To determine how to transform [tex]\( f(x) = x^2 \)[/tex] into [tex]\( g(x) = (x-4)^2 \)[/tex], we need to identify the type of transformation and the direction of the shift.
1. Recall the Transformation Rule:
- When you shift a function horizontally, subtracting a constant [tex]\(\text{c}\)[/tex] from [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex] will shift the graph to the right by [tex]\(\text{c}\)[/tex] units.
- Conversely, adding a constant [tex]\(\text{c}\)[/tex] to [tex]\( x \)[/tex] will shift the graph to the left by [tex]\(\text{c}\)[/tex] units.
2. Analyze the Given Functions:
- The original function is [tex]\( f(x) = x^2 \)[/tex].
- The transformed function is [tex]\( g(x) = (x-4)^2 \)[/tex].
3. Identify the Transformation:
- Compare [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = x^2 \)[/tex] is transformed into [tex]\( g(x) = (x-4)^2 \)[/tex].
- Here, [tex]\( (x-4) \)[/tex] implies that [tex]\( x \)[/tex] is replaced by [tex]\( x-4 \)[/tex].
4. Determine the Shift:
- According to the transformation rule, [tex]\( (x - 4) \)[/tex] indicates a shift to the right.
- The value [tex]\( 4 \)[/tex] is the number of units by which the graph is shifted.
So, [tex]\( f(x) = x^2 \)[/tex] needs to be shifted to the right by 4 units to obtain [tex]\( g(x) = (x-4)^2 \)[/tex].
Therefore, the direction is:
[tex]\[ \text{Right} \][/tex]
And the number of units is:
[tex]\[ 4 \][/tex]
In conclusion, [tex]\( f(x) = x^2 \)[/tex] should be shifted to the Right by 4 units to obtain [tex]\( g(x) = (x-4)^2 \)[/tex].
