Asymptote

Definition

An asymptote is a straight line that a curve approaches infinitely closely without ever reaching. Hyperbolas have a vertical and a horizontal asymptote; exponential graphs have a horizontal asymptote.

Worked example

For y = 2/(x − 3) + 1 the asymptotes are x = 3 and y = 1.

Where it appears in the exam

Paper 1 functions questions ask you to write down the asymptotes of a hyperbola or exponential graph, use them in a sketch, or find a graph's equation from them.

Step-by-step method

1. For y = a/(x − p) + q, the vertical asymptote is x = p.
2. The horizontal asymptote is y = q.
3. For y = a·b^(x − p) + q, the only asymptote is y = q.
4. Draw the asymptotes as dotted lines before sketching the curve.

Common mistake

Misreading the sign of p: y = 2/(x + 3) has its vertical asymptote at x = −3, not x = 3.

Related functions terms

• Domain and range
• Inverse function

Frequently asked questions

What does asymptote mean in Grade 12 Maths?

An asymptote is a straight line that a curve approaches infinitely closely without ever reaching. Hyperbolas have a vertical and a horizontal asymptote; exponential graphs have a horizontal asymptote.

Which NSC paper tests asymptote?

Asymptote belongs to the Functions section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving asymptote?

Snap a question that uses asymptote into the Snap&Learn AI solver, or work through the functions topic guide and NSC past papers.