To determine which of the given functions represents a vertically compressed version of the quadratic parent function [tex]\( F(x) = x^2 \)[/tex], we need to understand what it means for a function to be vertically compressed. A vertically compressed function has a coefficient between 0 and 1 when compared to the parent function [tex]\( x^2 \)[/tex].
Let's analyze each given function:
A. [tex]\( G(x) = \frac{3}{4} x^2 \)[/tex]
In this case, the coefficient in front of [tex]\( x^2 \)[/tex] is [tex]\( \frac{3}{4} \)[/tex] which is less than 1. Hence, this is a vertically compressed version of [tex]\( F(x) = x^2 \)[/tex].
B. [tex]\( G(x) = (2.5 x)^2 \)[/tex]
First, simplify this expression:
[tex]\[ G(x) = (2.5 x)^2 = 2.5^2 \cdot x^2 = 6.25 x^2 \][/tex]
Here, the coefficient 6.25 is greater than 1, which means this is a vertically stretched version of [tex]\( F(x) = x^2 \)[/tex], not compressed.
C. [tex]\( G(x) = 5 x^2 \)[/tex]
In this case, the coefficient in front of [tex]\( x^2 \)[/tex] is 5, which is greater than 1. Therefore, this is also a vertically stretched version of [tex]\( F(x) = x^2 \)[/tex].
D. [tex]\( G(x) = (14 x)^2 \)[/tex]
First, simplify this expression:
[tex]\[ G(x) = (14 x)^2 = 14^2 \cdot x^2 = 196 x^2 \][/tex]
Here, the coefficient 196 is much greater than 1, indicating this is a significantly vertically stretched version of [tex]\( F(x) = x^2 \)[/tex], not compressed.
Based on this analysis, the function that shows the quadratic parent function [tex]\( F(x) = x^2 \)[/tex] being vertically compressed is:
[tex]\[ \boxed{\text{A. } G(x) = \frac{3}{4} x^2} \][/tex]
