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.
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)
Practice questions for Reciprocal Graphs
The graph of y = 1/x has an asymptote along the x-axis. What does this mean?
Explain why the graph y = 5/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
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