The function f is of the form $f(x) = \frac{{x + a}}{{bx + c}}$, $x \ne - \frac{c}{b}$. Given that the graph of f has asymptotes x = −4 and y = −2 , and that the point $\left( {\frac{2}{3},{\text{ }}1} \right)$ lies on the graph, find the values of a , b and c .

vertical asymptote $x = - 4 \Rightarrow - 4b + c = 0$     M1

horizontal asymptote $y = - 2 \Rightarrow \frac{1}{b} = - 2$     M1

$b = - \frac{1}{2}{\text{ and }}c = - 2$     A1A1

$1 = \frac{{\frac{2}{3} + a}}{{ - \frac{1}{2} \times \frac{2}{3} - 2}}$     M1

$a = - 3$     A1

[6 marks]