College Algebra, Chapter 4, 4.6, Section 4.6, Problem 36

Use transformations of the graph of $\displaystyle y = \frac{1}{x}$ to graph the rational function $\displaystyle r(x) = \frac{-2}{x - 2}$.

If we let $\displaystyle f(x) = \frac{1}{x}$, then we can express $r$ in terms of $f$ as follows:


$
\begin{equation}
\begin{aligned}

r(x) =& \frac{-2}{x - 2}
&&
\\
\\
=& -2 \left( \frac{1}{x - 2} \right)
&& \text{Factor out } -2
\\
\\
=& -2 f(x - 2)
&& \text{Since } f(x) = \frac{1}{x}

\end{aligned}
\end{equation}
$



It shows that the graph of $r$ is obtained from the graph of $f$ by shifting the graph of $f$, 2 units to the right and reflecting about the $x$-axis. Then the result is stretched vertically by a factor of $2$. Thus, $r$ has vertical asymptote at $x = 2$ and horizontal asymptote at $y = 0$.