To determine which translation maps the vertex of the graph of [tex]\( f(x) = x^2 \)[/tex] onto the vertex of the graph of [tex]\( g(x) = x^2 - 10x + 2 \)[/tex], we need to follow these steps:
1. Identify the vertex of [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = x^2 \)[/tex] is a simple quadratic function where the vertex is at the origin [tex]\((0, 0)\)[/tex].
2. Determine the vertex form of [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) = x^2 - 10x + 2 \)[/tex] is also a quadratic function, but we need to find its vertex. To do that, let's rewrite [tex]\( g(x) \)[/tex] by completing the square.
[tex]\[
g(x) = x^2 - 10x + 2
\][/tex]
To complete the square, follow these steps:
- Take the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-10\)[/tex]), divide it by 2, and square it:
[tex]\[
\left( \frac{-10}{2} \right)^2 = (-5)^2 = 25
\][/tex]
- Add and subtract this square inside the equation to maintain the equality:
[tex]\[
g(x) = x^2 - 10x + 25 - 25 + 2
\][/tex]
- Rewrite the quadratic expression as a square:
[tex]\[
g(x) = (x - 5)^2 - 23
\][/tex]
3. Identify the vertex of [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) = (x - 5)^2 - 23 \)[/tex] is now in vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex. Here the vertex is at [tex]\( (5, -23) \)[/tex].
4. Determine the translation:
To map the vertex of [tex]\( f(x) \)[/tex] at [tex]\( (0, 0) \)[/tex] to the vertex of [tex]\( g(x) \)[/tex] at [tex]\( (5, -23) \)[/tex]:
- We need to move right by 5 units (from [tex]\( x = 0 \)[/tex] to [tex]\( x = 5 \)[/tex]).
- We need to move down by 23 units (from [tex]\( y = 0 \)[/tex] to [tex]\( y = -23 \)[/tex]).
Therefore, the translation that maps the vertex of the graph of [tex]\( f(x) \)[/tex] onto the vertex of the graph of [tex]\( g(x) \)[/tex] is right 5, down 23.
