Lesson 10: The Coordinate Plane and Graphing Points

Every treasure map, city grid and video game world runs on the same invention: the coordinate plane. Two number lines — one horizontal, one vertical — cross at a point called the origin, and suddenly every position on a flat surface can be named with just two numbers. In this lesson you will plot and read points, name the four quadrants, reflect points across the axes, and measure distances on the grid — the groundwork for every graph you will meet in algebra and beyond.

Ordered pairs and the axes

A point is named by an ordered pair (x, y). The first number, the x-coordinate, tells you how far to move horizontally from the origin: right if positive, left if negative. The second number, the y-coordinate, tells you the vertical move: up if positive, down if negative. The horizontal line is the x-axis; the vertical line is the y-axis. Memory hook: x comes before y in the alphabet, and you walk along the corridor (x) before you climb the stairs (y).

Worked example 1. Plot (3, 5): from the origin, move 3 right and 5 up. Plot (−2, 4): 2 left, 4 up. The point (3, −2) is 3 right and 2 down.

The four quadrants

The axes split the plane into four quadrants, numbered counter-clockwise from the top right:

Points on an axis belong to no quadrant. (0, −4) sits on the y-axis; (−6, 0) sits on the x-axis; and (0, 0) is the origin itself.

Reflections: mirror images on the grid

Reflecting a point in the x-axis flips the sign of its y-coordinate: (2, 3) → (2, −3). Reflecting in the y-axis flips the sign of its x-coordinate: (2, 3) → (−2, 3). The point stays the same distance from the mirror line — exactly how a mirror image behaves.

Distances on the grid

For two points sharing an x-coordinate, the distance is the difference of the y-coordinates. Between (1, 2) and (1, 6): |6 − 2| = 4 units. For two points sharing a y-coordinate, subtract the x-coordinates: between (−2, −1) and (3, −1): |3 − (−2)| = 5 units. (Diagonal distances need the Pythagorean theorem — lesson 14 has you covered.)

Worked example 2. On a town map, the library is at (4, 1) and the school is at (4, 5). They share x = 4, so they lie on the same vertical street, 5 − 1 = 4 blocks apart. The park at (−1, 5) shares the school's y-coordinate, so it is |4 − (−1)| = 5 blocks west of the school along the same horizontal street.

In the real world: grids everywhere

City planners lay out streets in grids (Manhattan's avenues and streets are a giant coordinate plane). Game designers place every character, coin and enemy at coordinates. Sat-nav systems use the same idea wrapped around the Earth: latitude and longitude are just (x, y) for the whole planet. Even your phone screen is a grid of pixels, each with its own coordinates — the top-left pixel of many screens is (0, 0).

Common mistakes

• Reversing the order. (2, 5) and (5, 2) are different points. The x-coordinate always comes first.
• Moving up for a negative y. (3, −2) goes 2 down, not up. The sign of y controls up versus down.
• Calling an axis point a quadrant. (0, −4) is on the y-axis, not in Quadrant III or IV. Quadrants contain only points strictly inside them.
• Counting grid squares instead of steps for distance. The distance between (1, 2) and (1, 6) is 4 units — subtract the coordinates; do not count the points themselves (there are 5 lattice points but 4 gaps).

Exam Experience

1. In which quadrant is the point (−3, 2)? A) Quadrant I B) Quadrant II C) Quadrant III D) Quadrant IV

2. Which statement about the point (0, −4) is true? A) It is in Quadrant III B) It is in Quadrant IV C) It lies on the y-axis D) It is the origin

3. What is the distance between the points (2, 1) and (2, 7)? A) 5 units B) 6 units C) 8 units D) 9 units

4. (Constructed response) A treasure map uses a coordinate grid. The start is at A(1, 1). The treasure path goes from A to B(5, 1), then from B to C(5, 4). (a) How far is it from A to B? (b) How far is it from B to C? (c) The path A → B → C is two sides of a rectangle. Give the coordinates of the fourth corner D that would complete the rectangle. (d) The treasure is buried at the point found by reflecting C in the y-axis. Give the coordinates of the treasure.

