Imagine driving toward a wall and halving your distance every second — you get closer and closer but mathematically you never reach the wall. That is exactly what a reciprocal graph describes: a curve that approaches a line forever but never crosses it. This mysterious line is called an asymptote, and it is the defining feature of y = 1/x — one of the most distinctive and beautiful curves in GCSE mathematics.
This introduction covers Closer and Closer, but Never Touching 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 1 of 11 in this topic. Use this introduction to connect the idea to the wider topic before moving on to questions and flashcards.
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