AHL 2.13—Rational functions

Rational Function: A function that can be expressed as the ratio of two polynomials, $f(x)=\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\ne 0$.

In AHL 2.13, you work with two specific , and more demanding , forms of rational functions:

Form 1 (Linear over Quadratic):
$f(x)=\frac{ax+b}{c{x}^2+dx+e}$

Form 2 (Quadratic over Linear):
$f(x)=\frac{a{x}^2+bx+c}{dx+e}$

These extend the simpler $\frac{ax+b}{cx+d}$ form you met in SL 2.8. The key step up here is that these functions introduce oblique (slant) asymptotes and require more sophisticated algebraic analysis , including polynomial long division.

Asymptote: A line that the graph of a function approaches arbitrarily closely but (in most cases) never reaches.

There are three types of asymptotes relevant to AHL 2.13:

1. Vertical Asymptotes
Occur where the denominator equals zero and the numerator does not also equal zero at that point. The function grows without bound near these values.

2. Horizontal Asymptotes
Determined by comparing the degrees of the numerator and denominator:

• Degree of numerator < degree of denominator → $y=0$
• Degree of numerator = degree of denominator → $y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$
• Degree of numerator > degree of denominator → no horizontal asymptote (may have oblique)

3. Oblique (Slant) Asymptotes
Occur when the degree of the numerator is exactly one more than the degree of the denominator. Found by performing polynomial long division , the quotient (ignoring the remainder) gives the oblique asymptote.