To determine which function [tex]\( g(x) \)[/tex] is a transformation of the quadratic parent function [tex]\( f(x) = x^2 \)[/tex], we need to evaluate the transformations described by each option. Each transformation involves either a vertical stretch/compression, a reflection over the x-axis, or both.
1. Option A: [tex]\( g(x) = -\frac{1}{2} x^2 \)[/tex]
- This represents a vertical compression (by a factor of [tex]\(\frac{1}{2}\)[/tex]) and a reflection over the x-axis.
- The function [tex]\( g(x) \)[/tex] is a downward-opening parabola that is wider than the parent function [tex]\( f(x) \)[/tex].
2. Option B: [tex]\( g(x) = 2 x^2 \)[/tex]
- This represents a vertical stretch (by a factor of 2).
- The function [tex]\( g(x) \)[/tex] is an upward-opening parabola that is narrower than the parent function [tex]\( f(x) \)[/tex].
3. Option C: [tex]\( g(x) = -2 x^2 \)[/tex]
- This represents a vertical stretch (by a factor of 2) and a reflection over the x-axis.
- The function [tex]\( g(x) \)[/tex] is a downward-opening parabola that is narrower than the parent function [tex]\( f(x) \)[/tex].
4. Option D: [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex]
- This represents a vertical compression (by a factor of [tex]\(\frac{1}{2}\)[/tex]).
- The function [tex]\( g(x) \)[/tex] is an upward-opening parabola that is wider than the parent function [tex]\( f(x) \)[/tex].
Given the characteristics of each transformation, Option D, [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex], represents the intended transformation.
Therefore, the correct function for [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{\frac{1}{2} x^2} \][/tex]
