Question

How do you graph `(-5x)/(x+3)`?

Answer 1

Establish the asymptotes and a few data points; then sketch the graph based on this information.

Explanation:

Note that `f(x)=(-5x)/(x+3)` is undefined for `x=-3`
and
`color(white)("XXXX")f(x) rarr +oo` as `(x+3) rarr -0`
`color(white)("XXXX")f(x) rarr -oo` as `(x+3) rarr +0`

Therefore `x=-3` is an asymptote
with `f(x)` positive if `x< -3`
and `f(x)` negative if `x > -3`

Furthermore,
`(-5x)/(x+3) = -5 + 15/(x+3)`

as `abs(x) rarr oo, 15/(x+3) rarr 0`

therefore `f(x) = -5` is an asymptotic value.

Combining this information with a few data points:
#color(white)("XXX") {: (x," ",f(x)), (-2," ",10), (0," ",0), (2," ",-2), (-4," ",-20), (-6," ",-10), (-8," ",-8) :}#
and drawing the graph should not be difficult.
graph{(-5x)/(x+3) [-30.68, 34.26, -20.04, 12.45]}