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

Jul 31, 2015

Graph as $y = \frac{2 x}{x - 4}$
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).

$y = \frac{2 x}{x - 4}$ is equivalent to $y = \frac{2}{1 - \frac{4}{x}}$ [provided we ignore the special case $x = 0$]
As $x \rightarrow \pm \infty$
$\textcolor{w h i t e}{\text{XXXX}}$$\frac{4}{x} \rightarrow 0$
and
$\textcolor{w h i t e}{\text{XXXX}}$$y = \frac{2}{1 - \frac{4}{x}} \rightarrow \frac{2}{1} = 2$
giving the horizontal asymptotic limit.

A few test values for $x$, such as
$\textcolor{w h i t e}{\text{XXXX}}$$x = 0 \rightarrow y = 0$
$\textcolor{w h i t e}{\text{XXXX}}$$x = 2 \rightarrow y = - 2$
$\textcolor{w h i t e}{\text{XXXX}}$$x = 5 \rightarrow y = 10$
$\textcolor{w h i t e}{\text{XXXX}}$$x = - 4 \rightarrow y = - 1$
help give shape to the graph

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