To determine the function [tex]\( h(x) \)[/tex] as a transformation of the square root parent function [tex]\( f(x) = \sqrt{x} \)[/tex], let's analyze the given option.
Option A: [tex]\( h(x) = \sqrt{x} + 6 \)[/tex]
The parent function here is [tex]\( f(x) = \sqrt{x} \)[/tex], which is the square root function. The transformation involves adding 6 to this parent function.
When we add a constant to a function, this translates to a vertical shift. Specifically, adding 6 to [tex]\( \sqrt{x} \)[/tex] means we shift the entire square root graph upwards by 6 units.
- The original parent function [tex]\( f(x) = \sqrt{x} \)[/tex] passes through points like [tex]\((0,0)\)[/tex], [tex]\((1,1)\)[/tex], [tex]\((4,2)\)[/tex], etc.
- By adding 6 to this function, these points are transformed as follows:
- [tex]\((0,0)\)[/tex] becomes [tex]\((0,6)\)[/tex]
- [tex]\((1,1)\)[/tex] becomes [tex]\((1,7)\)[/tex]
- [tex]\((4,2)\)[/tex] becomes [tex]\((4,8)\)[/tex]
Thus, the transformed function, [tex]\( h(x) = \sqrt{x} + 6 \)[/tex], represents the graph of the square root function shifted vertically upwards by 6 units.
Therefore, the function [tex]\( h(x) \)[/tex] is indeed [tex]\( \sqrt{x} + 6 \)[/tex]. So, the correct answer is:
Option A: [tex]\( h(x) = \sqrt{x} + 6 \)[/tex]
