Question

How do you graph `(2x)/(x-4)`?

Answer 1

Graph as `y = (2x)/(x-4)`
by establishing a few data points with random values of `x` and noting the asymptotic limits at `y=2` and `x=4`

Explanation:

The asymptotic limit `x=4` should be obvious from the expression (since division by 0 is undefined).

`y=(2x)/(x-4)` is equivalent to `y=2/(1-4/x)` [provided we ignore the special case `x=0`]
As `x rarr +-oo`
`color(white)("XXXX")` `4/x rarr 0`
and
`color(white)("XXXX")` `y=2/(1-4/x) rarr 2/1 = 2`
giving the horizontal asymptotic limit.

A few test values for `x`, such as
`color(white)("XXXX")` `x=0 rarr y=0`
`color(white)("XXXX")` `x=2 rarr y = -2`
`color(white)("XXXX")` `x=5 rarr y=10`
`color(white)("XXXX")` `x=-4 rarr y = -1`
help give shape to the graph

graph{(2x)/(x-4) [-25.3, 26, -11.27, 14.4]}