Key takeaway: An asymptote is a boundary line that a curve approaches forever without settling on it, appearing as a vertical, horizontal or slanted line on the graph.
Grade 11 – Algebra 2F-IF.C.7dFunctions and Graphs
Definition of Asymptote
An asymptote is a guide line for a curve. Travel along the graph far enough, or close enough to a forbidden input, and the gap between you and the line shrinks toward zero.
A vertical one is a line \(x=a\) where outputs blow up, often because a denominator becomes zero. A horizontal one is a line \(y=b\) that the outputs near when \(x\) grows large in either direction. A slanted one appears when the top degree is exactly one more than the bottom degree.
Formula and notation
These lines are named by their equations: \(x=a\) for vertical, \(y=b\) for horizontal, \(y=mx+k\) for slanted. In limit language:
\[\lim_{x\to a} f(x)=\pm\infty \quad\text{or}\quad \lim_{x\to\infty} f(x)=b.\]
Graphs draw such lines dashed.
Key facts
- A curve may cross a horizontal guide line, but never a vertical one.
- A quotient of polynomials has at most one horizontal or slanted guide line, never both.
- Vertical ones sit at zeros of the denominator that do not cancel.
- The exponential function 2^x levels off along the x-axis.
Asymptote example
Find every asymptote of \(h(x)=\frac{5x+1}{2x-7}\).
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The bottom is zero when \(2x-7=0\), that is \(x=\frac{7}{2}\). The top is not zero there, so a vertical asymptote lives at that input.
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Top and bottom both have degree 1. Divide the leading coefficients: \(\frac{5}{2}\).
The vertical line is \(x=\frac{7}{2}\) and the horizontal line is \(y=\frac{5}{2}\).
Common mistake
Watch out
Noor says \(k(x)=\frac{x^2+1}{x-3}\) levels off at \(y=1\), because both leading coefficients are 1.
Why it is wrong
The top has degree 2 and the bottom degree 1, so the outputs grow without bound and no horizontal line exists. Division gives \(x+3+\frac{10}{x-3}\), so the curve follows the slanted line \(y=x+3\) in the long run.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs