Which describes the graph of y = 1/x?

Two separate curves that never touch the axes

A 120 km journey takes t hours at s km/h, so t = 120/s. What does the graph of t against s look like?

A curve that falls, getting closer to the axis (reciprocal)

Cubic, reciprocal and exponential graphs

Not every graph is a straight line or a U shape. Learn to recognise cubic, reciprocal and exponential graphs and use them in real problems.

Key Stage 3 · KS3 Mathematics: Algebra
About 16 minutes

Practise this lesson free (7 questions)

Recognising shapes

Linear y = mx + c: a straight line.
Quadratic y = x²: a U-shaped curve (upside down if the x² term is negative).
Cubic y = x³: an S-shaped curve.
Reciprocal y = 1x: two separate curves that get closer and closer to the axes but never touch them.
Exponential y = 2^x: passes through (0, 1) and grows faster and faster.

Graph shapes: quadratic (U), cubic (S), reciprocal (two curves) and exponential.

Tables of values

Substitute each x value carefully, using brackets for negatives.

x	−2	−1	0	1	2
y = x³ − 2x	−4	1	0	−1	4
y = 2^x	0.25	0.5	1	2	4

Graphs in real life

Exponential growth: bacteria that double every hour. 50 bacteria become 100, 200, 400, ...
Reciprocal (inverse proportion): for a fixed 120 km journey, time = 120 ÷ speed. Doubling the speed halves the time.
Read approximate values from a graph by going across to the curve and then to the other axis.

Worked example

Find y when x = −2 for y = x³ − 2x.

x³ = (−2)³ = −8
−2x = −2 × (−2) = +4
−8 + 4

Answer: −4

Key idea

Straight line: linear. U shape: quadratic. S shape: cubic. Two curves avoiding the axes: reciprocal. Through (0, 1) and growing faster and faster: exponential.

Key words

Reciprocal: 1 divided by a number. The reciprocal of 4 is 1/4, and the reciprocal of 2/3 is 3/2.
Quadratic: An expression or equation where the highest power of x is 2, such as x^{2} + 3x − 4. Its graph is a U shape (or upside-down U).
Term: One part of an expression or sequence. In 3x + 5, the terms are 3x and 5.
Substitute: Replace each letter in an expression or formula with a number, then work it out.
Inverse proportion: Two quantities where one goes up as the other goes down at the same rate: if one doubles, the other halves. y = k ÷ x.
Time: 60 seconds = 1 minute, 60 minutes = 1 hour. 0.5 hours = 30 minutes, 0.25 hours = 15 minutes.

All KS3 Mathematics key terms and definitions →

Common mistakes

Not “A U-shaped curve”. That is a quadratic.
Not “An S-shaped curve through the origin”. That is a cubic like y = x³.
Not “y = x³”. x³ is a cubic.
Not “y = 3x”. This is a reciprocal graph.

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

Which equation has a graph made of two separate curves that never touch the axes?
1. y = x² + 5
2. y = 5x
3. y = x³
4. y = 5x
Show the answer
Answer: y = 5x
y = 5x has no value at x = 0, so it splits into two curves.
Which of these equations gives an exponential graph?
1. y = 2x
2. y = 2^x
3. y = 2x
4. y = x²
Show the answer
Answer: y = 2^x
In y = 2^x the x is the power, so it grows faster and faster.

Check you have got it

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Practise this lesson free (7 questions)All KS3 Mathematics lessons

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Sources and references

National Curriculum in England, Key Stage 3: Mathematics programme of study, topic KS3 Mathematics: Algebra

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.