graph of $f(x)$ from graph of $(f(x))^2$ for a rational function

Question

$f(x)=\dfrac{ax+1}{bx+c}$ we have been given graph of $(f(x))^2$ , Pls see below

we are asked to make 2 sketches of graph of $f(x)$ since we have rational function , I know that our horizontal asymptote will be $y= \pm\sqrt q$ and vertical asymptote wil be $x=p$ and the graph will pass thru point $A$

Please help me complete the sketch of $f(x)$, also any resource that helps to learn about such sketches is appreciated.

thanks!

Answer 1

Since we pass through $A$, we can determine $a$. From knowledge of the horizontal asymptote, we can determine that $a/b=\pm \sqrt{q}$. From the vertical asymptote, we know $bp+c=0$, This allows you to determine all of the variables up to a sign.