Chapter 3.7, Problem 40E

To determine

To graph the rational function using transformation of the graph of $y=\frac{1}{x}$ .

Explanation of Solution

Given information:

The rational function is $r\left(x\right)=\frac{2x-9}{x-4}$

Graph:

Let, $f\left(x\right)=\frac{1}{x}$ , then express $r\left(x\right)=\frac{2x-9}{x-4}$ in term of $f$ ,

Use long division to reduce the original function into standard form,

  $x-4\begin{array}{c}\hfill 2\\ \hfill \overline{)\begin{array}{l}\underset{\_}{\begin{array}{l}2x-9\\ 2x-8\end{array}}\\ -1\end{array}}\end{array}$

Therefore, $r\left(x\right)$ can be expressed as:

  $\begin{array}{l}r\left(x\right)=2-\frac{1}{x-4}\\ r\left(x\right)=-\frac{1}{x-4}+2\left[\text{Rearrange terms}\right]\\ r\left(x\right)=-f\left(x-4\right)+2\left[\text{since, }f\left(x\right)=\frac{1}{x}\right]\end{array}$

From this form, the graph of $r\left(x\right)$ can be easily obtainedfrom $f$ by shifting 4 units to the right, reflecting in the $x$ -axis and shifting upward 2 units.

Interpretation:

The graph of $r\left(x\right)=\frac{2x-9}{x-4}$ is obtained by shifting the graph of $f\left(x\right)=\frac{1}{x}$ by 4 units to the right, reflecting in the $x$ -axis and shifting upward 2 units.