To solve the problem, we need to understand the transformation being applied to the quadratic parent function [tex]\( F(x) = x^2 \)[/tex].
Vertical compression involves scaling the output (y-values) of the function. Given that we're asked to vertically compress by multiplying by [tex]\( \frac{1}{4} \)[/tex], we will scale the y-values of the function [tex]\( F(x) \)[/tex] by a factor of [tex]\( \frac{1}{4} \)[/tex].
Here are the steps to find the new function [tex]\( G(x) \)[/tex]:
1. Identify the parent function:
[tex]\[ F(x) = x^2 \][/tex]
2. Apply the vertical compression factor: The y-values of [tex]\( F(x) \)[/tex] need to be multiplied by [tex]\( \frac{1}{4} \)[/tex].
3. Write the transformed equation:
[tex]\[ G(x) = \frac{1}{4} \cdot F(x) \][/tex]
Since [tex]\( F(x) = x^2 \)[/tex], substituting this into the equation gives:
[tex]\[ G(x) = \frac{1}{4} \cdot x^2 \][/tex]
Thus, the equation for the new function [tex]\( G(x) \)[/tex] after applying the vertical compression is:
[tex]\[ G(x) = \frac{1}{4} x^2 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B. \ G(x) = \frac{1}{4} x^2} \][/tex]
