Answer :
To determine which function represents a horizontal stretch of the parent function [tex]\( y = x^2 \)[/tex], let's analyze the transformations of each given function.
1. [tex]\( y = 3x^2 \)[/tex]:
This represents a vertical stretch of the parent function. Multiplying by 3 makes the graph steeper, but it does not affect the horizontal component.
2. [tex]\( y = \frac{1}{3}x^2 \)[/tex]:
This represents a vertical compression of the parent function. Dividing by 3 makes the graph wider vertically, but it does not affect the horizontal component.
3. [tex]\( y = 5 + x^2 \)[/tex]:
This represents a vertical shift of the parent function. The +5 shifts the graph upwards by 5 units, but it does not stretch horizontally or vertically.
4. [tex]\( y = \left(\frac{1}{2}x\right)^2 \)[/tex]:
This represents a horizontal stretch of the parent function. To see this, rewrite the equation as [tex]\( y = \left(\frac{1}{2}x\right)^2 = \left(\frac{1}{2}\right)^2 x^2 = \frac{1}{4}x^2 \)[/tex]. Here, the x-term is scaled by [tex]\(\frac{1}{2}\)[/tex] before squaring, effectively stretching the graph horizontally by a factor of 2.
Given that a horizontal stretch involves changing the [tex]\(x\)[/tex]-variable inside the function, the correct transformation is:
[tex]\[ y = \left(\frac{1}{2}x\right)^2 \][/tex]
Thus, the function representing a horizontal stretch of the parent function [tex]\( y = x^2 \)[/tex] is:
[tex]\[ \boxed{y = \left(\frac{1}{2}x\right)^2} \][/tex]
So, the correct option is:
[tex]\[ \boxed{4} \][/tex]
1. [tex]\( y = 3x^2 \)[/tex]:
This represents a vertical stretch of the parent function. Multiplying by 3 makes the graph steeper, but it does not affect the horizontal component.
2. [tex]\( y = \frac{1}{3}x^2 \)[/tex]:
This represents a vertical compression of the parent function. Dividing by 3 makes the graph wider vertically, but it does not affect the horizontal component.
3. [tex]\( y = 5 + x^2 \)[/tex]:
This represents a vertical shift of the parent function. The +5 shifts the graph upwards by 5 units, but it does not stretch horizontally or vertically.
4. [tex]\( y = \left(\frac{1}{2}x\right)^2 \)[/tex]:
This represents a horizontal stretch of the parent function. To see this, rewrite the equation as [tex]\( y = \left(\frac{1}{2}x\right)^2 = \left(\frac{1}{2}\right)^2 x^2 = \frac{1}{4}x^2 \)[/tex]. Here, the x-term is scaled by [tex]\(\frac{1}{2}\)[/tex] before squaring, effectively stretching the graph horizontally by a factor of 2.
Given that a horizontal stretch involves changing the [tex]\(x\)[/tex]-variable inside the function, the correct transformation is:
[tex]\[ y = \left(\frac{1}{2}x\right)^2 \][/tex]
Thus, the function representing a horizontal stretch of the parent function [tex]\( y = x^2 \)[/tex] is:
[tex]\[ \boxed{y = \left(\frac{1}{2}x\right)^2} \][/tex]
So, the correct option is:
[tex]\[ \boxed{4} \][/tex]
