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 compression by a factor of $b .$ If $0 < b < 1,$ it is a horizontal stretch. If $1 < b,$ it is a horizontal compression.
$g (x) = \frac{1}{- x}$	Reflection across the $y -$ axis.

Now, let's consider the given function. $g (x) = \frac{1}{- 0 . 5 ( x - 3 )} + 1 ⇕ g (x) = \frac{1}{- 0 . 5 ( x - 3 )} + 1$ We can see that $b = - 0.5,$ $h = 3,$ and $k = 1 .$ Also, there is a $negative$ sign in the denominator. From here, we can determine the transformations.

A horizontal stretch by a factor of $2 .$
A reflection across the $y -$ axis.
A horizontal translation $3$ units to the right.
A vertical translation $1$ unit up.

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

Feature	$f (x) = \frac{1}{x}$	$g (x) = (\frac{1}{- 0 . 5 ( x - 3 )}) + 1$
Vertical asymptote	$x = 0$	$x = 0 + 3$ $⇕$ $x = 3$
Horizontal asymptote	$y = 0$	$y = 0 + 1$ $⇕$ $y = 1$
Reference point	$(- 1, - 1)$	$(- 2 (- 1) + 3, - 1 + 1)$ $⇕$ $(5, 0)$
Reference point	$(1, 1)$	$(- 2 (1) + 3, 1 + 1)$ $⇕$ $(1, 2)$

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

Describing Transformations of Rational Functions 1.3 - Solution