Asymptote
In short: An asymptote is a straight guide line that a curve approaches ever more closely, either near a forbidden input value or far out to the left or right, without being reached at that spot.
Algebra 2Functions and GraphsF-IF.C.7d
What is Asymptote?
An asymptote is a boundary the curve leans toward. The graph races to it, bends alongside it and gets as near as you like, yet the picture never settles exactly on it in the vertical case.
A vertical one at \(x = a\) means the outputs grow without bound as \(x\) approaches \(a\). A horizontal one at \(y = b\) means the outputs settle toward \(b\) when \(x\) becomes enormously positive or enormously negative. A slanted version appears when a fraction’s top degree is exactly one larger than its bottom degree.
How to spot it
- A graph has a dashed line that the curve approaches but seems never to cross at that input.
- A formula divides by an expression that equals zero for some x-value, as in 1/(x – 8).
- A quantity levels off toward a ceiling or a floor over a long time, like a cooling cup of tea.
- The question asks what happens to f(x) when x gets very large or very close to a number.
The method
- Factor the fraction and cancel shared factors; a cancelled factor leaves only a hole in the graph.
- Set each remaining denominator factor equal to zero: every root marks a vertical guide line.
- Compare degrees of top and bottom: a smaller top gives \(y = 0\), equal degrees give the quotient of the lead coefficients.
- If the top degree is exactly one more than the bottom, divide to obtain the slanted line.
Asymptote: solved example
Problem
Find every asymptote of \(f(x) = \frac{4x^2 + 1}{x^2 – 9}\).
Solution
The bottom equals \((x – 3)(x + 3)\), and the top shares no factor with it. Vertical lines therefore sit at \(x = 3\) and \(x = -3\).
Top and bottom both have degree 2, so far to the left and right the outputs approach \(\frac{4}{1}\). The full list is \(x = 3\), \(x = -3\) and \(y = 4\).
Try it
Question 1
Name the guide lines of \(g(x) = \frac{5}{x – 8} + 2\).
Show the answer
The fraction blows up at \(x = 8\), and it fades to 0 for large \(|x|\), so the lines are \(x = 8\) and \(y = 2\).
Question 2
Does \(h(x) = \frac{x^2 – 1}{x – 1}\) have a vertical asymptote at \(x = 1\)?
Show the answer
No. It simplifies to \(x + 1\) for \(x \neq 1\), so there is only a hole at \((1, 2)\).
Asymptote vs. Rational Function
A guide line describes how a graph behaves; a rational function is the rule that creates the graph. Fractions of polynomials usually come with vertical and horizontal lines of this kind, but exponential and logarithmic curves have them too, so the line and the formula are different ideas.
Keep going
Lessons
Worksheets and quizzes
- Rational Functions and Asymptotes – Worksheet
- Rational Functions and Asymptotes – Quiz
- Infinite Limits and Asymptotes – Worksheet
- Infinite Limits and Asymptotes – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs