Graphing Rational Functions

Recall that rational functions of the form $f (x) = \frac{a}{x - h} + k$ can be called simple rational functions. In this form, $x = h$ is the vertical asymptote and $y = k$ is the horizontal asymptote. With this in mind, let's begin by determining the asymptotes of the given function. $f (x) = \frac{- 1}{x + 1} + 1 \Leftrightarrow f (x) = \frac{- 1}{x - ( - 1 )} + 1$ The vertical asymptote is $x = - 1$ and the horizontal asymptote is $y = 1 .$ To graph the function, we will first graph these asymptotes on the coordinate plane.

Next, we will make a table of values. To do so, we will consider $x -$ values both to the left and to the right of the vertical asymptote.

$x$	$\frac{- 1}{x + 1} + 1$	$f (x)$
$- 3$	$\frac{- 1}{- 3 + 1} + 1$	$1.5$
$- 2$	$\frac{- 1}{- 2 + 1} + 1$	$2$
$- 1.5$	$\frac{- 1}{- 1 . 5 + 1} + 1$	$3$
$- 0.5$	$\frac{- 1}{- 0 . 5 + 1} + 1$	$- 1$
$0$	$\frac{- 1}{0 + 1} + 1$	$0$
$1$	$\frac{- 1}{1 + 1} + 1$	$0.5$

Finally, we can plot the points $(x, f (x))$ on the same coordinate plane and connect them with smooth curves.

Graphing Rational Functions 1.1 - Solution