Sure, let's go through the transformations step-by-step:
1. Start with the base function: [tex]\( f(x) = \sqrt{x} \)[/tex]
2. Apply the horizontal transformation:
- The transformation [tex]\( x - 2 \)[/tex] inside the square root function shifts the graph horizontally.
- Since it is [tex]\( x - 2 \)[/tex], it means a shift to the right by 2 units.
So far, we have [tex]\( \sqrt{x - 2} \)[/tex].
3. Apply the vertical transformation:
- The transformation [tex]\( +3 \)[/tex] outside the square root function shifts the graph vertically.
- Since it is [tex]\( +3 \)[/tex], it means a shift up by 3 units.
Therefore, the final function after the transformations is [tex]\( f(x) = \sqrt{x - 2} + 3 \)[/tex].
Given these steps, the correct transformation for the function [tex]\( f(x) = \sqrt{x - 2} + 3 \)[/tex] is a shift to the right by 2 units and a shift up by 3 units.
Thus, the correct choice is:
- right 2, up 3
