To find the correct function representing the transformation of the parent function by shifting it three units to the left and up two, let's analyze the transformations step by step.
1. Shifting three units to the left:
- A horizontal shift to the left is achieved by adding to the [tex]\(x\)[/tex]-value inside the function. For the function [tex]\(f(x) = |x|\)[/tex], this means replacing [tex]\(x\)[/tex] with [tex]\(x + 3\)[/tex]. So the function becomes [tex]\(f(x) = |x + 3|\)[/tex].
2. Shifting two units up:
- A vertical shift upward is achieved by adding to the entire function. Therefore, [tex]\(f(x) = |x + 3|\)[/tex] becomes [tex]\(f(x) = |x + 3| + 2\)[/tex].
Now, let's compare this with the given options:
- [tex]\(f(x) = |x+3|+2\)[/tex]
- [tex]\(f(x) = |x-3|-2\)[/tex]
- [tex]\(f(x) = -|x-3|+2\)[/tex]
- [tex]\(f(x) = -|x+3|+2\)[/tex]
From these options, we see that the function that correctly represents the transformation of the parent function three units to the left and up two is:
[tex]\[ f(x) = |x + 3| + 2 \][/tex]
Therefore, the correct option is the first function.
