To determine which function represents a horizontal translation of the parent quadratic function [tex]\( f(x) = x^2 \)[/tex], let's analyze each given option.
1. Understand the parent function:
The parent function is [tex]\( f(x) = x^2 \)[/tex].
2. Horizontal translation:
A horizontal translation involves shifting the graph left or right. A horizontal shift to the right by [tex]\( h \)[/tex] units is represented by the function [tex]\( f(x-h) = (x-h)^2 \)[/tex]. Conversely, a shift to the left by [tex]\( h \)[/tex] units is represented by [tex]\( f(x+h) = (x+h)^2 \)[/tex].
Now, let's examine each option:
A. [tex]\( j(x) = x^2 - 4 \)[/tex]
- This function represents a vertical translation (shift down by 4 units) rather than a horizontal translation. It does not match the form [tex]\( (x-h)^2 \)[/tex] or [tex]\( (x+h)^2 \)[/tex].
B. [tex]\( g(x) = (x-4)^2 \)[/tex]
- This function represents a horizontal translation of the parent function to the right by 4 units. The correct form for a horizontal translation to the right is [tex]\( (x-h)^2 \)[/tex] where [tex]\( h \)[/tex] is positive. Therefore, this is a horizontal shift to the right by 4 units.
C. [tex]\( h(x) = 4x^2 \)[/tex]
- This function represents a vertical stretch (scaling) of the parent function and does not represent a horizontal translation. It does not take the form [tex]\( (x-h)^2 \)[/tex] or [tex]\( (x+h)^2 \)[/tex].
D. [tex]\( k(x) = -x^2 \)[/tex]
- This function represents a reflection across the x-axis, not a horizontal translation. It does not have the form [tex]\( (x-h)^2 \)[/tex] or [tex]\( (x+h)^2 \)[/tex].
Given these analyses, the only function that represents a horizontal translation of [tex]\( f(x) = x^2 \)[/tex] is:
B. [tex]\( g(x) = (x-4)^2 \)[/tex]
