Line of Symmetry for Hyperbolas

Question

How might I find the equation for one of the lines of symmetry for the hyperbola $$y= 2 + \frac 6{x-4},\,\text{ where x cannot equal}\; 4.$$

I know that the lines of symmetry for the rational function $y=A/x$ are $y=x$ and $y=-x$...and that to find the lines of symmetry of $y=A/(x-h) + k$, where x cannot equal $h$...the equations become $y-k=x-h$ and $y-k = -(x-h)$...but I don't know how to apply that for the problem above...any ideas?

Answer 1

Graph your function, $$y = 2 + \dfrac 6{x - 4}$$ along with the lines $$y - 2 = x - 4 \iff y = x - 2\tag{1}$$ and $$ y - 2 = -(x - 4) \iff y =-x + 6\tag{2}$$

and see what you get.

$(1), (2)$ follow from the information you provide:

I know ... and that to find the lines of symmetry of $y=A/(x-h) + k$, where x cannot equal $h$...the equations become $y-k=x-h$ and $y-k = -(x-h)$

In your equation, $A = 6,\; k= 2,\; h = 4.$