Full study of a homographic function

Text

Given the homographic function $y=\dfrac{2-3x}{x+4}$:

• (a) find centre and asymptotes;
• (b) write it in implicit form;
• (c) find vertices and foci;
• (d) find the equation of the inverse.

Solution

(a) Vertical asymptote $x=-4$; horizontal asymptote $y=-3$. Centre $C(-4;-3)$.

(b) $y(x+4)=2-3x\Rightarrow (x+4)(y+3)=14$.

(c) It is a rectangular hyperbola with $k=14>0$: the vertices lie on the bisector $y+3=x+4$, with $(x+4)^2=14$, i.e. $V\left(-4\pm\sqrt{14};-3\pm\sqrt{14}\right)$. Semi-axes $a=b=\sqrt{2k}=2\sqrt7$, $c=2\sqrt{14}$; foci $F\left(-4\pm2\sqrt7;-3\pm2\sqrt7\right)$.

(d) Swapping $x\leftrightarrow y$ and solving: $y=\dfrac{2-4x}{x+3}$.

$\boxed{C(-4;-3);\ (x+4)(y+3)=14;\ y_{inv}=\tfrac{2-4x}{x+3}}$