Answer key and explanations

• 1. B — Negative x and positive y is Quadrant II.
• 2. C — x = 0 means the point sits on the y-axis; axis points belong to no quadrant.
• 3. B — Same x-coordinate, so subtract the y-coordinates: 7 − 1 = 6.
• 4. (a) |5 − 1| = 4 units. (b) |4 − 1| = 3 units. (c) D(1, 4) — same x as A, same y as C. (d) Reflecting C(5, 4) in the y-axis flips the x-sign: (−5, 4).

Scoring guide for question 4 (4 points)

• 1 point for distance A to B = 4 units, from subtracting the x-coordinates.
• 1 point for distance B to C = 3 units, from subtracting the y-coordinates.
• 1 point for D(1, 4), correctly combining A's x with C's y.
• 1 point for the treasure at (−5, 4), with the x-sign flipped and the y-coordinate unchanged.

Going further: patterns hiding in the grid

Worked example 3 (midpoints). The midpoint of two points is found by averaging their coordinates. The midpoint of (2, 3) and (6, 7) is ((2 + 6)/2, (3 + 7)/2) = (4, 5) — exactly halfway along and halfway up. Surveyors, game designers (placing a camera between two characters) and map apps use midpoints constantly, and it is really just the coordinate plane shaking hands with lesson 15's mean.

Worked example 4 (double reflection). Reflect (−3, 2) in the x-axis: (−3, −2). Reflect the result in the y-axis: (3, −2). Doing both flips lands diagonally opposite the start — equivalent to a half-turn (180° rotation) about the origin. Two mirrors, one rotation: transformations compose, and coordinates make the composition visible in the signs.

Worked example 5 (points that form a pattern). Plot (1, 2), (2, 4), (3, 6). Each y is double its x — the points line up on the rule y = 2x, and the next point is (4, 8). Coordinates, patterns and equations are three views of the same object; lesson 17 and 18 pick up exactly this thread.

Worked example 6 (a journey). A delivery robot starts at (0, 0), drives to (4, 0), then to (4, 3). Leg one: 4 units. Leg two: 3 units. Total journey: 7 units. Warehouse robots plan thousands of such legs a day, always in coordinates, always summing horizontal and vertical moves.

Quick practice

• 1. In which quadrant is (5, −1)?
• 2. Reflect (6, −3) in the x-axis.
• 3. Find the distance between (−1, 4) and (5, 4).
• 4. A point is 3 units left and 4 units down from the origin. Give its coordinates.

Answers. 1. Quadrant IV (+, −). 2. (6, 3) — the y-sign flips. 3. 5 − (−1) = 6 units. 4. (−3, −4) — left is negative x, down is negative y.

Investigation: the delivery robot's day

A warehouse robot starts at its dock, the point (−3, −2) on the warehouse grid. It drives to a shelf at (4, −2) — same y, so a straight horizontal run of 4 − (−3) = 7 units — then turns and drives to the packing bench at (4, 3), a vertical run of 3 − (−2) = 5 units. Morning deliveries total 7 + 5 = 12 units of travel. Two questions for the robot's programmer. First: a second dock is the reflection of the first in the y-axis — where is it? Flip the x-sign: (3, −2). Second: a charging point should sit exactly halfway between the dock (−3, −2) and the shelf (4, −2). Average the coordinates: ((−3 + 4)/2, −2) = (0.5, −2). Half a unit is a perfectly respectable coordinate — grids are continuous, not just the lattice points. Everything in this lesson was used: axis-parallel distance, reflection, midpoints, and coordinates with decimals. When a video game reports your position as (0.5, −2), it is doing precisely this mathematics, sixty times a second.

Challenge corner

Three points: A(0, 0), B(6, 0), C(6, 4). They form a right triangle: AB runs 6 units along the x-axis, BC climbs 4 units, and the right angle sits at B. Now two reflections to finish the topic properly. Reflect A in the y-axis: A is on the axis (x = 0), so it does not move — (0, 0). Points on a mirror line are their own reflections, a small fact that exam questions love. Reflect D(−2, 5) first in the x-axis, then in the y-axis: (−2, −5), then (2, −5). Each reflection flips exactly one sign; two reflections flip both, landing diagonally opposite — the same half-turn you met earlier. If you can predict the signs before you compute them, the coordinate plane has become a place you see rather than a set of rules you recite.