College Algebra, Chapter 4, 4.6, Section 4.6, Problem 38

Use transformations of the graph of $\displaystyle y = \frac{1}{x}$ to graph the rational function $\displaystyle r(x) = \frac{3x - 3}{x + 2}$.

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

$\displaystyle r(x) = \frac{3x - 3}{x + 2}$

By performing division








$
\begin{equation}
\begin{aligned}

=& 3 - \frac{9}{x + 2}
&&
\\
\\
=& 3 - 9 \left( \frac{1}{x + 2} \right)
&& \text{Factor out } 9
\\
\\
=& 3 - 9 f(x + 2)
&& \text{Since } f(x) = \frac{1}{x}

\end{aligned}
\end{equation}
$


It shows that the graph of $r$ is obtained by shifting the graph of $f$, 2 units to the left, reflecting about the $x$-axis and stretching vertically by a factor of $9$. Then the result is shifted $3$ units upward. Thus, $r$ has vertical asymptote at $x = -2$ and horizontal asymptote at $y = 3$.