Describing Transformations of Rational Functions

Let's start by recalling possible transformations of the parent function $f (x) = \frac{1}{x} .$

Function	Transformation of the Graph of $f (x) = \frac{1}{x}$
$g (x) = \frac{1}{x - h}$	Horizontal translation by $h$ units. If $h > 0,$ the translation is to the right. If $h < 0,$ the translation is to the left.
$g (x) = \frac{1}{x} + k$	Vertical translation by $k$ units. If $k > 0,$ the translation is up. If $k < 0,$ the translation is down.
$g (x) = \frac{1}{b ( x )}$	Horizontal stretch or shrunk by a factor of $b .$ If $0 < b < 1,$ it is a horizontal stretch. If $1 < b,$ it is a horizontal shrink.

Now, let's consider the given function. $g (x) = \frac{1}{2 ( x + 2 )} + 3 ⇕ g (x) = \frac{1}{2 ( x - ( - 2 ) )} + 3$ We can see that $b = \frac{1}{2},$ $h = - 2,$ and $k = 3 .$ From here, we can determine the transformations.

A horizontal shrink by a factor of $\frac{1}{2} .$
A horizontal translation $2$ units to the left.
A vertical translation $3$ units up.

Using these transformations, we can find the asymptotes and the reference points of the graph of $g (x) .$ Note that the horizontal compression and the horizontal translation affect only the $x -$ coordinates, while the vertical translation affects only the $y -$ coordinates.

Feature	$f (x) = \frac{1}{x}$	$g (x) = \frac{1}{2 ( x - ( - 2 ) )} + 3$
Vertical asymptote	$x = 0$	$x = 0 + (- 2)$ $⇕$ $x = - 2$
Horizontal asymptote	$y = 0$	$y = 0 + 3$ $⇕$ $y = 3$
Reference point	$(- 1, - 1)$	$(\frac{1}{2} (- 1) + (- 2), - 1 + 3)$ $⇕$ $(- 2.5, 2)$
Reference point	$(1, 1)$	$(\frac{1}{2} (1) + (- 2), 1 + 3)$ $⇕$ $(- 1.5, 4)$

Next, we will use the table above to graph $f (x)$ and $g (x) .$

Describing Transformations of Rational Functions 1.8 - Solution