GCSE · Maths · AQA · Spec 8300 · Foundation

Plotting straight-line graphs

Grab any point on this line and its coordinates fit the equation perfectly. Once you see why, plotting stops being copying and starts checking itself.

Every point on this line obeys y = 2x + 1

x: 0. y: 1

Drag the point anywhere along the line — go right down into the negatives. Then test it: double the x, add 1. You land on the y every single time, even at points like (1.3, 3.6).

Watch out: The line doesn't stop at the last value in your table. The points carry on, so the line carries on — draw it across the whole range the question asks for.

Your turn: build the table for y = 3 − 2x

Complete a table of values for y = 3 − 2x, for x from −2 to 3, ready to plot.

x-values: −2, −1, 0, 1, 2, 3. Substitute each one into y = 3 − 2x, with brackets round the negatives.Brackets stop a minus sign getting lost.
missing step

Which line is step 2?

x = −2: y = 3 − 2 × (−2) = 3 − 4 = −1x = −2: y = 3 − 2 × (−2) = 3 + 4 = 7x = −2: y = (3 − 2) × (−2) = −2

Spot the slip: why does this 'straight line' bend?

Draw the graph of y = 2x − 3 for values of x from −1 to 3.

A student's answer — which line goes wrong?

x = −1: y = 2 × (−1) − 3 = −5
x = 0: y = 0 − 3 = −3
x = 1: y = 2 − 3 = 1
x = 2: y = 4 − 3 = 1
x = 3: y = 6 − 3 = 3
Points: (−1, −5), (0, −3), (1, 1), (2, 1), (3, 3)
Joined the points dot to dot — the graph kinks at (1, 1).

What does the graph of x = 3 look like?

Here's an equation with only one letter in it: x = 3. There's no y anywhere.

Which is closest to what you think the graph looks like?

A horizontal line, because the 3 is on the x-axis.A vertical line through 3 on the x-axis — every point on it has x-coordinate 3.It's just the single point (3, 0).You can't draw it — without a y there's nothing to plot.

How sure are you?

I'm sureI think soI'm guessing

A horizontal line, because the 3 is on the x-axis.

This is a common misconception

Really tempting — but test it. A horizontal line keeps the same height all the way along, like (0, 3), (1, 3), (2, 3). Their x-coordinates are 0, 1 and 2, not 3, so they don't fit x = 3. That horizontal line is actually y = 3. For x = 3 it's the x-coordinate that's stuck at 3, and points that all share one x stack up vertically.

A vertical line through 3 on the x-axis — every point on it has x-coordinate 3.

This matches the evidence

Yes. The equation only makes one demand: x must be 3. It says nothing about y, so y can be anything. (3, −2), (3, 0), (3, 5) all fit — and those points stack up into a vertical line crossing the x-axis at 3.

It's just the single point (3, 0).

This is a common misconception

(3, 0) is on the graph — but so are (3, 1), (3, −4) and (3, 2.5). Because the equation says nothing about y, y is free to be anything. That's infinitely many points, all with x = 3: a whole line, not a dot.

You can't draw it — without a y there's nothing to plot.

This is a common misconception

No y in the equation doesn't mean nothing to plot. It means y can be any value at all. Pick any y you like, pair it with x = 3, and that point is on the graph.

x = a is a vertical line through a on the x-axis. y = b is a horizontal line through b on the y-axis. Not sure which way it goes? Write down one point that fits and see where it lands.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The table of values finds a few of those points. The ruler finds all the rest.

What you need to know

A point is on a line when its coordinates make the line's equation true.
Build a table of values: choose x-values (including negatives), substitute each one, and work out y.
Write each pair as (x, y) — x first, then y — and plot it accurately.
Rule one straight line through the points across the full range asked for; never join them dot to dot.
A point that won't line up means a substitution has gone wrong — recheck it.
x = a is a vertical line; y = b is a horizontal line.

The big picture

To plot a straight-line graph, you find some points whose coordinates make the equation true, plot them, and rule one straight line through them across the whole range you're asked for. A table of values does the finding: pick x-values, substitute each into the equation, and write each result as an (x, y) pair. Because every point of a straight-line equation lies on the same line, the plotted points should line up — and one that doesn't is a sign to recheck the arithmetic. Equations with just one letter, like x = 3 or y = −2, give vertical and horizontal lines.

Key points

1Every point on a straight-line graph fits its equation, and every point that fits the equation is on the graph.

2A table of values finds a few of those points; the ruled line joins them and carries on through all the rest.

3Put brackets round negative x-values and multiply before you add or subtract.

4Plot at least three points: two fix where the line goes, and the third checks your working.

5The graph of x = a is vertical, through a on the x-axis; the graph of y = b is horizontal, through b on the y-axis.

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Worked example

Problem

Draw the graph of y = 3x − 1 for values of x from −2 to 2.

⚠ Watch out

Losing the minus sign when x is negative. In y = 5 − 2x at x = −3, 2 × (−3) = −6, and 5 − (−6) = 11, not −1. Put brackets round every negative x before you substitute.

🧠

Memory hook

Every point fits, so every point lines up. One point out of line? Your maths has slipped — the line hasn't bent.

✓

Check yourself

Without drawing anything: is (4, 9) on the line y = 2x + 1? What about (−3, −7)? Substitute the x, see what y the equation gives, and compare.

Flashcards

(7)

What does it mean for a point to be on the line y = 2x + 1?

Its coordinates make the equation true: the y-value equals 2 × the x-value + 1.

Why is a table of values enough to draw a straight-line graph?

All the points of a straight-line equation lie on one line, so a few correct points fix where it goes. The ruled line fills in every point in between and beyond.

Why plot at least three points for a straight line?

Two points fix the line; the third is your check. If it doesn't line up, one of your values is wrong.

In a coordinate pair, which value comes first?

x first (across), then y (up or down): (x, y).

What is y = 3 − 2x when x = −2?

3 − 2 × (−2) = 3 + 4 = 7. Minus a negative is plus.

What do the graphs of x = 4 and y = −2 look like?

x = 4 is a vertical line through 4 on the x-axis. y = −2 is a horizontal line through −2 on the y-axis.

How far should you rule a straight-line graph?

Across the whole range the question asks for. The points carry on past your table, so the line doesn't stop at the last one you plotted.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Every point on this line obeys y = 2x + 1

Your turn: build the table for y = 3 − 2x

Spot the slip: why does this 'straight line' bend?

What does the graph of x = 3 look like?

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