College Algebra, Chapter 4, 4.6, Section 4.6, Problem 40

Use transformations of the graph of $\displaystyle y = \frac{1}{x}$ to graph the rational function $\displaystyle r(x) = \frac{2x - 9}{x - 4}$.

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

$\displaystyle r(x) = \frac{2x - 9}{x - 4}$

By performing division








$
\begin{equation}
\begin{aligned}
r(x) =& \frac{2x - 9}{x - 4}\\
\\
=& 2 - \frac{1}{x - 4}
&&
\\
\\
=& 2 - f(x - 4)
&& \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 $f$ 4 units to the right and reflecting about the $x$-axis. Then the result is shifted $2$ units upward. Thus, $r$ has vertical asymptote at $x = 4$ and horizontal asymptote at $y = 2$.