the pair (x, y), the x-coordinate is the signed distance from the y-axis (right positive) and the y-coordinate is the signed distance from the x-axis (up positive).

The order is fixed by convention: across first, then up. (3, 5) is three right and five up; (5, 3) is five right and three up. These are two different points, and writing a pair backwards is the most common way a correct calculation ends up drawn in the wrong place.

Example · Six points

(3, 5) right 3, up 5. (−4, 2) left 4, up 2. (−3, −1) left 3, down 1. (2, −4) right 2, down 4. (0, −3) on the y-axis, 3 below the origin. (−5, 0) on the x-axis, 5 to the left.

Quadrants

The two axes cut the plane into four regions, numbered anticlockwise starting from the upper right. Each is fixed by the pair of signs, and nothing else needs to be remembered:

Quadrant I — x > 0, y > 0. Upper right.
Quadrant II — x < 0, y > 0. Upper left.
Quadrant III — x < 0, y < 0. Lower left.
Quadrant IV — x > 0, y < 0. Lower right.

Every inequality there is strict, so a point with a zero coordinate lies on an axis and in no quadrant. The origin lies on both axes. This is not pedantry: in “Domain and Range” the same strictness decides whether an endpoint belongs to a domain.

The graph of an equation

Definition · Graph

The graph of an equation in x and y is the set of all points (x, y) whose coordinates satisfy it. A point is on the graph if and only if substituting makes the equation true.

So membership is decided by arithmetic, never by how the picture looks. Take 2x + 3y = 12. Substituting (3, 2) gives 6 + 6 = 12, true, so (3, 2) is on the graph. Substituting (1, 4) gives 2 + 12 = 14, false, so it is not — and no amount of it looking close changes that.

Example · Testing four pairs against 2x + 3y = 12

(6, 0): 12 + 0 = 12 — on. (0, 4): 0 + 12 = 12 — on. (−3, 6): −6 + 18 = 12 — on. (1, 4): 2 + 12 = 14 — off.

One more practical point: the scale on the two axes need not match, and a graph drawn with a stretched axis is still correct. What a changed scale does distort is the appearance of steepness, which is exactly why “Slope” defines slope as a ratio of numbers rather than as an angle you could measure with a protractor.

Interactive

The coordinate plane

Points, quadrants, midpoints and exact distances

Type points, then read their coordinates

Predict each quadrant before choosing a preset or typing points. The panel plots the list, classifies every point from its signs, and computes exact midpoint and distance. Those last two rows are previews, not completion work here; the points are entered in a text box, not dragged.

Example

Points, as (x, y)

Window

Points read—

Quadrants used—

P1 to P2—

Write each point in brackets: (2, -3). Fractions are allowed, and every distance below is exact.

Method

Plotting a point

Four moves. The first one is where the errors are.

Read the pair in order

First number is horizontal, second is vertical. If the pair came out of a table or a substitution, confirm which column was which before moving.

Start at the origin

Both coordinates are measured from (0, 0), not from the last point you plotted.

Move along x, then parallel to y

Right for positive, left for negative; then up for positive, down for negative. Doing it in the other order lands in the same place, but doing it in a mixed order does not.

Check the signs against the quadrant

A point you expected in the lower left had better have two negative coordinates. This catches a swapped pair immediately.

Points and regions

Score 0 / 0

Choose the answer you can justify.

Six points, placed and classified

Plot each point, then name the region from the signs alone.

( 3, 5) right 3, up 5 QI (+, +) (-4, 2) left 4, up 2 QII (-, +) (-3, -1) left 3, down 1 QIII (-, -) ( 2, -4) right 2, down 4 QIV (+, -) ( 0, -3) on the y-axis no quadrant (-5, 0) on the x-axis no quadrant and the pair that is not the same point: ( 3, 5) ≠ ( 5, 3) both in QI, different points

The last line is worth the space it takes. Both points are in quadrant I and both are built from the digits 3 and 5, so a swap survives every check that looks only at signs. The only defence is reading the pair in order the first time.

For a faded rehearsal, plot (−2, 4), (3, −5) and (0, 6). The first move is supplied: read each pair as across, then up. Finish by naming the region of each point and testing all three in x − y = −6; show the substitution that supports every verdict.

Common mistakes

Reading the pair as (up, across)(2, 7) is 2 across and 7 up. The swap produces a point that is often still plausible-looking, which is what makes it dangerous.

Putting an axis point in a quadrant"Closest quadrant" is not a thing. If either coordinate is zero the point is on an axis, and the honest answer is that it is in none.

Deciding membership by eyeA point that looks like it is on the line may not be. Substitute. The picture is a summary of the arithmetic, never a replacement for it.

Check your understanding

✓

Finish when you can plot a point, name its quadrant, and decide membership of a graph by substituting rather than by looking.

Given six pairs and an equation you should be able to place all six, classify each by sign, and say which are on the graph with the arithmetic that proves it. That substitution test is the definition every later lesson leans on, from the intercepts in “Graphing a Linear Equation” to the inverse in “Inverse Functions”.

Note: The convention that x comes first is arbitrary but universal, and it is the same convention that makes f(x) in “Function Notation” an input and the value it returns an output. Once a pair is written (input, output) the whole course reads in one direction.