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

Aug 30, 2015

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

#### Explanation:

Note that $f \left(x\right) = \frac{- 5 x}{x + 3}$ is undefined for $x = - 3$
and
$\textcolor{w h i t e}{\text{XXXX}} f \left(x\right) \rightarrow + \infty$ as $\left(x + 3\right) \rightarrow - 0$
$\textcolor{w h i t e}{\text{XXXX}} f \left(x\right) \rightarrow - \infty$ as $\left(x + 3\right) \rightarrow + 0$

Therefore $x = - 3$ is an asymptote
with $f \left(x\right)$ positive if $x < - 3$
and $f \left(x\right)$ negative if $x > - 3$

Furthermore,
$\frac{- 5 x}{x + 3} = - 5 + \frac{15}{x + 3}$

as $\left\mid x \right\mid \rightarrow \infty , \frac{15}{x + 3} \rightarrow 0$

therefore $f \left(x\right) = - 5$ 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]}