Working with Coordinates in All Four Quadrants Revision Notes | GCSE AQA Higher Maths

1. The Grid That Covers Everything

You have been plotting points on grids since primary school — but those grids only showed positive numbers. The full Cartesian grid extends in all four directions, and that opens up a much richer set of problems. Negative coordinates appear constantly in GCSE questions on shapes, straight lines, and transformations.

The key insight is this: the $x$-axis and $y$-axis divide the plane into four regions called quadrants. Every point on the grid sits in one of those quadrants — or exactly on an axis. Knowing which quadrant a point is in tells you the signs of its coordinates before you even look at the numbers.

2. Reading and Plotting Coordinates

To plot a point, start at the origin $(0, 0)$. Move left or right along the $x$-axis by the $x$-value — right for positive, left for negative. Then move up or down by the $y$-value — up for positive, down for negative. Mark the point where you land.

Points that sit exactly on the $x$-axis have a $y$-value of zero, for example $(5, 0)$. Points on the $y$-axis have an $x$-value of zero, for example $(0, -3)$. These points do not belong to any quadrant — they lie on the boundary between them.

3. Plotting and Reading Points — Worked Examples

Plotting is a skill that rewards careful, methodical reading. The most reliable approach is to deal with the $x$-value and $y$-value one at a time rather than trying to locate both simultaneously.

Worked example: Plotting a Point in a Negative Quadrant

Plot the point $P(-5, 3)$ on a Cartesian grid and state which quadrant it lies in.

1. $x = -5$ — Negative $x$ means we move 5 units to the LEFT of the origin along the $x$-axis. [M1 — correct horizontal movement]
2. $y = 3$ — Positive $y$ means we move 3 units UP from the $x$-axis position found in Step 1.
3. Move to $x = -5$ on the horizontal axis, then count 3 units upward. Mark $P$. — The point is now plotted at $(-5, 3)$. [A1 — correct position]
4. $x < 0$ and $y > 0$, so $P$ is in Quadrant II. — Check the sign pattern: negative $x$, positive $y$ always gives Quadrant II.

Answer: $P(-5, 3)$ is in Quadrant II.

Worked example: Reading Coordinates from a Grid

A point $Q$ is marked on a grid. Starting from the origin, it is 4 units to the right and 6 units below the $x$-axis. Write down the coordinates of $Q$ and name its quadrant.

1. 4 units to the right $\Rightarrow x = +4$ — Right of the origin is always positive $x$.
2. 6 units below the $x$-axis $\Rightarrow y = -6$ — Below the $x$-axis is always negative $y$. [M1 — correct sign applied]
3. $Q = (4, -6)$ — Horizontal value first, vertical value second. [A1 — correct coordinate]
4. $x > 0$ and $y < 0$ $\Rightarrow$ Quadrant IV — Positive $x$, negative $y$ is always Quadrant IV.

Answer: $Q = (4, -6)$, Quadrant IV.

4. Using Coordinates to Describe Shapes

Coordinates become powerful when you use them to describe and analyse geometric shapes. A common exam question gives you the vertices of a shape spread across multiple quadrants and asks you to find lengths, midpoints, or missing vertices.

Two results are particularly useful here. The midpoint of a line segment is found by averaging the $x$-values and averaging the $y$-values of the two endpoints. The length of a horizontal or vertical line segment is found by subtracting the smaller coordinate from the larger — but you must be careful when the segment crosses an axis.

Worked example: Finding the Midpoint of a Segment Crossing the $y$-axis

Find the midpoint $M$ of the line segment joining $A(-6, 2)$ and $B(4, 2)$.

1. $x_1 = -6,\ y_1 = 2,\ x_2 = 4,\ y_2 = 2$ — Write out both coordinate pairs clearly before substituting.
2. $\frac{-6 + 4}{2} = \frac{-2}{2} = -1$ — $-6 + 4 = -2$, then $-2 \div 2 = -1$. [M1 — correct midpoint method]
3. $\frac{2 + 2}{2} = \frac{4}{2} = 2$ — Both $y$-values are equal, so the midpoint $y$-value stays at 2.
4. $M = (-1, 2)$ — [A1 — correct midpoint coordinates] The midpoint lies in Quadrant II since $x < 0$ and $y > 0$.

Answer: $M = (-1, 2)$

Worked example: Finding the Length of a Vertical Segment Crossing the $x$-axis

Find the length of the vertical line segment joining $C(3, -5)$ and $D(3, 4)$.

1. Both points have $x = 3$, so the segment is vertical. — A vertical segment has the same $x$-value at both ends.
2. $y_2 - y_1 = 4 - (-5) = 4 + 5 = 9$ — Subtracting a negative is the same as adding. $4 - (-5) = 9$. [M1 — correct subtraction across zero]
3. $|9| = 9$ — Length is always positive. Here the result was already positive, so no sign change needed.
4. Length $= 9$ units — [A1 — correct length] The segment crosses the $x$-axis: 5 units below and 4 units above, totalling 9 units.

Answer: Length $= 9$ units

5. Coordinates on Straight Lines

Coordinates across all four quadrants also appear when you work with straight-line graphs. A line like $y = 2x - 3$ passes through multiple quadrants, and you need to be confident substituting negative $x$-values to find the corresponding $y$-values.

Worked example: Completing a Table of Values Across All Four Quadrants

Complete the table of values for $y = 2x - 3$ for $x = -2, -1, 0, 1, 2$.

1. $y = 2 \times (-2) - 3 = -4 - 3 = -7$ — Multiply first: $2 \times (-2) = -4$. Then subtract 3: $-4 - 3 = -7$.
2. $y = 2 \times (-1) - 3 = -2 - 3 = -5$ — $2 \times (-1) = -2$, then $-2 - 3 = -5$.
3. $y = 2 \times 0 - 3 = 0 - 3 = -3$ — The $y$-intercept of this line is $(0, -3)$.
4. $y = 2 \times 1 - 3 = 2 - 3 = -1$ — $2 - 3 = -1$. The point $(1, -1)$ is in Quadrant IV.
5. $y = 2 \times 2 - 3 = 4 - 3 = 1$ — $4 - 3 = 1$. The point $(2, 1)$ is in Quadrant I. [A1 — all five values correct]

Answer: $(x, y)$: $(-2, -7),\ (-1, -5),\ (0, -3),\ (1, -1),\ (2, 1)$

Notice how the line crosses three quadrants: it passes through Quadrant III (bottom-left), Quadrant IV (bottom-right), and Quadrant I (top-right). The crossing point on the $x$-axis — where $y = 0$ — can be found by setting $2x - 3 = 0$, giving $x = 1.5$. That point, $(1.5, 0)$, sits on the boundary between Quadrants I and IV.

6. Practice Problems

The following problems cover the full range of coordinate skills from this lesson. Work through each one before checking the solution — the process of attempting the question first is what builds exam confidence.

7. Exam Technique and Common Mistakes

Coordinate questions on GCSE Higher papers are reliable mark-earners — but only if you avoid the traps that appear repeatedly. The mistakes below account for a large proportion of dropped marks in this topic.

Mastering coordinates in all four quadrants means being fluent with negative numbers in a spatial context — and that fluency transfers directly to graph work, transformations, and vector questions throughout the rest of the GCSE course.