Portion of a hyperbola from a function with a square root

Text

Consider the function $y = 1 - 2\sqrt{x^2 - 4x - 5}$.

• (a) find the domain (existence condition);
• (b) find the sign condition on $y$;
• (c) square and complete the square to identify the conic;
• (d) identify type, centre, semi-axes, vertices and asymptotes;
• (e) sketch only the portion matching the function.

Solution

(a) $(x-5)(x+1)\ge0 \Rightarrow x\le-1 \ \vee\ x\ge5$.

(b) $1-y=2\sqrt{\cdots}\ge0 \Rightarrow y\le1$.

(c) $\left(\dfrac{1-y}{2}\right)^2=(x-2)^2-9$, hence $\dfrac{(x-2)^2}{9}-\dfrac{(y-1)^2}{36}=1$.

(d) Hyperbola, horizontal, centre $C(2;1)$, $a=3$, $b=6$, vertices $(-1;1)$ and $(5;1)$, asymptotes $y-1=\pm2(x-2)$; $c=3\sqrt5$, foci $(2\pm3\sqrt5;1)$.

(e) The function gives only the lower half ($y\le1$) of both branches.

$\boxed{C(2;1),\ a=3,\ b=6,\ \text{vertices }(-1;1),(5;1)}$