Coordinates & Plotting Straight-Line Graphs

SPEC A8, A9, A10 • ALGEBRA — GRAPHS

The gateway to every graph skill — master (x, y) plotting, midpoints and the distance formula.

The Cartesian Plane

Every graph sits on a coordinate grid — two perpendicular number lines meeting at the origin.

Coordinates

A point is written (x, y). The first number is the x-coordinate (horizontal). The second is the y-coordinate (vertical). The origin is (0, 0).

The Four Quadrants

The axes split the plane into four quadrants. The sign pattern of (x, y) tells you which one you're in.

Order Matters

Reading a coordinate: x first, then y. The point A(3, 2) means "3 across, 2 up" — not "3 up, 2 across". Swapping them puts you in a different quadrant entirely.

Quick Check — A8 Skill

Given any point, you must be able to name its quadrant from the signs alone: (−, +) → Q2, (−, −) → Q3, and so on. Both coordinates positive? Q1. Both negative? Q3.

Plotting a Straight Line

To plot a line like y = mx + c, build a table of values, plot the points, and join them with a ruler.

The 4-Step Method

Pick at least 3 x-values — a good default is −2, 0, and 2 (one negative, zero, one positive).

Substitute each x into the equation to find the matching y.

Plot the 3 points on the grid.

Join them with a ruler and extend the line across the grid.

Why three points?

Two points define a line — but a third point is your check. If all three don't sit on the same straight line, you've made a calculation error. Fix it before you draw.

Worked Example: Plot y = 2x − 3

Choose x = −2, 0, 2 and substitute into y = 2x − 3.

x−202

y = 2x − 32(−2) − 3 = −72(0) − 3 = −32(2) − 3 = 1

The third point (−2, −7) lies below the visible grid but still sits on the extended line. The ruler-drawn line passes cleanly through both plotted points — that's your check that the calculation was correct.

Horizontal & Vertical Lines

Two special cases every GCSE student must recognise on sight:

Horizontal: y = b

Every point has the same y-coordinate. The line y = 3 is a horizontal line passing through (0, 3).

Think: "y is always 3, no matter what x is."

Vertical: x = a

Every point has the same x-coordinate. The line x = −2 is a vertical line passing through (−2, 0).

Think: "x is always −2, no matter what y is."

The Midpoint Formula

Given two points, the midpoint is the point exactly halfway between them.

Midpoint Formula

M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )Average the x-coordinates. Average the y-coordinates.

Mental shortcut

"Midpoint" → add and halve, both coordinates. That's it.

⚠️ Common error: Students use subtraction — (x₂ − x₁)/2 — instead of addition. Subtraction gives you half the horizontal gap, not the midpoint. Always add, then halve.

Worked Example — Midpoint01

Find the midpoint of A(1, 4) and B(5, 8). [1 mark]

M

Method: apply the midpoint formula — average the x-coordinates, then average the y-coordinates.

C

Calculate: x-mid = (1 + 5)/2 = 6/2 = 3. y-mid = (4 + 8)/2 = 12/2 = 6.

E

Evaluate: the midpoint is (3, 6). Check — (3, 6) does sit halfway between (1, 4) and (5, 8) on a sketch. ✓

Distance Between Two Points

To find the straight-line distance between two points, use Pythagoras' theorem — the horizontal gap and the vertical gap are the two legs of a right triangle, and the distance is the hypotenuse.

Distance Formula

d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]

The distance formula is just Pythagoras in disguise — same idea, different notation.

Worked Example — Distance02

Find the distance between A(−2, 3) and B(4, 11). [2 marks]

M

Method: use d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Take A as (x₁, y₁) and B as (x₂, y₂).

C

Calculate: x₂ − x₁ = 4 − (−2) = 6. y₂ − y₁ = 11 − 3 = 8. So d = √(6² + 8²) = √(36 + 64) = √100.

E

Evaluate: d = 10. Sense-check — (6, 8, 10) is a classic Pythagorean triple. ✓

Examiner Alert: The Axis Swap

⚠️ EXAMINER ALERT: Students plot (3, 4) where 4 should be — reading the coordinates as "up 3, across 4" instead of "across 3, up 4". This lands the point in the wrong place and costs every mark tied to that plot.

