To determine the translation that maps the graph of the function [tex]\( f(x) = x^2 \)[/tex] onto the function [tex]\( g(x) = x^2 - 6x + 6 \)[/tex], we need to analyze the transformation of the vertex of the parabola described by these functions.
1. Identify the vertex of [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = x^2 \)[/tex] is a basic parabola with its vertex at [tex]\((0,0)\)[/tex].
2. Rewrite [tex]\( g(x) \)[/tex] to find its vertex:
The function [tex]\( g(x) = x^2 - 6x + 6 \)[/tex] can be written in vertex form by completing the square.
Start with the given function:
[tex]\[
g(x) = x^2 - 6x + 6
\][/tex]
Group the [tex]\( x \)[/tex]-terms together and complete the square:
[tex]\[
g(x) = x^2 - 6x + 6
\][/tex]
To complete the square, add and subtract [tex]\(\left(\frac{-6}{2}\right)^2 = 9\)[/tex]:
[tex]\[
g(x) = x^2 - 6x + 9 - 9 + 6
\][/tex]
[tex]\[
g(x) = (x - 3)^2 - 3
\][/tex]
Now, [tex]\( g(x) = (x-3)^2 - 3 \)[/tex].
3. Identify the vertex of [tex]\( g(x) \)[/tex]:
The vertex form [tex]\( g(x) = (x - 3)^2 - 3 \)[/tex] shows that the vertex of [tex]\( g(x) \)[/tex] is [tex]\((3, -3)\)[/tex].
4. Determine the translation required:
To map the vertex of [tex]\( f(x) = x^2 \)[/tex] at [tex]\((0,0)\)[/tex] onto the vertex of [tex]\( g(x) = x^2 - 6x + 6 \)[/tex] at [tex]\((3, -3)\)[/tex], you must:
- Move 3 units to the right (since [tex]\(0 + 3 = 3\)[/tex])
- Move 3 units down (since [tex]\(0 - 3 = -3\)[/tex])
Therefore, the translation that maps the graph of [tex]\( f(x) = x^2 \)[/tex] onto the graph of [tex]\( g(x) = x^2 - 6x + 6 \)[/tex] is right 3 units, down 3 units.
