Answer:
g(x) = |x| + 8**
Step-by-step explanation:
The transformation that represents a vertical translation in the context of function notation is an addition or subtraction of a constant term outside of the function's primary operation. In the case of the function \( f(x) = |x| \), a vertical translation upwards by 8 units would add 8 to the value of the function for all \( x \).
Here are the breakdowns of the options given:
- **g(x) = |x + 8|**: This is a horizontal shift, not a vertical translation. Specifically, it moves the graph of \( f(x) \) 8 units to the left.
- **g(x) = |x| + 8**: This is a vertical translation, moving the graph of \( f(x) \) up by 8 units.
- **g(x) = |8x|**: This transformation alters the slope or "stretch" of the function, compressing it horizontally by a factor of 8, not translating it vertically.
- **g(x) = 8|x|**: This scales the function vertically by a factor of 8, again not translating it but stretching or compressing it vertically.
Given these transformations, the function that represents a vertical translation of 8 units is:
**g(x) = |x| + 8**