# Transformation of Rational Functions

Rational functions are characterised by the presence of both a horizontal asymptote and a vertical asymptote.
Any graph of a rational function can be obtained from the reciprocal function $f(x)=\dfrac{1}{x}$ by a combination of transformations including a translation, stretches and compressions.

$y=\dfrac{a}{x}$: horizontal compression, $0 \lt a \lt 1$

$y=\dfrac{a}{x}$: horizontal stretch, $a \gt 1$

$y=\dfrac{1}{bx}$: vertical stretch, $0 \lt b \lt 1$

$y=\dfrac{1}{bx}$: vertical compression, $b \gt 1$

$y=\dfrac{1}{x-c}, c \gt 0$: horizontal translate by $c$ units to the right

$y=\dfrac{1}{x+c}, c \gt 0$: horizontal translate by $c$ units to the left

$y=\dfrac{1}{x}+d, d \gt 0$: vertical translate by $d$ units upwards

$y=\dfrac{1}{x}-d, d \gt 0$: vertical translate by $d$ units downwards