To determine which of the given functions shifts the graph of the function [tex]\( f(x) = 5(x+2)^2 - 3 \)[/tex] upward by 4 units, we need to understand how vertical shifts work.
Shifting a function upward by 4 units means adding 4 to the original function.
Given the function:
[tex]\[ f(x) = 5(x+2)^2 - 3 \][/tex]
To shift this function upward by 4 units, we add 4 to the entire function:
[tex]\[ f(x) + 4 = 5(x+2)^2 - 3 + 4 \][/tex]
Simplify this expression:
[tex]\[ f(x) + 4 = 5(x+2)^2 + 1 \][/tex]
Thus, the new function after shifting the graph of [tex]\( f(x) \)[/tex] upward by 4 units is:
[tex]\[ f_{new}(x) = 5(x+2)^2 + 1 \][/tex]
Given the options:
A) [tex]\( 5(x+2)^2 + 1 \)[/tex]
B) [tex]\( 5(x+6)^2 - 3 \)[/tex]
C) [tex]\( 5(x-2)^2 - 3 \)[/tex]
D) [tex]\( 5(x+2)^2 - 7 \)[/tex]
The correct choice is option A:
[tex]\[ f(x) = 5(x+2)^2 + 1 \][/tex]
Therefore, the function that shifts the graph of [tex]\( f(x) \)[/tex] upward by 4 units is:
[tex]\[ \boxed{f(x) = 5(x+2)^2 + 1} \][/tex]
