GraphsTopic Summary

Knowledge Organiser: Reciprocal Graphs

Part of Reciprocal Graphs · GCSE GCSE Mathematics revision

This topic summary covers Knowledge Organiser: Reciprocal Graphs within Reciprocal Graphs for GCSE Mathematics. Revise Reciprocal Graphs in Graphs for GCSE Mathematics with 11 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 11 of 11 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 11 of 11

Practice

11 questions

Recall

10 flashcards

Knowledge Organiser: Reciprocal Graphs

Key Facts

General form: y = k/x (k ≠ 0)
k > 0: branches in Q1 and Q3
k < 0: branches in Q2 and Q4
Product xy = k (constant at all points)
y = k/x shows inverse proportion: y ∝ 1/x

Asymptotes

Vertical asymptote: x = 0 (y-axis)
Horizontal asymptote: y = 0 (x-axis)
Curve NEVER touches or crosses either asymptote
x = 0 is undefined (division by zero)

Key Vocabulary

Hyperbola: the shape of y = k/x (two branches)
Asymptote: a line the curve approaches but never reaches
Inverse proportion: y ∝ 1/x, so xy = constant
Undefined: y = k/0 has no value

Common Errors

Drawing branches meeting at or near the origin
Allowing the curve to touch or cross the axes
Including x = 0 in a table of values
Confusing which quadrants the branches appear in

Key Formulas

y = k/x (or y = k × x⁻¹) — two branches, never touches axes
k > 0: branches in quadrants 1 and 3
k < 0: branches in quadrants 2 and 4
Asymptotes: x = 0 (y-axis) and y = 0 (x-axis)

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Keep building this topic

Read this section alongside the surrounding pages in Reciprocal Graphs. That gives you the full topic sequence instead of a single isolated revision point.

Common Misconceptions Closer and Closer, but Never Touching Reciprocal Graph Essentials

Practice Questions for Reciprocal Graphs

The graph of y = 1/x has an asymptote along the x-axis. What does this mean?

A. The graph touches the x-axis at x = 0
B. The graph crosses the x-axis at x = 1
C. The graph gets closer and closer to the x-axis but never reaches it
D. The graph is a straight line along the x-axis

1 markfoundation

Explain why the graph y = 5/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

2 markshigher

Quick Recall Flashcards

What does the graph of y = 1/x look like?

A hyperbola — two separate curved branches. - One branch in the top-right (positive x, positive y) - One branch in the bottom-left (negative x, negative y) The curve gets closer and closer to both axes but never touches them.

What is an asymptote and where are they on y = k/x?

An asymptote is a line the curve approaches but never reaches or crosses. For y = k/x: - Vertical asymptote: x = 0 (the y-axis) - The function is UNDEFINED when x = 0 - Horizontal asymptote: y = 0 (the x-axis) - y never equals zero for any finite x

11 questions on Reciprocal Graphs — practise free

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Common Misconceptions

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Reciprocal Graphs