To determine the equation of the new function after applying the given transformations to the absolute value parent function [tex]\( f(x) = |x| \)[/tex], follow these steps:
1. Shift 4 units left:
- Shifting a function to the left by [tex]\( k \)[/tex] units is achieved by adding [tex]\( k \)[/tex] inside the function's argument. For instance, if we want to shift [tex]\( f(x) \)[/tex] left by 4 units, we modify it to [tex]\( f(x + 4) \)[/tex].
- So, the function after this step becomes [tex]\( |x + 4| \)[/tex].
2. Shift 2 units up:
- Shifting a function up by [tex]\( k \)[/tex] units is achieved by adding [tex]\( k \)[/tex] to the entire function. For instance, if we want to shift [tex]\( f(x) \)[/tex] up by 2 units, we modify it to [tex]\( f(x) + 2 \)[/tex].
- Applying this to [tex]\( |x + 4| \)[/tex], the function becomes [tex]\( |x + 4| + 2 \)[/tex].
Hence, after applying both transformations, the new function is:
[tex]\[ g(x) = |x + 4| + 2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \text{B. } g(x) = |x+4|+2 \][/tex]
