Precalculus / Unit 2: Polynomial and Rational Functions
2.5 Rational Functions and Asymptotes
Notes
A rational function is a ratio of polynomials $\displaystyle \frac{p(x)}{q(x)}$.
- Vertical asymptotes: zeros of the denominator that remain after canceling common factors.
- A canceled factor leaves a hole instead.
- Horizontal asymptote: if degree of top < degree of bottom, $y = 0$. If the degrees are equal, $y = $ ratio of leading coefficients.
- If the top's degree is exactly one more than the bottom's, there is a slant asymptote found by division.
- Example: $\displaystyle \frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x - 1}$, so the slant asymptote is $y = x + 1$.
Practice
What are the asymptotes of $\displaystyle f(x) = \frac{2x + 1}{x - 3}$?
The denominator is zero at $x = 3$; equal degrees give $\displaystyle y = \frac{2}{1} = 2$.
Where is the hole in the graph of $\displaystyle g(x) = \frac{x^2 - 9}{x - 3}$?
It simplifies to $x + 3$ for $x \ne 3$, so the hole is at $x = 3$, $y = 6$.
What is the slant asymptote of $\displaystyle h(x) = \frac{x^2 + 1}{x - 1}$?
Division gives $x + 1$ with remainder 2.
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