To determine which equation represents a vertical stretch of the parent function [tex]\( f(x)=x^2 \)[/tex], let's examine each option by evaluating them at [tex]\( x = 1 \)[/tex].
1. Option [tex]\( y = 4x^2 \)[/tex]:
[tex]\[
y = 4 \cdot (1)^2 = 4
\][/tex]
This equation multiplies the original parent function [tex]\( x^2 \)[/tex] by 4, making the graph steeper. Hence, it represents a vertical stretch.
2. Option [tex]\( y = \frac{1}{4} + x^2 \)[/tex]:
[tex]\[
y = \frac{1}{4} + (1)^2 = \frac{1}{4} + 1 = 1.25
\][/tex]
This equation adds a constant to the parent function [tex]\( x^2 \)[/tex] which translates the graph vertically but does not stretch it.
3. Option [tex]\( y = \left(\frac{1}{4}x\right)^2 \)[/tex]:
[tex]\[
y = \left(\frac{1}{4} \cdot 1\right)^2 = \left(\frac{1}{4}\right)^2 = 0.0625
\][/tex]
This equation compresses the graph horizontally by a factor of 4 (since the x-term is multiplied by [tex]\( \frac{1}{4} \)[/tex]) and then squares the result. This does not represent a vertical stretch.
4. Option [tex]\( y = x^2 - 4 \)[/tex]:
[tex]\[
y = (1)^2 - 4 = 1 - 4 = -3
\][/tex]
This equation subtracts a constant from the parent function [tex]\( x^2 \)[/tex], which translates the graph downward but does not change its shape in terms of stretching.
Therefore, the equation that represents a vertical stretch of the parent function [tex]\( f(x) = x^2 \)[/tex] is [tex]\( y = 4x^2 \)[/tex].
Hence, the correct choice is:
[tex]\( y = 4x^2 \)[/tex].
