Question

how does the graph of $g(x) = \frac{1}{x - 5} + 2$ compare to the graph of the parent function $f(x) = \frac{1}{x}$?\
\
\bigcirc\\ $g(x)$ is shifted 5 units left and 2 units up from $f(x)$.\
\bigcirc\\ $g(x)$ is shifted 5 units right and 2 units up from $f(x)$.\
\bigcirc\\ $g(x)$ is shifted 5 units left and 2 units down from $f(x)$.\
\bigcirc\\ $g(x)$ is shifted 5 units right and 2 units down from $f(x)$.

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( f(x) \), the transformation \( f(x - h) \) shifts the graph \( h \) units to the right if \( h>0 \), and \( h \) units to the left if \( h<0 \). In \( g(x)=\frac{1}{x - 5}+2 \), comparing to \( f(x)=\frac{1}{x} \), the argument of \( x \) is \( x - 5 \), so \( h = 5>0 \), which means a shift 5 units to the right.

Step2: Recall Vertical Shift Rule

For a function \( f(x) \), the transformation \( f(x)+k \) shifts the graph \( k \) units up if \( k>0 \), and \( k \) units down if \( k<0 \). In \( g(x)=\frac{1}{x - 5}+2 \), we have \( +2 \), so \( k = 2>0 \), which means a shift 2 units up.

Answer:

g(x) is shifted 5 units right and 2 units up from f(x).