Let's analyze the given functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = (x-5)^2 \)[/tex].
1. Understanding Function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 \)[/tex] is a basic quadratic function, whose graph is a parabola opening upwards with its vertex at the origin [tex]\((0, 0)\)[/tex].
2. Understanding Function [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = (x-5)^2 \)[/tex] is also a quadratic function. However, it has been modified compared to [tex]\( f(x) \)[/tex]. Specifically, the expression inside the parentheses [tex]\((x-5)\)[/tex] suggests that a transformation has been applied to [tex]\( f(x) = x^2 \)[/tex].
3. Determining the Transformation:
- The general form of a horizontal shift for a function [tex]\( f(x) \)[/tex] is [tex]\( f(x-h) \)[/tex], where [tex]\( h \)[/tex] represents the units of shift. When [tex]\( h > 0 \)[/tex], the shift is to the right, and when [tex]\( h < 0 \)[/tex], the shift is to the left.
- In [tex]\( g(x) = (x-5)^2 \)[/tex], we see that the term [tex]\( x-5 \)[/tex] indicates a horizontal shift.
4. Identifying the Direction and Units of Shift:
- The term [tex]\( x-5 \)[/tex] means that [tex]\( x \)[/tex] in [tex]\( f(x) = x^2 \)[/tex] is replaced by [tex]\( x-5 \)[/tex]. This corresponds to a shift of the graph of [tex]\( f(x) = x^2 \)[/tex] to the right by 5 units.
In conclusion, to obtain [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex], [tex]\( f(x) \)[/tex] should be shifted right by 5 units.
