Sure, let's analyze the translation step-by-step.
We start with the original equation of the function:
[tex]\[ y = (x - 5)^2 + 5 \][/tex]
We are translating this function to a new equation:
[tex]\[ y = (x - 0)^2 + 0 \][/tex]
or simply:
[tex]\[ y = x^2 \][/tex]
### Understanding the Translation
The general form of a translated equation is:
[tex]\[ y = (x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
Here, in the initial function [tex]\( y = (x - 5)^2 + 5 \)[/tex], the vertex of the parabola is at the point [tex]\((5, 5)\)[/tex].
In the translated function [tex]\( y = x^2 \)[/tex], the vertex of the parabola moves to the origin, [tex]\((0, 0)\)[/tex].
### Determining the Translation Vector
To find the translation vector, we need to determine how far and in what direction the vertex has moved:
1. Horizontal Translation:
- The original x-coordinate of the vertex is 5.
- The new x-coordinate of the vertex is 0.
- The horizontal translation is [tex]\( 0 - 5 = -5 \)[/tex].
2. Vertical Translation:
- The original y-coordinate of the vertex is 5.
- The new y-coordinate of the vertex is 0.
- The vertical translation is [tex]\( 0 - 5 = -5 \)[/tex].
Therefore, the translation moves the graph 5 units to the right (positive x-direction) and 5 units down (negative y-direction).
In vector notation, the translation can be represented as:
[tex]\[ \langle 5, -5 \rangle \][/tex]
### Selecting the Correct Option
Now, let's match this translation vector to the options provided:
A. [tex]\( T_{<-5, -5>} \)[/tex] — translates the graph left 5 units and down 5 units.
B. [tex]\( T_{<5, 5>} \)[/tex] — translates the graph right 5 units and up 5 units.
C. [tex]\( T_{<5, -5>} \)[/tex] — translates the graph right 5 units and down 5 units.
D. [tex]\( T_{<-5, 5>} \)[/tex] — translates the graph left 5 units and up 5 units.
Based on our analysis, the correct option is:
[tex]\[ \boxed{C} \][/tex]