Weak Answer

"For (3, 4), I went up 3 and across 4."

✗ Point lands at (4, 3) — wrong quadrant, wrong answer.

Strong Answer

"For (3, 4), x = 3 so go 3 right. Then y = 4 so go 4 up."

✓ Correct position every time.

Memory Hook

"Along the corridor, then up the stairs." You always walk the corridor (x) before climbing the stairs (y).

Examiner Alert: Ruler & Check

⚠️ EXAMINER ALERT:Freehand straight lines lose marks. Examiners check whether your drawn line passes through the specified points — a wobbly line can miss the points by enough to cost the accuracy mark. Always use a ruler.

Three-Point Sanity Check

After plotting, look at your three points. If they do not lie on a single straight line, do not draw anything yet — you have a calculation error somewhere in your table. Recompute each y-value before you pick up the ruler.

Mark-scheme language

Typical wording: "Correct values in table B1 ; correct plot B1 ; line drawn with a ruler B1 ." Three independent marks — the ruler mark is as real as the plotting mark.

The MCE Framework for Graph Questions

Every coordinate-geometry question — plot a line, find a midpoint, compute a distance — follows the same three-step shape.

MMethod: Name what's asked (plot, midpoint, distance) and write the formula you'll use before touching numbers.

CCalculate: Substitute values carefully. Show each step — examiners award method marks even if you slip in the final line.

EEvaluate: Sense-check your answer. Is the midpoint between the two points? Is the distance positive? Does a sketch agree?

Why it works here

MCE maps directly onto the three marks GCSE examiners typically allocate: M gets the method mark, C gets the calculation mark, E gets the accuracy mark by catching slips before they cost you.

Practice Question 1

(a) Complete the table of values for y = 3x − 1.

x−1012

y????

(b) On a set of axes, draw the graph of y = 3x − 1 for −1 ≤ x ≤ 2. [3 marks]

M

Method: substitute each x into y = 3x − 1 to fill the table, then plot and join.

C

Calculate each y: x = −1 → 3(−1) − 1 = −4; x = 0 → −1; x = 1 → 2; x = 2 → 5.

E

Evaluate: points are (−1, −4), (0, −1), (1, 2), (2, 5). Gaps between y-values are all +3 — consistent with gradient 3. Plot and join with a ruler. B1 correct table values, B1 all points plotted, B1 straight line drawn with ruler.

Practice Question 2

P has coordinates (−3, 2) and Q has coordinates (5, 8).

(a) Find the midpoint M of PQ. [1 mark]

(b) Find the length of PQ. [2 marks]

M

Method: (a) average the x-coordinates and the y-coordinates. (b) use d = √[(x₂ − x₁)² + (y₂ − y₁)²].

C

Calculate (a): x-mid = (−3 + 5)/2 = 2/2 = 1. y-mid = (2 + 8)/2 = 10/2 = 5. So M = (1, 5).

Calculate (b): x₂ − x₁ = 5 − (−3) = 8. y₂ − y₁ = 8 − 2 = 6. d = √(64 + 36) = √100 = 10.

E

Evaluate: (a) Midpoint (1, 5) sits between P and Q ✓. (b) PQ = 10 units. Sense-check — this is a (6, 8, 10) Pythagorean triple. ✓

Summary

Key Points

(x, y) order: always x first (horizontal), then y (vertical). Origin = (0, 0).

Plot y = mx + c from a table of 3+ values; draw with a ruler.

Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ) — add and halve , both coordinates.

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²] — Pythagoras on the grid.

Horizontal lines: y = b. Vertical lines: x = a.

Definitely Do

Use a ruler for every straight line.

Compute 3 points as a built-in check.

Add (don't subtract) for midpoints.

Avoid

Swapping x and y when you plot.

Freehand lines — they lose the accuracy mark.

Drawing a line when your 3 points don't agree.

Exam Tip

GCSE graph questions always reward visible method. Write the formula out before substituting — M1 is yours before you've done any arithmetic.