Rational Functions
Analyse graphs and algebra of rational functions.
A rational function is a quotient of two polynomials.
General form\[f(x)=\frac{P(x)}{Q(x)},\qquad Q(x)\ne0\]
Key graph features
- Zeros come from the numerator, provided the denominator is not also zero.
- Vertical asymptotes often come from zeros of the denominator.
- Long-term behaviour is found by comparing polynomial degrees or using division.
Example
For \(f(x)=\frac{2x+1}{x-3}\), the vertical asymptote is \(x=3\), because the denominator is zero there.