Sure! Let's break down the problem step-by-step to understand which function [tex]\( g(x) \)[/tex] represents a transformation of the quadratic parent function [tex]\( f(x) = x^2 \)[/tex].
1. Identify the Parent Function:
The given parent function is [tex]\( f(x) = x^2 \)[/tex]. This is a standard quadratic function whose graph is a parabola opening upwards.
2. Understand the Transformation:
The correct transformation given tells us that the transformed function is either one of the following:
- [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex]
- [tex]\( g(x) = -\frac{1}{2} x^2 \)[/tex]
- [tex]\( g(x) = 2 x^2 \)[/tex]
- [tex]\( g(x) = -2 x^2 \)[/tex]
3. Analyze the Transformation:
- For [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex], the parabola is stretched vertically by a factor of [tex]\(\frac{1}{2}\)[/tex], meaning it is wider than the parent function.
- For [tex]\( g(x) = -\frac{1}{2} x^2 \)[/tex], the parabola is also stretched vertically by a factor of [tex]\(\frac{1}{2}\)[/tex] but it opens downwards.
- For [tex]\( g(x) = 2 x^2 \)[/tex], the parabola is compressed vertically by a factor of [tex]\(2\)[/tex], making it narrower than the parent function.
- For [tex]\( g(x) = -2 x^2 \)[/tex], the parabola is compressed vertically by a factor of [tex]\(2\)[/tex] but it opens downwards.
4. Determine the Correct Transformation:
Given that the answer is [tex]\(1\)[/tex], we correspond this to the function:
- [tex]\( g(x) = \frac{1}{2} x^2 \)[/tex]
This means that the correct function [tex]\( g(x) \)[/tex] is a vertically stretched version of the parent function by a factor of [tex]\( \frac{1}{2} \)[/tex], making the parabola wider.
Therefore, the correct answer is:
[tex]\[ \boxed{g(x) = \frac{1}{2} x^2} \][/tex]
