Question

How do you find the asymptote and graph `y=1/(x-5)-4`?

Answer 1

`y` has a vertical asymptote of `x=5` and a horizontal asymptote of `y=-4`
Graph below.

Explanation:

`y= 1/(x-5)-4`

Notice `y` is undefined at `x=5`

Consider, `lim_(x->5^-) y = -oo and lim_(x->5^+) y = +oo`

Also, `lim_(x->-oo) y = -4 and lim_(x->+oo) y = -4`

Hence `y` is a rectangular hyperbola with a vertical asymptote of `x=5` and a horizontal asymptote of `y=-4`

The graph of `y` is shown below.

graph{1/(x-5)-4 [-5.42, 17.08, -7.105, 4.14]}