Let's analyze the function transformation step-by-step:
1. Understanding the Parent Function:
- The parent function given is [tex]\( f(x) = x^2 \)[/tex]. This is a parabola that opens upwards with its vertex at the origin, [tex]\((0,0)\)[/tex].
2. Transforming the Function:
- We are comparing it to the function [tex]\( g(x) = (x+8)^2 \)[/tex].
3. Horizontal Translations:
- Adding or subtracting a constant inside the parentheses (inside the argument of the function) affects the horizontal position of the graph.
- If we have [tex]\( f(x) = (x - h)^2 \)[/tex], the graph would shift [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive.
- Conversely, if we have [tex]\( f(x) = (x + h)^2 \)[/tex], the graph shifts [tex]\( h \)[/tex] units to the left if [tex]\( h \)[/tex] is positive.
4. Identifying the Shift:
- In the function [tex]\( g(x) = (x+8)^2 \)[/tex], we see [tex]\( +8 \)[/tex] inside the parentheses.
- This indicates a horizontal translation to the left by 8 units.
Conclusion: The graph [tex]\( f (x) = (x+8)^2 \)[/tex] has been translated 8 units to the left compared to the parent function [tex]\( f(x) = x^2 \)[/tex].
Therefore, the correct answer is:
B) It has been translated 8 units to the left.
