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:
- The term [tex]\( x+5 \)[/tex] inside the denominator of the function [tex]\( \frac{1}{x+5} \)[/tex] causes a horizontal shift.
- Specifically, [tex]\( y = \frac{1}{x+5} \)[/tex] represents a shift to the left by 5 units. This is because the [tex]\( +5 \)[/tex] shifts the graph in the opposite direction.
2. Vertical Shift:
- The constant [tex]\( +2 \)[/tex] outside of the fraction causes a vertical shift.
- The function [tex]\( y = \frac{1}{x} \)[/tex] shifted by [tex]\( +2 \)[/tex] means that the whole graph is moved up by 2 units.
Putting this all together, the graph of [tex]\( y = \frac{1}{x+5} + 2 \)[/tex] is:
- Shifted to the left by 5 units.
- Shifted up by 2 units.
Thus, the correct description of the transformation is:
- It is shifted left 5 units and up 2 units from the parent function.
Therefore, the answer is:
- It is shifted left 5 units and up 2 units from the parent function.
