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:

Identify the horizontal shift

The input variable \(x\) in the parent function \(f(x) = \frac{1}{x}\) is replaced by \(x - 5\) in \(g(x) = \frac{1}{x-5} + 2\). A replacement of \(x\) with \(x - h\) represents a horizontal shift of \(h\) units. Since \(h = 5\), the graph is shifted \(5\) units to the right.

Identify the vertical shift

The constant \(+2\) is added to the entire rational expression. A transformation of the form \(y = f(x) + k\) represents a vertical shift of \(k\) units. Since \(k = 2\), the graph is shifted \(2\) units up.

Combine the transformations

Combining both transformations, the graph of \(g(x)\) is shifted \(5\) units right and \(2\) units up from the parent function \(f(x)\).

Answer:

• 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). (Correct answer)
• 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).