If you apply the changes below to the absolute value parent function [tex]\( f(x) = |x| \)[/tex], what is the equation of the new function?

- Shift 4 units left.
- Shift 2 units up.

A. [tex]\( g(x) = |x-4| + 2 \)[/tex]
B. [tex]\( g(x) = |x+4| + 2 \)[/tex]
C. [tex]\( g(x) = |x+2| - 4 \)[/tex]
D. [tex]\( g(x) = |x+2| + 4 \)[/tex]



Answer :

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]