GCSE Maths · Algebra

Cubic and reciprocal graphs GCSE Questions and Worked Answers

A cubic graph has x³ as its highest power; the simplest is y = x³. A reciprocal graph such as y = 4/x has two separate branches and is undefined at x = 0. Make a value table before sketching either shape.

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Start with the meaning

What you need to know about cubic and reciprocal graphs

A graph records pairs of numbers that follow a rule. If the rule says 'use the input three times in a product', input 2 gives 2 × 2 × 2 = 8. We write the rule as y = x³ and plot the point (2, 8), with input x across and output y upwards. Negative inputs work too: (−2)³ = −8.

See the idea first

Two rules, two very different shapes

For y = x³, make a table on both sides of zero and join the points with a smooth curve. For y = 4/x, divide 4 by each input instead. At x = 0 that would divide by zero, which is undefined: leave a gap and draw two separate branches. A line that a branch approaches without reaching in these examples is called an asymptote; both axes are asymptotes of y = 4/x.

y = x³ passes through the origin and has negative outputs for negative inputs. This particular cubic rises throughout.

The two branches of y = 4/x never meet either axis. Zero cannot be an input, and the output never equals zero.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Complete a table and plot

What the problem asks: Find y when x = −1, 0 and 2 for y = x³.

How to solve it: Cube each input: −1, 0 and 8. Plot (−1, −1), (0, 0), (2, 8), with further points for a reliable smooth sketch.

Recognise a reciprocal curve

What the problem asks: Find points on y = 6/x and explain the missing value at x = 0.

How to solve it: Inputs 1, 2, 3 give outputs 6, 3, 2. Negative inputs give negative outputs. Zero is not allowed; the two branches must not be joined through it.

Find intercepts

What the problem asks: Where does y = x³ − 8 cross the axes?

How to solve it: For the y-axis, put x = 0 to get y = −8. For the x-axis, put y = 0 and solve x³ = 8 to get x = 2.

A reliable routine

Build and check a curve from its equation

Use this when asked to plot, recognise or sketch a cubic or reciprocal equation. Each calculated pair must satisfy the equation; the curve must also respect where the rule is undefined.

Choose positive, negative and zero inputs where allowed.
Substitute carefully, keeping negative inputs in brackets.
Plot the pairs using labelled, evenly spaced scales.
Draw a smooth curve, keeping reciprocal branches separate; check intercepts and end behaviour.

Check: Not every cubic rises throughout: y = x³ − 3x has a rise, a fall, then a rise. Do not assume it has the exact shape of y = x³. A sketch shows essential features; an accurate plot uses the requested scales.

Fully worked

Cubic and reciprocal graphs GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Cube negative inputs

2 marks

Question

Complete the y-values for $y=x^3$ at x = −2, −1, 0, 1, 2.

(-2)^3=-8,\quad(-1)^3=-1

The values, in order, are −8, −1, 0, 1, 8. Plot all five pairs; use a smooth S-shaped curve, not straight segments.

Example 2

A cubic with a constant term

3 marks

Question

For $y=x^3+2$ , find the y-intercept and the points with x = −1 and x = 2.

At x = 0,

y=0^3+2=2

so the y-intercept is (0, 2). At x = −1, y = 1; at x = 2, y = 10. The additional points are (−1, 1) and (2, 10).

Example 3

Cubic x-intercept

2 marks

Question

Find the x-intercept of $y=x^3-27$ .

At an x-intercept, y = 0.

x^3-27=0

x^3=27

x=3

The intercept is (3, 0).

Example 4

Reciprocal table

3 marks

Question

Find the y-values for $y=8/x$ at x = −4, −2, 1, 2, 4. Describe the two branches.

Dividing 8 by each input gives −2, −4, 8, 4, 2. Negative inputs give negative outputs, and positive inputs give positive outputs. The branches lie in the lower-left and upper-right quadrants and approach the axes.

