To determine the transformation needed to get from the graph of [tex]\( f(x) = x^2 \)[/tex] to the graph of [tex]\( g(x) = (x-5)^2 \)[/tex], let's analyze the given function step by step.
1. Understand the Vertical and Horizontal Shifts:
The standard quadratic function is [tex]\( f(x) = x^2 \)[/tex].
For a function in the form [tex]\( g(x) = (x - h)^2 \)[/tex]:
- If [tex]\( h \)[/tex] is positive (i.e., [tex]\( x - h \)[/tex]), the graph of [tex]\( f(x) = x^2 \)[/tex] is shifted [tex]\( h \)[/tex] units to the right.
- If [tex]\( h \)[/tex] is negative (i.e., [tex]\( x + |h| \)[/tex]), the graph of [tex]\( f(x) = x^2 \)[/tex] is shifted [tex]\( |h| \)[/tex] units to the left.
2. Identify the Transformation:
- The function [tex]\( g(x) = (x - 5)^2 \)[/tex] has the form [tex]\( f(x - h) \)[/tex] where [tex]\( h = 5 \)[/tex] (positive).
3. Determine the Direction of the Shift:
- Since [tex]\( h = 5 \)[/tex] is positive, this corresponds to a horizontal shift to the right.
Therefore, the transformation required to move from [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = (x - 5)^2 \)[/tex] is a horizontal shift to the right by 5 units.
The correct answer is:
A. The graph of [tex]\( f(x) = x^2 \)[/tex] should be shifted 5 units to the right.
