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.
Closer and Closer, but Never Touching
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.
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