Sure! Let's 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].
1. Identify the Parent Function and the Given Function:
- The parent function is [tex]\( f(x) = x^2 \)[/tex].
- The given function is [tex]\( g(x) = (x + 5)^2 + 3 \)[/tex].
2. Analyze the Transformations Involved:
- The function [tex]\( g(x) = (x + 5)^2 + 3 \)[/tex] has two obvious transformations from the parent function [tex]\( f(x) = x^2 \)[/tex]:
- The term [tex]\( (x + 5) \)[/tex] inside the squared term indicates a horizontal shift.
- The term [tex]\( +3 \)[/tex] outside the squared term indicates a vertical shift.
3. Focus on Vertical Translations:
- We are interested in the vertical translation, which is indicated by the constant term added outside the squared term.
4. Determine the Vertical Translation:
- The term [tex]\( +3 \)[/tex] outside of the squared term moves the graph of the function upward by 3 units.
Therefore, the value that represents 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] is [tex]\( \boxed{3} \)[/tex].
