Question

which of the following is the function represented by the graph? consider the graph below
\\( y = \frac{1}{(x + 3)} - 5 \\)
\\( y = \frac{1}{(x - 3)} + 5 \\)
\\( y = \frac{1}{(x + 5)} - 3 \\)
\\( y = \frac{1}{(x - 5)} + 3 \\)

Explanation:

Step1: Identify vertical asymptote

The vertical asymptote is $x=5$. For a rational function $y=\frac{1}{x-h}+k$, vertical asymptote is $x=h$, so $h=5$.

Step2: Identify horizontal asymptote

The horizontal asymptote is $y=3$. For the form $y=\frac{1}{x-h}+k$, horizontal asymptote is $y=k$, so $k=3$.

Step3: Substitute h,k into the form

Substitute $h=5$ and $k=3$ into $y=\frac{1}{x-h}+k$.
<Expression>
$y=\frac{1}{x-5}+3$
</Expression>

Answer:

$\boldsymbol{y=\frac{1}{x-5}+3}$ (the fourth option)