To determine how the graph of the function [tex]\( y = \frac{1}{x+5} + 2 \)[/tex] compares with the graph of the parent function [tex]\( y = \frac{1}{x} \)[/tex], we need to analyze the transformations applied to the parent function.
1. Horizontal Shift:
Consider the term [tex]\( \frac{1}{x+5} \)[/tex]. The addition of 5 inside the function's argument indicates a horizontal shift:
- The function [tex]\( y = \frac{1}{x+5} \)[/tex] shifts the graph of [tex]\( y = \frac{1}{x} \)[/tex] to the left by 5 units.
2. Vertical Shift:
Now, look at the addition of 2 outside the fraction:
- The term [tex]\( +2 \)[/tex] shifts the graph vertically upward by 2 units.
Putting these transformations together:
- The graph of [tex]\( y = \frac{1}{x+5} + 2 \)[/tex] is shifted left 5 units and up 2 units compared to the graph of the parent function [tex]\( y = \frac{1}{x} \)[/tex].
Therefore, the correct answer is:
- It is shifted left 5 units and up 2 units from the parent function.
