Coordinates in all four quadrants
Coordinates are foundational throughout OCR J560. Every Foundation paper has at least one coordinate question; Higher tests fluent navigation in all four quadrants.
The four quadrants
The Cartesian plane is divided by the x-axis and y-axis into four quadrants, numbered anticlockwise from the top-right:
| Quadrant | x | y | Example |
|---|---|---|---|
| 1 | + | + | (3, 5) |
| 2 | − | + | (−2, 4) |
| 3 | − | − | (−1, −6) |
| 4 | + | − | (5, −3) |
The origin O is at (0, 0). Points on the axes belong to neither quadrant.
Reading and writing coordinates
(x, y) — x first (horizontal, "across"), y second (vertical, "up"). The pair is ordered: (3, 5) ≠ (5, 3).
Distance between two points
Distance from (x₁, y₁) to (x₂, y₂):
distance = √((x₂ − x₁)² + (y₂ − y₁)²)
This is Pythagoras applied to the right triangle whose legs are the horizontal and vertical separations.
Example: from (1, 2) to (4, 6).
- Δx = 3, Δy = 4.
- Distance = √(9 + 16) = √25 = 5.
Midpoint
Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)/2, (y₁ + y₂)/2) — average each coordinate.
Example: midpoint of (−2, 4) and (6, −2) = (2, 1).
Reflecting a point
| Reflection | (x, y) becomes |
|---|---|
| in x-axis (y = 0) | (x, −y) |
| in y-axis (x = 0) | (−x, y) |
| in y = x | (y, x) |
| in y = −x | (−y, −x) |
Translating a point
A translation by vector (a, b) sends (x, y) to (x + a, y + b).
OCR mark scheme conventions
- B1 for plotting/identifying coordinates correctly.
- M1 for using the distance formula or midpoint formula.
- A1 cao for the answer.
- For "show that" questions, full working from formula to answer is needed.
⚠Common mistakes
- Reversing x and y in the coordinate pair.
- Forgetting the negative signs in quadrants 2, 3, 4.
- Subtracting in the wrong order in distance/midpoint (doesn't matter for midpoint, only for sign-checking distance).
- Missing the square root at the end of the distance formula.
AI-generated · claude-opus-4-7 · v3-ocr-maths-leaves