Consider the function [tex]f(x)=5(x+2)^2-3[/tex]. Which of the following functions shifts the graph of [tex]f(x)[/tex] upward by 4 units?

A) [tex]f(x)=5(x+2)^2+1[/tex]

B) [tex]f(x)=5(x+6)^2-3[/tex]

C) [tex]f(x)=5(x-2)^2-3[/tex]

D) [tex]f(x)=5(x+2)^2-7[/tex]



Answer :

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]