To determine which function [tex]\( g(x) \)[/tex] represents a transformation of the quadratic parent function [tex]\( f(x) = x^2 \)[/tex], let's analyze the types of transformations each given option represents:
1. Option A: [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex]
This function shows a vertical compression of the parent function [tex]\( f(x) = x^2 \)[/tex] by a factor of [tex]\(\frac{1}{2}\)[/tex]. In a vertical compression, the graph of the function becomes 'flatter' compared to the original quadratic parent function.
2. Option B: [tex]\( g(x) = -\frac{1}{2} x^2 \)[/tex]
This function includes both a vertical reflection and a vertical compression. The negative sign indicates a reflection across the x-axis, making the parabola open downwards. The factor [tex]\(\frac{1}{2}\)[/tex] represents a vertical compression.
3. Option C: [tex]\( g(x) = 2 x^2 \)[/tex]
This indicates a vertical stretch of the parent function [tex]\( f(x) = x^2 \)[/tex] by a factor of 2. In a vertical stretch, the graph of the function becomes 'steeper' compared to the original quadratic function.
4. Option D: [tex]\( g(x) = -2 x^2 \)[/tex]
This function represents both a vertical reflection and a vertical stretch. The negative sign reflects the parabola across the x-axis, causing it to open downwards. The factor 2 indicates that the function is vertically stretched.
Given these transformations, if we consider the correct transformation to match [tex]\( g(x) \)[/tex], we find that the appropriate function is:
Option C: [tex]\( g(x) = 2 x^2 \)[/tex]
This function is a vertical stretch of the parent function [tex]\( f(x) = x^2 \)[/tex] by a factor of 2, making the parabola steeper while still opening upwards.
