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

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = \frac{1}{x - h} \), the graph shifts \( h \) units right if \( h>0 \), left if \( h<0 \). In \( y=\frac{1}{x + 5}=\frac{1}{x-(-5)} \), \( h=-5 \), so shift left 5 units.

Step2: Recall Vertical Shift Rule

For a function \( y=\frac{1}{x}+k \), the graph shifts \( k \) units up if \( k>0 \), down if \( k<0 \). Here \( k = 2>0 \), so shift up 2 units.

Answer:

It is shifted left 5 units and up 2 units from the parent function.