This deep dive covers Understanding Asymptotes 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 4 of 11 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Understanding Asymptotes
An asymptote is a line that the curve approaches but never reaches or crosses.
For y = k/x, there are two asymptotes:
- Vertical asymptote: x = 0 — when x = 0, we get k ÷ 0, which is undefined. The curve gets infinitely close to the y-axis but never touches it.
- Horizontal asymptote: y = 0 — as x becomes very large (positive or negative), y = k/x approaches zero but never reaches it. The curve gets infinitely close to the x-axis but never crosses it.
Important: a curve can NEVER cross its asymptote. If you draw the curve touching or crossing the axes, you will lose marks.
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.