Exercise 25 Page 512

We want to draw the asymptotes and the graph of the given function. We will start by considering some possible transformations.

Transformations of $f(x)=\frac{1}{x},x\ne 0$
Vertical Translations	$$\begin{array}{c}\text{Translation up }k\text{ units, }k>0\\ y=\frac{1}{x}+k\end{array}$$
	$$\begin{array}{c}\text{Translation down }k\text{ units, }k>0\\ y=\frac{1}{x}-k\end{array}$$
Horizontal Translations	$$\begin{array}{c}\text{Translation right }h\text{ units, }h>0\\ y=\frac{1}{x-h}\end{array}$$
	$$\begin{array}{c}\text{Translation left }h\text{ units, }h>0\\ y=\frac{1}{x+h}\end{array}$$
Vertical Stretch or Shrink	$$\begin{array}{c}\text{Vertical stretch, }a>1\\ y=\frac{a}{x}\end{array}$$
	$$\begin{array}{c}\text{Vertical shrink, }0<a<1\\ y=\frac{a}{x}\end{array}$$
Reflection	$$\begin{array}{c}\text{In the }x\text{-axis}\\ y=\frac{\text{-}1}{x}\end{array}$$

Note that if the graph of the function is translated, the asymptotes are also translated in the same distance and direction. Consider the function. The given function is a combination of transformations.

• Horizontal translation $5$ units left
• Vertical stretch by a factor of $8$
• Reflection in the $x\text{-axis}$
• Vertical translation down $6$ units

Let's apply these transformations one at a time. We will start by translating the parent function, $f(x)=\frac{1}{x},$ $5$ units left.

Now, let's apply a stretch by a factor of $8.$

The third transformation is a reflection in the $x\text{-axis}.$

The last transformation is a vertical translation $6$ units down.

Finally, let's look at the graph of the given function and its asymptotes alone.

We can see that the vertical asymptote is the line $x=\text{-}5,$ and the equation of the horizontal asymptote is $y=\text{-}6.$ Using this information, we can state the domain and range of the function.