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 the transformation rules

For a function \(y = f(x - h)+k\), if \(h>0\), the graph of \(y = f(x)\) is shifted \(h\) units to the right; if \(k>0\), the graph of \(y = f(x)\) is shifted \(k\) units up.
For the function \(g(x)=\frac{1}{x - 5}+2\) compared to \(f(x)=\frac{1}{x}\), here \(h = 5\) and \(k=2\).

Step2: Apply the transformation rules

Since \(h = 5>0\), the graph of \(y=\frac{1}{x}\) is shifted \(5\) units to the right. Since \(k = 2>0\), the graph of \(y=\frac{1}{x}\) is shifted \(2\) units up.

Answer:

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