To determine the transformation needed to graph the function [tex]\( g(x) = (x-3)^2 \)[/tex] from the base function [tex]\( f(x) = x^2 \)[/tex], we can follow these steps:
1. Identify the Base Function:
The base function is given by [tex]\( f(x) = x^2 \)[/tex]. This is a simple quadratic function with its vertex at the origin [tex]\((0,0)\)[/tex] and it opens upwards.
2. Understand the Transformation:
The function [tex]\( g(x) = (x-3)^2 \)[/tex] can be viewed as a transformation of the base function [tex]\( f(x) \)[/tex]. Specifically, we recognize that the argument of [tex]\( f(x) \)[/tex] has been modified to [tex]\( x-3 \)[/tex].
3. Horizontal Shift:
When a function [tex]\( f(x) \)[/tex] is transformed to [tex]\( f(x-h) \)[/tex], it results in a horizontal shift of the graph of [tex]\( f(x) \)[/tex] to the right by [tex]\( h \)[/tex] units if [tex]\( h \)[/tex] is positive, and to the left by [tex]\( h \)[/tex] units if [tex]\( h \)[/tex] is negative.
- In this case, the function [tex]\( g(x) = (x-3)^2 \)[/tex] can be written in the form [tex]\( f(x-h) \)[/tex] where [tex]\( h = 3 \)[/tex]. Thus, this represents a horizontal shift to the right by 3 units.
4. Conclusion:
Therefore, to graph the function [tex]\( g(x) = (x-3)^2 \)[/tex], you need to take the graph of [tex]\( f(x) = x^2 \)[/tex] and shift it horizontally to the right by 3 units.
The transformation needed is:
- Shift the graph of [tex]\( f(x) = x^2 \)[/tex] horizontally right by 3 units.
