We want to explain the relation between the given two functions.

y_{1} = \frac{1}{x - 1} + 2 and y_{2} = \frac{1}{x + 1} - 2

Notice that

y_{2}

is the transformation of

y_{1} .

To define this transformation, we will recall the possible translations.

Translations of $f (x) = \frac{1}{x}, x \neq = 0$
Vertical Translations	$Translation up k units, k > 0 y = \frac{1}{x} + k$
Vertical Translations	$Translation down k units, k > 0 y = \frac{1}{x} - k$
Horizontal Translations	$Translation right h units, h > 0 y = \frac{1}{x - h}$
Horizontal Translations	$Translation left h units, h > 0 y = \frac{1}{x + h}$

Using this table, let's indicate the translation from the graph of the parent function $f (x) = \frac{1}{x}$ to the graph of $y_{1}$ and $y_{2} .$

Function	Translation
$y_{1} = \frac{1}{x - 1} + 2$	Translation up $2$ units and right $1$ unit
$y_{2} = \frac{1}{x + 1} - 2$	Translation down $2$ units and left $1$ unit

Let's draw the graphs of these two functions on the coordinate plane.

Finally, let's consider a point on the graph of $y_{1} .$ We will see the translation applied to obtain its corresponding point on the graph of $y_{2} .$ For simplicity, let's use the point $(0, 1) .$

Therefore, the graph of $y_{2}$ is the graph of $y_{1}$ translated $2$ units to the left and $4$ units down.

Exercise