To determine which function \( h(x) \) is a transformation of the parent function \( f(x) = \sqrt{x} \), we need to understand the effects of various transformations on the square root function. The transformations can generally shift the graph either vertically or horizontally. Let's analyze each option given.
1. Vertical Shifts: These involve adding or subtracting a constant from the function.
- Downward shift: Given by \( h(x) = \sqrt{x} - k \)
- Upward shift: Given by \( h(x) = \sqrt{x} + k \)
2. Horizontal Shifts: These involve adding or subtracting a constant within the square root.
- Right shift: Given by \( h(x) = \sqrt{x - k} \)
- Left shift: Given by \( h(x) = \sqrt{x + k} \)
Now, we will analyze each option:
A. \( h(x)=\sqrt{x}-2 \)
This represents a vertical shift of the graph of \( \sqrt{x} \) downward by 2 units.
B. \( h(x)=\sqrt{x-2} \)
This represents a horizontal shift of the graph of \( \sqrt{x} \) to the right by 2 units.
C. \( h(x)=\sqrt{x+2} \)
This represents a horizontal shift of the graph of \( \sqrt{x} \) to the left by 2 units.
D. \( h(x)=\sqrt{x}+2 \)
This represents a vertical shift of the graph of \( \sqrt{x} \) upward by 2 units.
Based on this analysis, the function \( h(x) \) is a transformation of \( f(x) \) where the graph of \( h(x) \) is shifted upward by 2 units from the graph of \( \sqrt{x} \). Therefore, the correct answer is:
[tex]\[ \boxed{h(x)=\sqrt{x}+2} \][/tex]
In other words, the function [tex]\( h(x) \)[/tex] is achieved by shifting the parent function [tex]\( \sqrt{x} \)[/tex] vertically upward by 2 units.
