Question

consider the following function:
\\y = \frac{1}{(x + 5)} + 2\\

how does the graph of this function compare with the graph of the parent function, \\(y = \frac{1}{x}\\)?

• it is shifted right 5 units and up 2 units from the parent function.
• it is shifted left 5 units and up 2 units from the parent function.
• it is shifted right 5 units and down 2 units from the parent function.
• it is shifted left 2 units and down 5 units from the parent function.
• it is shifted right 2 units and up 5 units from the parent function.
• it is shifted left 2 units and up 5 units from the parent function.

Explanation:

Identify the transformation parameters

Comparing the given function \(y = \frac{1}{x+5} + 2\) to the parent function \(y = \frac{1}{x}\), we identify the horizontal and vertical shifts from the form \(y = \frac{1}{x-h} + k\):

$$ h = -5, \quad k = 2 $$

Determine horizontal and vertical shifts

Using the values of \(h\) and \(k\):

• Since \(h = -5\), the graph is shifted left by 5 units.
• Since \(k = 2\), the graph is shifted up by 2 units.

Match with the correct option

The transformation is a shift left 5 units and up 2 units, which corresponds to the second option.

Answer:

• It is shifted right 5 units and up 2 units from the parent function.
• It is shifted left 5 units and up 2 units from the parent function. (Correct answer)
• It is shifted right 5 units and down 2 units from the parent function.
• It is shifted left 2 units and down 5 units from the parent function.
• It is shifted right 2 units and up 5 units from the parent function.
• It is shifted left 2 units and up 5 units from the parent function.