Question

How do you graph `y= x/(x-4)`?

Answer 1

Have a look:

Explanation:

Let us study our function in steps:

1) Domain (`=x` values allowed)
we need the denominator different from zero:
`x-4!=0`
`x!=4`
`x=4` will be a vertical asymptote.

2) Intercepts:
set: `x=0` then `y=0`
set: `y=0` then `x=0`

3) Limits:
`lim_(x->4^+)y=+oo`
`lim_(x->4^-)y=-oo`
`lim_(x->+oo)y=1`
`lim_(x->+oo)y=1`
`y=1` will be a horizontal asymptote.

4) Derivatives:
FIRST: `y'=(x-4-x)/(x-4)^2=-4/(x-4)^2` (Quotient Rule)
set: `y'=0` then `-4=0` never so we have NO max or min.
SECOND: `y''=(-2(-4)(x-4))/(x-4)^4=8/(x-4)^3`
set `y''>0`
you get:
`8>0` always
`(x-4)^3>0` when `x>4`
graphically:

graph{x/(x-4) [-10, 10, -5, 5]}