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} )?

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

Explanation:

For rational parent functions $f(x)=\frac{1}{x}$, horizontal shifts follow $f(x-h)$ (right by $h$ units when $h>0$) and vertical shifts follow $f(x)+k$ (up by $k$ units when $k>0$). Here, $g(x)=\frac{1}{x-5}+2$ uses $h=5$ and $k=2$, so it shifts right 5 units and up 2 units from $f(x)$.

Answer:

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