How Do You Graph a Rational Function of the Form (ax+b)/(cx+d)?

To graph $f(x)=\dfrac{ax+b}{cx+d}$, find and draw its vertical asymptote, horizontal asymptote, and intercepts, then determine where the two branches lie. This is IB Mathematics AA sub-topic SL 2.8: Reciprocal and simple rational functions.

Assuming $c\neq 0$, the denominator is zero when

$$cx+d=0 \Rightarrow x=-\frac{d}{c}.$$

Therefore, $x=-\dfrac{d}{c}$ is the vertical asymptote and is excluded from the domain. Since the numerator and denominator have the same degree, the horizontal asymptote is the ratio of their leading coefficients: $y=\dfrac{a}{c}$.

A useful algebraic rewrite is

$$f(x)=\frac{a}{c}+\frac{bc-ad}{c(cx+d)}.$$

This shows the graph as a translated reciprocal function. The asymptotes intersect at its centre, and the sign of the reciprocal coefficient determines the branch positions. If $bc-ad>0$, the branches lie above-right and below-left of the centre; if $bc-ad<0$, they lie above-left and below-right.

After plotting these features, choose one test value on each side of the vertical asymptote. Plot the resulting points and draw smooth branches with the correct limiting behaviour. Do not assume a horizontal asymptote can never be crossed; asymptotes describe limiting behaviour, not a universal barrier.

For example,

$$f(x)=\frac{2x+1}{x-3}=2+\frac{7}{x-3}.$$

Its asymptotes are $x=3$ and $y=2$, and its intercepts are $\left(-\dfrac{1}{2},0\right)$ and . The right branch lies above , while the left branch lies below it.

A common misconception is that asymptotes are the coordinate axes. They must instead be calculated and labelled. Also check for cancellation: if $ad=bc$, the function simplifies to a constant with a hole.

Exam technique: Show exact asymptote and intercept calculations on Paper 1. On Paper 2, technology may support the sketch, but label both asymptotes and all required intercepts.