Question

How do you graph `y= (2x)/(x-1)`?

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=1`

Explanation:

The asymptotic limit `x=1` should be obvious from the expression (since division by 0 is undefined).

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

A few test values for `x`, such as
`color(white)("XXXX")` `x=-1 rarr y = -1`
`color(white)("XXXX")` `x=0 rarr y=0`
`color(white)("XXXX")` `x=2 rarr y = 4`
`color(white)("XXXX")` `x=3 rarr y= 3`

help give shape to the graph

graph{(2x)/(x-1) [-25.3, 26, -11.27, 14.4]}