Question

how does the graph of $g(x) = \frac{1}{x + 4} - 6$ compare to the graph of the parent function $f(x) = \frac{1}{x}$?
$g(x)$ is shifted 4 units right and 6 units up from $f(x)$.
$g(x)$ is shifted 4 units left and 6 units up from $f(x)$.
$g(x)$ is shifted 4 units left and 6 units down from $f(x)$.
$g(x)$ is shifted 4 units right and 6 units down from $f(x)$.

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = f(x + h) \), if \( h>0 \), the graph shifts \( h \) units left; if \( h<0 \), shifts \( h \) units right. In \( g(x)=\frac{1}{x + 4}-6 \), comparing to \( f(x)=\frac{1}{x} \), here \( h = 4>0 \), so horizontal shift is 4 units left.

Step2: Recall Vertical Shift Rule

For a function \( y = f(x)+k \), if \( k>0 \), the graph shifts \( k \) units up; if \( k<0 \), shifts \( |k| \) units down. In \( g(x)=\frac{1}{x + 4}-6 \), \( k=-6<0 \), so vertical shift is 6 units down.

Answer:

\( g(x) \) is shifted 4 units left and 6 units down from \( f(x) \) (the option with this description, e.g., the third option in the original layout if we consider the order: first row third, or as per the visible text " \( g(x) \) is shifted 4 units left and 6 units down from \( f(x) \)").