To determine the vertical translation from the graph of the parent function [tex]\( f(x) = x^2 \)[/tex] to the graph of the function [tex]\( g(x) = (x+5)^2 + 3 \)[/tex], follow these steps:
1. Understand the Parent Function:
The parent function in this case is [tex]\( f(x) = x^2 \)[/tex].
2. Analyze the New Function:
The given function is [tex]\( g(x) = (x+5)^2 + 3 \)[/tex].
3. Identify the Components of the Function:
- [tex]\( (x+5)^2 \)[/tex] represents a horizontal shift.
- The constant [tex]\( +3 \)[/tex] at the end represents a vertical shift.
4. Determine the Horizontal Shift:
- The term [tex]\( (x+5) \)[/tex] indicates a horizontal shift 5 units to the left, but this shift does not affect the vertical translation we are interested in.
5. Determine the Vertical Shift:
- The addition of [tex]\( +3 \)[/tex] at the end of the function means the graph is shifted 3 units up vertically.
Therefore, the value that represents the vertical translation from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( g(x) = (x+5)^2 + 3 \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
