Certainly! To describe the translation from [tex]\( y = (x-5)^2 + 5 \)[/tex] to [tex]\( y = (x-0)^2 + 0 \)[/tex], let's carefully examine how each component of the function changes.
1. Translation in the x-axis:
- Initially, the function has [tex]\( (x-5) \)[/tex], indicating that the vertex of the parabola is shifted 5 units to the right from the origin.
- In the final function, it is [tex]\( (x-0) \)[/tex] or simply [tex]\( x \)[/tex], which means the vertex is now at the origin.
This shift represents a movement of 5 units to the right on the x-axis.
2. Translation in the y-axis:
- Initially, the function has a constant term [tex]\( +5 \)[/tex] which means the vertex of the parabola is 5 units up from the origin.
- In the final function, it is [tex]\( +0 \)[/tex], so the vertex is at the origin on the y-axis.
This shift represents a movement of 5 units down on the y-axis.
Combining these two translations, the overall translation vector is:
- 5 units to the right (positive direction)
- 5 units down (negative direction)
Hence, the translation vector is [tex]\( <5, -5> \)[/tex].
Now, let’s match this vector with the given options:
A. [tex]\( T_{<-5,5>} \)[/tex]
B. [tex]\( T_{<-5,-5>} \)[/tex]
C. [tex]\( T_{<5,5>} \)[/tex]
D. [tex]\( T_{<5,-5>} \)[/tex]
The correct choice corresponding to the translation vector [tex]\( <5, -5> \)[/tex] is:
[tex]\[ \boxed{D} \][/tex]
