Is there a way to manipulate a hyperbola like $\frac{x^2 - 4}{x^2 + 4}$ into a readily sketch-able form?

Question

$\frac{2x}{2x-1}$ cannot be sketched but can be manipulated into $\frac{2x-1+1}{2x-1}= 1+\frac{1}{2x-1}$ which can be easily sketched.However, is there a way to to this when situations like $\frac{x^2 - 4}{x^2 + 4}$ occur? The only method I know is the lengthy process of y and x int, asymptotes via limits, domain and range, odd or even etc.

Answer 1

Why don't you do the same procedure: $$\frac{x^2-4}{x^2+4}=1-\frac8{x^2+4}$$ The last term is a Lorentzian function. So this will be like a horizontal line at $y=1$, but has a dip around $0$ to $y=-1$.