Example 5

A negative reciprocal

3 marks

Question

For $y=-6/x$ , find y at x = −3 and x = 2. State the quadrants containing the graph.

y=\frac{-6}{-3}=2

y=\frac{-6}{2}=-3

The points are (−3, 2) and (2, −3), so the branches are in the upper-left and lower-right quadrants. The axes are asymptotes.

Example 6

Read the reciprocal constant

3 marks

Question

A graph has equation $y=k/x$ and passes through (3, 4). Find k and the point with x = −2.

4=\frac{k}{3}

k=12

Then

y=\frac{12}{-2}=-6

The required point is (−2, −6). Multiplying x and y gives the same constant 12 everywhere on this curve.

10 original questions · total 20 marks

Cubic and reciprocal graphs GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 25 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

A cube

1 mark

Find y for $y=x^3$ at x = −3.

Show worked answer

y=(-3)^3=-27

Coefficient

2 marks

Find y for $y=2x^3$ at x = −2.

Show worked answer

y=2(-2)^3=2(-8)=-16

Cube before multiplying by 2.

Y-intercept

1 mark

Find the y-intercept of $y=x^3-5$ .

Show worked answer

x=0\implies y=-5

The point is (0, −5).

X-intercept

2 marks

Find the x-intercept of $y=x^3+8$ .

Show worked answer

x^3=-8\implies x=-2

The point is (−2, 0).

Three roots

3 marks

Find all x-intercepts of $y=x^3-4x$ .

Show worked answer

0=x(x^2-4)

0=x(x-2)(x+2)

Thus x = −2, 0, 2. The intercepts are (−2, 0), (0, 0), (2, 0).

Reciprocal values

2 marks

For $y=10/x$ , find y at x = −5 and x = 4.

Show worked answer

10/(-5)=-2

10/4=2.5

The points are (−5, −2) and (4, 2.5).

Undefined input

2 marks

Why is there no point with x = 0 on $y=7/x$ ?

Show worked answer

It would require 7 ÷ 0, which is undefined. No number multiplied by zero gives 7, so there is no finite output.

No x-intercept

2 marks

Explain why $y=5/x$ never crosses the x-axis.

Show worked answer

At a crossing, y would equal 0, so 5/x = 0. For an allowed non-zero x, multiplying by x would give 5 = 0, impossible. The curve approaches y = 0 but never reaches it.

Find k

2 marks

$y=k/x$ passes through (−4, 3). Find k.

Show worked answer

3=k/(-4)

k=-12

The product xy is −12.

Shifted reciprocal

Higher only3 marks

For $y=4/x+2$ , find its horizontal asymptote and y when x = 2.

Show worked answer

As the size of x grows, 4/x approaches 0, so the horizontal asymptote is y = 2.

y=4/2+2=4

The point is (2, 4). The vertical asymptote remains x = 0.

Examiner-style feedback

Common cubic and reciprocal graphs mistakes

Negative cubes

A negative input cubed is negative: (−2)³ = −8.

Joining through zero

A reciprocal graph has a forbidden input. Draw separate branches, never a vertical connecting segment.

Confusing a curve with its asymptote

For y = k/x with non-zero k, neither axis is part of the graph.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Tables establish real points.
Cube signs carefully.
Never divide by zero.
Check intercepts and separate branches.

Quick answers

Cubic and reciprocal graphs FAQ

Does every cubic cross the x-axis three times?

No. A cubic may have one, two or three distinct real x-intercepts. y = x³ has only one.

Are reciprocal graphs straight lines?

No. As one positive variable increases, the other decreases at a changing rate, producing a curve.

Build connected skills

What to revise next

Next skill

Graph transformations

Shift and reflect familiar curves.

Revise graph transformations

Next skill

Inverse proportion

Use the reciprocal relationship in context.

Revise inverse proportion

Content standards

Curriculum and rights review

A12 simple cubic and reciprocal graphs across tiers. Translated reciprocal graphs are identified as a Higher A13 extension. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content