To determine the vertical translation from the parent function \( f(x) = x^2 \) to the function \( g(x) = (x + 5)^2 + 3 \), we can analyze how the graph has been transformed.
The parent function \( f(x) = x^2 \) has its vertex at (0, 0). The function \( g(x) = (x + 5)^2 + 3 \) can be understood in two parts:
- Horizontal Shift: The term \( (x + 5)^2 \) indicates a leftward shift of 5 units.
- Vertical Shift: The \( + 3 \) at the end indicates an upward shift of 3 units.
Since we are interested in the vertical translation, we focus on the \( +3 \) part.
Thus, the vertical translation from \( f(x) \) to \( g(x) \) is \( 3 \).
Therefore, the answer is:
3.