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Closer and Closer, but Never Touching
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.
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.
Mathematics glossary
- 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
Earn the mark scheme marks
💡 Exam Tips for Reciprocal Graphs
- Two separate branches: always draw two distinct curves — never a single connected curve that passes through or near the origin
- Never touch the axes: the curves must approach but never reach the x-axis and y-axis
- Identify quadrants from k: k > 0 → branches in Q1 and Q3; k < 0 → branches in Q2 and Q4
- Find k using xy = k: any point on the curve gives the value of k
- Never include x = 0 in a table of values — state it is undefined
Now try it yourself
Quiz · Question 1 of 11
The graph of y = 1/x has an asymptote along the x-axis. What does this mean?
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This topic in real past papers
Every real exam question we've found on reciprocal graphs, with a full worked answer.