A rational function is one algebraic expression divided by another. The interesting part is where it breaks: the curve can never touch the x-values that make the denominator zero, and it settles towards a fixed line as x gets very large. Find those lines and the shape of the graph follows.
Syllabus points covered
- Find the asymptotes of a rational function with linear or quadratic denominator
- Sketch the graph of a rational function, showing asymptotes and intercepts
- Determine the set of values a rational function can take
- Sketch graphs of related curves, including y², |y| and 1/y
Common mistakes
- Finding the vertical asymptotes and forgetting the horizontal or oblique one.
- Assuming a curve never crosses its horizontal asymptote — it often does.
- Not testing the sign of the function either side of a vertical asymptote before sketching.
- Using the discriminant to find the range but forgetting to exclude values where the denominator is zero.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheet
- Worksheet answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers