To solve the problem of vertically shifting the square root parent function [tex]\( F(x) = \sqrt{x} \)[/tex] up nine units, we need to understand the effect of vertical shifts on a function.
A vertical shift involves adding or subtracting a constant value to the entire function. In this case, we want to shift [tex]\( \sqrt{x} \)[/tex] upwards by 9 units.
Let's break down the steps:
1. Understand the Vertical Shift: When we shift a function [tex]\( f(x) \)[/tex] vertically by [tex]\( k \)[/tex] units, the new function [tex]\( g(x) \)[/tex] is given by:
[tex]\[
g(x) = f(x) + k
\][/tex]
If moving upwards, [tex]\( k \)[/tex] will be positive; if moving downwards, [tex]\( k \)[/tex] will be negative.
2. Apply the Shift to the Given Function:
- The original function is [tex]\( F(x) = \sqrt{x} \)[/tex].
- We want to shift this function up by 9 units, so [tex]\( k = 9 \)[/tex].
3. Form the New Function: Adding 9 to the original function gives us:
[tex]\[
G(x) = \sqrt{x} + 9
\][/tex]
So, the equation of the new function after shifting the square root function up by nine units is:
[tex]\[
G(x) = \sqrt{x} + 9
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{D} \)[/tex].
