To solve the problem, we need to analyze the transformations applied to the function [tex]\( f(x) = x^2 \)[/tex] to transform it into a new function [tex]\( g \)[/tex].
1. Identify the Type of Transformation:
- The function [tex]\( f(x) = x^2 \)[/tex] has undergone a transformation resulting in a function [tex]\( g \)[/tex].
- The type of transformation applied is a "translation". This is the correct term for the movement of the entire graph horizontally or vertically without altering its shape or orientation.
2. Recognize the Expression of the Transformed Function:
- The general form for function transformation [tex]\( g \)[/tex] of [tex]\( f(x) \)[/tex] involving translation can be expressed as [tex]\( g(x) = f(x) ± k \)[/tex].
- For the function [tex]\( f(x) = x^2 \)[/tex], a vertical translation involves adding or subtracting a constant [tex]\( k \)[/tex].
3. Determine the Specific Translation Applied:
- According to the given anwer, the specific translation applied to [tex]\( x^2 \)[/tex] is a vertical translation by 3 units.
- The transformation [tex]\( g(x) \)[/tex] can be written as [tex]\( g(x) = f(x) ± 3 \)[/tex].
4. Construct the Final Form of the Transformed Function:
- Applying the translation to [tex]\( f(x) = x^2 \)[/tex]:
[tex]\[
g(x) = x^2 ± 3
\][/tex]
5. Fine-Tune the Equation Based on the Given Clues:
- The presence of [tex]\( ± \)[/tex] suggests that there might be flexibility in whether it is [tex]\( +3 \)[/tex] or [tex]\( -3 \)[/tex]. In our case, let's assume a positive translation:
[tex]\[
g(x) = x^2 + 3
\][/tex]
Therefore, the detailed selection is as follows:
Function [tex]\( g \)[/tex] is a "translation" of function [tex]\( f \)[/tex].
[tex]\[
g(x) = x^2 + 3
\][/tex]
