To find the equation of the new function after the specified transformations, follow these steps carefully:
1. Start with the absolute value parent function:
[tex]\[ f(x) = |x| \][/tex]
2. Shift 4 units to the right:
- When you shift a function [tex]\( f(x) \)[/tex] horizontally to the right by [tex]\( h \)[/tex] units, you replace [tex]\( x \)[/tex] with [tex]\( x-h \)[/tex]. In this case, [tex]\( h = 4 \)[/tex].
- Therefore, shifting [tex]\( f(x) = |x| \)[/tex] to the right by 4 units results in:
[tex]\[ f(x-4) = |x-4| \][/tex]
3. Shift 6 units up:
- When you shift a function [tex]\( f(x) \)[/tex] vertically upward by [tex]\( k \)[/tex] units, you add [tex]\( k \)[/tex] to the entire function. In this case, [tex]\( k = 6 \)[/tex].
- Therefore, shifting [tex]\( |x-4| \)[/tex] upward by 6 units results in:
[tex]\[ g(x) = |x-4| + 6 \][/tex]
Hence, after applying the transformations, the new function is:
[tex]\[ g(x) = |x-4| + 6 \][/tex]
Comparing this with the given options:
- A. [tex]\( g(x) = |x+6| + 4 \)[/tex]
- B. [tex]\( g(x) = |x-6| + 4 \)[/tex]
- C. [tex]\( g(x) = |x+4| + 6 \)[/tex]
- D. [tex]\( g(x) = |x-4| + 6 \)[/tex]
The correct equation of the new function is [tex]\( \boxed{g(x) = |x-4| + 6} \)[/tex], which corresponds to option D.
