Skip to content
VirtusAcademy

Reciprocal Graphs

FoundationHigherAQAEdexcelOCR

Reciprocal Graphs is a key algebra topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on working with reciprocals, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Reciprocal graphs approach but never touch the axes (asymptotes).

Free downloads

These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A reciprocal graph comes from an equation of the form \(y = \frac{k}{x}\). It produces two separate curves, one in each of two opposite quadrants, and the shape is called a hyperbola.

The defining feature is what happens near zero. Since you cannot divide by zero, the curve never touches either axis. As \(x\) gets very small the value of \(y\) becomes enormous, and as \(x\) grows large \(y\) approaches zero without ever reaching it.

Those axes are called asymptotes: lines the curve approaches but never meets. Drawing the curve touching an axis is the classic error, and it misrepresents the fundamental behaviour of the function.

Revision notes

The shape

For positive \(k\), one branch sits in the top-right quadrant and the other in the bottom-left.

For negative \(k\), the branches occupy the top-left and bottom-right instead. The two branches are always separate, never joined.

Asymptotes

The curve approaches both axes but never touches them, because \(y\) is undefined at \(x = 0\) and can never equal zero.

This means the graph has a gap at \(x = 0\), and the table of values will have no entry there.

Plotting accurately

Choose \(x\) values close to zero and far from it to show both behaviours.

For \(y = \frac{6}{x}\): at \(x = 1\), \(y = 6\); at \(x = 6\), \(y = 1\); at \(x = 0.5\), \(y = 12\). Include negative values for the second branch.

Key points

  • A reciprocal graph has the form \(y = \frac{k}{x}\).
  • It has two separate branches.
  • The curve never touches either axis.
  • The axes are asymptotes.
  • There is no value at \(x = 0\).
  • Positive \(k\) gives branches in the top-right and bottom-left.

Worked examples

Example 1

Find \(y\) when \(x = 4\) for \(y = \dfrac{12}{x}\).

Working

\[\frac{12}{4}\]substitute x = 4
\[= 3\]divide to find y

Example 2

Find \(y\) when \(x = -3\) for \(y = \dfrac{6}{x}\).

Working

\[\frac{6}{-3}\]substitute the negative value
\[= -2\]a positive divided by a negative is negative

Example 3

Explain why \(y = \dfrac{5}{x}\) has no value at \(x = 0\).

Working

\[\frac{5}{0} \text{ is undefined}\]division by zero has no meaning
\[\text{The curve has an asymptote at } x = 0\]the graph approaches but never reaches the y-axis

Common mistakes

  • Drawing the curve touching an axis.

    The axes are asymptotes, so the curve gets close but never meets them.

  • Joining the two branches.

    They are entirely separate, with a gap at x = 0.

  • Including x = 0 in the table.

    The function is undefined there, so the entry is left blank.

  • Forgetting the negative branch.

    Negative x values produce the second branch, which is part of the graph.

Exam tips

  • Include x values both close to and far from zero.
  • Leave the x = 0 entry blank in your table.
  • Draw the two branches separately.
  • Make sure the curve clearly approaches without touching the axes.

Key terms

Reciprocal
One divided by a value.
Asymptote
A line the curve approaches but never touches.
Hyperbola
The two-branch curve of a reciprocal function.
Undefined
Having no valid value, as with division by zero.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.