To determine the translation that maps the vertex of the function [tex]\( f(x) = x^2 \)[/tex] onto the vertex of the function [tex]\( g(x) = x^2 - 10x + 2 \)[/tex], we need to follow these steps:
1. Find the Vertex of [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = x^2 \)[/tex] is a basic parabola with its vertex at the origin, i.e., the vertex is at [tex]\((0, 0)\)[/tex].
2. Find the Vertex of [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) = x^2 - 10x + 2 \)[/tex] is a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex].
To find the vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex], we use the vertex formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 2 \)[/tex].
- The x-coordinate of the vertex is [tex]\( x = -\frac{-10}{2 \cdot 1} = 5 \)[/tex].
Now, we need the y-coordinate of the vertex.
- Substitute [tex]\( x = 5 \)[/tex] back into the function [tex]\( g(x)\)[/tex]:
[tex]\[
g(5) = (5)^2 - 10(5) + 2 = 25 - 50 + 2 = -23
\][/tex]
Therefore, the vertex of [tex]\( g(x) \)[/tex] is at [tex]\((5, -23)\)[/tex].
3. Determine the Translation:
To move the vertex of [tex]\( f(x) = x^2 \)[/tex] at [tex]\((0, 0)\)[/tex] to the vertex of [tex]\( g(x) = x^2 - 10x + 2 \)[/tex] at [tex]\((5, -23)\)[/tex], we need to determine the horizontal and vertical shifts.
- Horizontal Translation:
From [tex]\( x = 0 \)[/tex] to [tex]\( x = 5 \)[/tex] means a translation of 5 units to the right.
- Vertical Translation:
From [tex]\( y = 0 \)[/tex] to [tex]\( y = -23 \)[/tex] means a translation of 23 units down.
Therefore, the translation that maps the vertex of [tex]\( f(x) = x^2 \)[/tex] to the vertex of [tex]\( g(x) = x^2 - 10x + 2 \)[/tex] is:
- Right 5 units,
- Down 23 units.
Hence, the correct translation is:
[tex]\[ \text{Right 5, Down 23} \][/tex]
