To determine which function is a horizontal translation of the parent quadratic function [tex]\( f(x)=x^2 \)[/tex], we need to understand what horizontal translation means in the context of graph transformations.
A horizontal translation involves shifting the graph of a function left or right. For the quadratic function [tex]\( f(x) = x^2 \)[/tex], the general form of a horizontal translation is given by [tex]\( g(x) = (x-h)^2 \)[/tex], where [tex]\( h \)[/tex] is the number of units the graph is shifted horizontally. If [tex]\( h \)[/tex] is positive, the graph shifts to the right, and if [tex]\( h \)[/tex] is negative, the graph shifts to the left.
Let’s examine each of the given options:
A. [tex]\( h(x) = 4x^2 \)[/tex]
- This function represents a vertical stretch of the parent function by a factor of 4. It does not involve any horizontal translation.
B. [tex]\( k(x) = -x^2 \)[/tex]
- This function represents a reflection of the parent function across the x-axis. It does not involve any horizontal translation.
C. [tex]\( g(x)=(x-4)^2 \)[/tex]
- This function represents a horizontal translation of the parent function [tex]\( f(x) = x^2 \)[/tex] by 4 units to the right. This matches the general form of a horizontal translation [tex]\( (x-h)^2 \)[/tex] with [tex]\( h = 4 \)[/tex].
D. [tex]\( j(x) = x^2 - 4 \)[/tex]
- This function represents a vertical translation of the parent function by 4 units down. It does not involve any horizontal translation.
Based on the analysis, the correct answer is:
C. [tex]\( g(x) = (x-4)^2 \)[/tex]
This function is a horizontal translation of the parent quadratic function [tex]\( f(x) = x^2 \)[/tex].
