To identify which function represents a horizontal translation of the parent quadratic function [tex]\( f(x) = x^2 \)[/tex], let's analyze the options given:
### Understanding Horizontal Translation:
A horizontal translation of a function involves shifting the graph of the function left or right. For the quadratic function [tex]\( f(x) = x^2 \)[/tex], if we want to translate it horizontally by [tex]\( k \)[/tex] units:
- To the right by [tex]\( k \)[/tex] units, we'd have [tex]\( g(x) = (x - k)^2 \)[/tex].
- To the left by [tex]\( k \)[/tex] units, we'd have [tex]\( g(x) = (x + k)^2 \)[/tex].
### Analyzing Each Option:
Let's evaluate each given function to see if it represents a horizontal translation of [tex]\( f(x) = x^2 \)[/tex].
#### Option A: [tex]\( g(x) = (x - 4)^2 \)[/tex]
- This function involves a horizontal translation to the right by 4 units.
- Therefore, [tex]\( g(x) = (x - 4)^2 \)[/tex] is indeed a horizontal translation of [tex]\( f(x) = x^2 \)[/tex].
#### Option B: [tex]\( h(x) = 4x^2 \)[/tex]
- This function involves a vertical scaling by multiplying the original function by 4.
- Vertical scaling changes the steepness (or width) of the parabola but does not translate it horizontally.
- Therefore, [tex]\( h(x) = 4x^2 \)[/tex] is not a horizontal translation.
#### Option C: [tex]\( j(x) = x^2 - 4 \)[/tex]
- This function involves a vertical translation downward by 4 units.
- Vertical translations affect the position of the graph along the y-axis but not the x-axis.
- Therefore, [tex]\( j(x) = x^2 - 4 \)[/tex] is not a horizontal translation.
#### Option D: [tex]\( k(x) = -x^2 \)[/tex]
- This function involves a vertical reflection over the x-axis.
- While the parabola opens downward because of the negative sign, there is no horizontal movement.
- Therefore, [tex]\( k(x) = -x^2 \)[/tex] is not a horizontal translation.
### Conclusion:
Based on the analysis, the function that represents a horizontal translation of the parent quadratic function [tex]\( f(x) = x^2 \)[/tex] is:
- Option A: [tex]\( g(x) = (x - 4)^2 \)[/tex]
Thus, the correct answer is:
A. [tex]\( g(x) = (x - 4)^2 \)[/tex].
