Drawing Straight-Line Graphs

Draw a straight line from its equation with a table of values, including lines like y = 5 − 2x, x = 4 and y = −2.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 67 real past-paper questions below

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What a Straight-Line Graph Is

An equation like $y = 3 x - 2$ is a rule. Every point whose x and y fit the rule sits on one straight line.

To draw it, work out some points, plot them, and join them with a ruler.

Rule

Work out at least three points. Two fix the line; the third checks you have not made a slip.

Draw the Line From a Table

The 3-mark question: “Draw the graph of $y = 3 x - 2$ for values of x from −2 to 3.” Make a table first, one x at a time.

Worked example

1
$x : - 2, - 1, 0, 1, 2, 3$ — Write every whole number from −2 to 3 along the top of a table. These are the x values the question asks for.
2
$x = - 2 : 3 \times (- 2) - 2 = - 6 - 2 = - 8$ — Put x = −2 into the rule. Keep it in brackets: 3 × (−2) = −6, then take away 2.
3
$x = - 1 : 3 \times (- 1) - 2 = - 3 - 2 = - 5$ — Same two moves for the next x: times by 3, then take away 2.
4
$x = 0 : 3 \times 0 - 2 = - 2$ — 3 × 0 = 0, so y = −2. This is where the line crosses the y-axis, because x = 0 there.
5
$x = 1 : 3 - 2 = 1$ — 3 × 1 = 3, then take away 2.
6
$x = 2 : 6 - 2 = 4$ — 3 × 2 = 6, then take away 2.
7
$x = 3 : 9 - 2 = 7$ — 3 × 3 = 9, then take away 2. That is the last x in the range.
8
$- 8, - 5, - 2, 1, 4, 7$ — Check the y row: it goes up 3 each time, the number in front of x. A value that breaks the pattern is a slip — work it out again.
9
$(- 2, - 8) \dots (3, 7)$ — Plot each pair as a point. Join them with one ruled line from x = −2 right to x = 3.

Common mistake

Stopping the line early. Draw it right from x = −2 to x = 3 with a ruler, through every point.

Key idea

Show the table and plot every point. Even if your line goes wrong, two or three correctly plotted points still earn marks.

When the x Term Is Negative

Many questions use a rule like $y = 6 - 3 x$ , with a minus in front of the x term. The line goes down from left to right. Take care with the signs.

Worked example

1
$x = - 2 : 6 - 3 \times (- 2)$ — Put x = −2 in, inside brackets, so its minus is not lost.
2
$3 \times (- 2) = - 6$ — Do the times first: 3 × (−2) = −6.
3
$6 - (- 6) = 6 + 6 = 12$ — Taking away a negative is the same as adding. So the first y is 12.
4
$x = - 1 : 6 - 3 \times (- 1) = 6 + 3 = 9$ — Same moves: 3 × (−1) = −3, and 6 − (−3) = 9.
5
$x = 0 : y = 6$ — 3 × 0 = 0, so y = 6. The line crosses the y-axis at 6.
6
$x = 1 : 6 - 3 = 3$ — 3 × 1 = 3, and 6 − 3 = 3.
7
$x = 2 : 6 - 6 = 0$ — y = 0, so the line crosses the x-axis at (2, 0).
8
$x = 3 : 6 - 9 = - 3$ — 6 − 9 goes below zero: y = −3.
9
$12, 9, 6, 3, 0, - 3$ — Check: y goes down 3 each time. The minus in −3x is what makes the line slope down.

Common mistake

Writing 6 − 3(−2) = 0. Three times −2 is −6, and 6 − (−6) = 12.

Flat and Upright Lines

The 1-mark question: “Draw the line $y = - 3$ ” or “Draw the line $x = 5$ .” Only one letter appears, so you need no table.

Worked example

1
$y = - 3$ — Every point has y = −3, whatever x is: (0, −3), (2, −3), (4, −3). So it is a flat line through −3 on the y-axis.
2
$x = 5$ — Every point has x = 5, whatever y is: (5, 0), (5, 2), (5, −4). They stack up, so it is an upright line through 5 on the x-axis.
3
$y = x$ — Here y always equals x: (0, 0), (1, 1), (3, 3). So it is a sloping line through the origin, going up.
4
$y = - x$ — Now y is x with its sign changed: (1, −1), (−2, 2). It also goes through the origin, but slopes down.

Memory trick

y = a number: flat. x = a number: upright.

Common mistake

Drawing x = 5 flat. Points like (5, 0) and (5, 4) all have x = 5, and they stack straight up.

Key idea

When you draw more than one line on a grid, label each with its equation, so the examiner knows which is which.

Lines Like 2x + 3y = 12

Some questions give a line as $2 x + 3 y = 12$ , with x and y on the same side. There is no “y =” to read, so use the two easy points: x = 0, then y = 0.

Worked example

1
$2 x + 3 y = 12$ — The question. x and y are on the same side, so a table needs a little more work.
2
$x = 0 : 2 \times 0 + 3 y = 12$ — Put x = 0. The x term becomes 0, so only y is left.
3
$3 y = 12, so y = 4$ — Divide both sides by 3. So (0, 4) is on the line: where it crosses the y-axis.
4
$y = 0 : 2 x + 3 \times 0 = 12$ — Now put y = 0. The y term becomes 0, so only x is left.
5
$2 x = 12, so x = 6$ — Divide both sides by 2. So (6, 0) is on the line: where it crosses the x-axis.
6
$x = 3 : 2 \times 3 + 3 y = 12$ — Find a third point as a check. Put x = 3, so 2 × 3 = 6.
7
$3 y = 12 - 6 = 6$ — Take 6 from both sides, so the y term is alone.
8
$y = 2$ — Divide by 3. Plot (0, 4), (6, 0) and (3, 2): if they lie in one ruled line, your points are right.

Common mistake

Stopping the line at the axes. If the question says x from −3 to 6, carry on: at x = −3, y = 6. Draw the ruled line across the whole range.

Key idea

If a y value comes out as a decimal, plot it exactly. For $2 y = x + 3$ , x = 2 gives y = 2.5: put the point halfway between the gridlines, not on 2 or 3.

Key idea

“Sketch the line” (Extended) means no table and no grid. Draw a straight line sloping the right way, and write the numbers where it crosses each axis — here 4 on the y-axis and 6 on the x-axis.

Which Method? Read the Question

Look at which letters the equation has.

You see	You do
“Draw y = … for x from … to …”	table, plot, ruler, full range
“Complete the table”	put each x into the rule
Only y = a number, or x = a number	flat line, or upright line
y = x or y = −x	sloping line through (0, 0)
x and y on one side, like 2x + 3y = 12	put x = 0, then y = 0, then one check point
“Does this point lie on the line?”	put its x in, compare the y
Gradient or y-intercept asked	The Equation y = mx + c

Key idea

Does (100, 297) lie on $y = 3 x - 2$ ? $3 (100) - 2 = 298$ , not 297, so no. Always write the reason.

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