Equations of Asymptotes Made Easy

If you are asked to sketch an exponential graph, a log graph or a reciprocal graph in your A level maths exam the you will need to correctly identify at least one asymptote – reciprocal and rational functions have two asymptotes!

You may think this is difficult, but it’s actually very simple to find the equations of asymptotes on a graph if you remember two simple rules. Watch the video to learn! 

In your A level maths exams you need to be able to find vertical asymptotes – the bottom of a fraction or the … in log(…) is 0 – and horizontal asymptotes – found by subbing in a massive x.

Stating the equation of an asymptote on a graph is worth 1 mark in the A level Maths exam and it’s an easy mark so no excuses for dropping it!  

The only graphs which have asymptotes are reciprocal graphs y = 1/x, y = 1/x^2, exponential graphs y = e^x, y = a^x, log graphs y = ln(x), y = log(x) and transformations of these graphs. Polynomial graphs such as y = x^2 and y = x^3 do not have asymptotes.

Vertical asymptotes have equation x = … and they occur for values of x which are ‘not allowed’ ie) something that would make the bottom of a fraction zero or something which would make ln(0).

Horizontal asymptotes have equation y = … and they show the long term behaviour of the function. The functions gets closer to the value of the asymptote but never reaches it. To find the position of the horizontal asymptote, sub a massive x into the equation. That’s it!

Try to avoid the most common mistakes made by students when sketching asymptotes:

  • Don’t sketch the asymptote at all
  • Sketching the asymptote but not showing where it is (no equation)
  • A graph which isn’t asymptotic to the asymptote
  • Labelling a horizontal asymptote as x = instead of y=
  • Labelling a vertical asymptote as y = instead of x =
  • Labelling the asymptote with a number on the axis instead of an equation

KEY POINTS: A level Maths + Asymptotes + Graph Sketching + Sketching Reciprocal Functions + Sketching Log Graphs + Sketching Exponential Graphs

EXAM BOARDS: AQA, EDEXCEL, OCR, WJEC