Plotting Curves from a Table of Values

Complete a table of values for any curve, plot the points, join them with one smooth curve and read answers off it.

In the exam for Edexcel Further Pure Maths 4PM1, Cambridge O Level Maths 4024, Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1 · 33 real past-paper questions below

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Table, Plot, Curve, Read

Most curve questions on the calculator paper follow the same four parts. The curve changes; the routine does not.

Steps

1
Complete the table: put each x into the equation.
2
Plot every pair from the table on the grid.
3
Join the points with one smooth curve.
4
Read answers off your curve.

Key idea

Together these parts are often worth 8 marks or more. Most of them need no new idea, only care.

Completing the Table

The 1 to 3 mark question: a table for a cubic like $y = x^{3} - 3 x + 1$ with a few gaps. Work out each term on its own, with negatives in brackets.

Worked example

1
$x = - 1.5$ — the first gap
2
$(- 1.5)^{3} = - 3.375$ — a cube keeps the minus
3
$- 3 \times (- 1.5) = + 4.5$ — minus times minus
4
$- 3.375 + 4.5 + 1 = 2.125$ — add the three terms
5
$x = 0.5 : 0.125 - 1.5 + 1$ — the next gap
6
$= - 0.375$ — keep every decimal

Common mistake

Mixing up the signs. A square of a negative is positive, but a cube of a negative stays negative: $(- 2)^{2} = 4$ but $(- 2)^{3} = - 8$ .

Key idea

Check with the table: values that are already printed must come out of your method too.

Fractions and Rounding in the Table

Some tables mix powers and fractions, like $y = x^{2} + \frac{2}{x}$ . The table skips $x = 0$ because you cannot divide by 0.

Worked example

1
$x = - 0.5$ — the gap
2
$(- 0.5)^{2} = 0.25$ — square: positive
3
$\frac{2}{- 0.5} = - 4$ — dividing by a half doubles
4
$0.25 - 4 = - 3.75$ — add the terms
5
$x = 3 : 9 + 0.666 \dots$ — the next gap
6
$\approx 9.7$ — round like the table, here 1 decimal place

Common mistake

Giving more or fewer decimals than the table uses. Read the line above the table: it says how to round.

Plotting the Points

The 3 to 4 mark question: “On the grid, draw the graph”. First read the scale. One small square is often not 1.

Worked example

1
$10 squares = 2 units$ — read two labels on the axis
2
$1 square = 0.2$ — one small square
3
$y = 2.125 = 2 + 0.125$ — whole units first
4
$0.125 \div 0.2 \approx 0.6$ — a bit over half a square

Rule

Go across to x first, then up or down to y. Mark each point with a small, neat cross.

Common mistake

Counting every small square as 1, or as 0.1, without reading the axis. Always divide the labelled gap by the number of squares.

One Smooth Curve

Join the points freehand, in one smooth curve that passes through every point.

Rule

No ruler. The curve bends everywhere, not only at the points you worked out.

Common mistake

Short straight lines from point to point, or a flat bottom between two equal values. Both lose the curve mark.

Common mistake

Joining across a gap. If the table skips $x = 0$ , draw two separate pieces.

Key idea

If one point will not sit on a smooth curve, check that column of the table again. It is almost always a slip.

Reading Answers Off the Curve

The 1 or 2 mark parts after the drawing: “use your graph to find …”. Draw faint lines to the curve and read the other axis.

Worked example

1
$x = 1.2$ — find y here
2
$up to the curve, across$ — to the y axis
3
$y \approx - 0.9$ — read to the grid
4
$x^{3} - 3 x + 1 = 0$ — now solve
5
$where y = 0$ — where it crosses the x axis
6
$x \approx - 1.9, 0.3, 1.5$ — a cubic can cross three times

Common mistake

Giving only one answer. Follow the whole line across the grid and count every crossing.

Key idea

For “by drawing a suitable straight line, solve …”, see Solving Equations by Graphical Methods.

Which Method? Read the Question

Each part of a curve question has its own move.

You see	You do
“Complete the table”	Each term on its own, negatives in brackets
“Correct to 1 decimal place”	Round your values to match
“On the grid, draw the graph”	Read the scale, plot all points, smooth curve
“Use your graph to find y”	Up from x to the curve, across
“Use your graph to solve”	Every crossing of the x axis
“By drawing a suitable line”	Solving Equations by Graphical Methods
Turning point or symmetry of a U	Quadratic Graphs and the Parabola

Key idea

For a fraction curve, its shape and its gaps are in Reciprocal Functions.

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