Revise Rational functions and their graphs for IB Mathematics: Analysis and Approaches (2.5)

Everything on this topic grows from one graph: the reciprocal function $y=\frac{1}{x}$. Because you can never divide by zero, the function is undefined at $x=0$, so the y-axis $x=0$ is a vertical asymptote. As $x$ grows very large the fraction $\frac{1}{x}$ shrinks towards zero, so the x-axis $y=0$ is a horizontal asymptote. The graph has two branches — one in the first quadrant where both $x$ and $y$ are positive, and one in the third quadrant where both are negative — and it has rotational symmetry of order 2 about the origin (it is an odd function). It has no intercepts, because it never reaches either axis.

A linear rational function has a linear (degree-1) polynomial on the top and on the bottom: $f(x)=\frac{ax+b}{cx+d}$ with $c \neq 0$. Its graph is a hyperbola — the reciprocal shape stretched and shifted. To analyse any such function you locate four things in a fixed order: the vertical asymptote, the horizontal asymptote, the y-intercept and the x-intercept. Get those, and the sketch draws itself.

$\text{Vertical asymptote: } cx+d=0 \;\Rightarrow\; x=-\tfrac{d}{c}\\text{\textdollar}$ $\text{Horizontal asymptote: } y=\tfrac{a}{c} \text{ (ratio of leading coefficients)}\\text{\textdollar}$ $\text{y-intercept: } f(0)=\tfrac{b}{d}\\text{\textdollar}$ $\text{x-intercept: } ax+b=0 \;\Rightarrow\; x=-\tfrac{b}{a}$

Two of these features are exactly the excluded values of the domain and range. The vertical asymptote $x=-\frac{d}{c}$ is the one value the domain cannot include, and the horizontal asymptote $y=\frac{a}{c}$ is the one value the range cannot include. So once you have the asymptotes, the domain and range come for free.

A vertical asymptote is a vertical line the graph approaches but never crosses. It occurs where the function is undefined — for a rational function, where the denominator is zero. Think of it as a wall the curve cannot pass through. To find it, set the denominator equal to zero and solve for $x$.

A horizontal asymptote tells you what $y$-value the function settles towards as $x$ becomes very large or very small ($x \to \pm\infty$). For a linear rational function the numerator and denominator both have degree 1, so for large $x$ only the leading terms $ax$ and $cx$ matter, and their ratio $\frac{ax}{cx}=\frac{a}{c}$ is the horizontal asymptote $y=\frac{a}{c}$. The general rule for comparing degrees is worth knowing because it appears in later work.

Sketching a rational function is systematic. The asymptotes form a 'scaffold' — two dashed guide lines — and the intercepts pin the curve to the axes. Draw the asymptotes first, plot the intercepts, then draw the two branches of the hyperbola so that each branch hugs both asymptotes. If you are unsure which regions the branches sit in, test one convenient point in each region.

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A). Follow-through (FT) means a correct step performed on your own earlier (possibly wrong) value still earns its mark, and ISW ('ignore subsequent working') means once you have written a correct answer, a clumsy later step will not be penalised again. Equivalent forms are accepted — $-\frac{1}{3}$ and $-0.333\ldots$ score the same. But that protection only exists if the method is written down. Study how each mark below is earned by a specific line.

The reciprocal and linear rational functions are your first meeting with asymptotic behaviour, and the habits transfer directly. Recognising $\frac{ax+b}{cx+d}$ as a shifted, scaled $\frac{1}{x}$ is a transformations-of-graphs skill; excluding the asymptote values sharpens your command of domain and range; and setting a denominator to zero to spot where a function breaks reappears throughout calculus, in limits and in curve sketching. Master the fixed routine — denominator to zero for the vertical asymptote, leading-coefficient ratio for the horizontal, $f(0)$ and numerator-zero for the intercepts, then sketch — and every rational-function question becomes a variation on a method you already own.