Certainly! Let's analyze the given transformation step by step to determine the correct translation.
We start with the equation of a parabola:
[tex]\[ y = (x + 3)^2 + 4 \][/tex]
and we want to transform it to:
[tex]\[ y = (x + 1)^2 + 6 \][/tex]
### Step-by-Step Analysis:
1. Identify the Transformation in the [tex]\(x\)[/tex]-term:
- In the original equation, we have the term [tex]\( (x + 3) \)[/tex].
- In the transformed equation, this term has changed to [tex]\( (x + 1) \)[/tex].
To find the horizontal shift:
- We compare [tex]\( x + 3 \)[/tex] with [tex]\( x + 1 \)[/tex].
- The transformation from [tex]\( x + 3 \)[/tex] to [tex]\( x + 1 \)[/tex] indicates a shift to the left by 2 units because [tex]\(3 - 2 = 1\)[/tex].
Thus, the horizontal translation is [tex]\(-2\)[/tex].
2. Identify the Transformation in the constant term:
- In the original equation, the constant term outside the squared part is [tex]\( +4 \)[/tex].
- In the transformed equation, this term has changed to [tex]\( +6 \)[/tex].
To find the vertical shift:
- We compare [tex]\( +4 \)[/tex] to [tex]\( +6 \)[/tex].
- The transformation from [tex]\( +4 \)[/tex] to [tex]\( +6 \)[/tex] indicates a shift upwards by 2 units because [tex]\(4 + 2 = 6\)[/tex].
Thus, the vertical translation is [tex]\(+2\)[/tex].
### Conclusion:
Combining both transformations, we get a translation vector [tex]\( T \)[/tex]. The horizontal shift is [tex]\(-2\)[/tex], and the vertical shift is [tex]\( +2 \)[/tex]. Therefore, the translation vector is:
[tex]\[ T_{<-2, 2\rangle} \][/tex]
This matches with option:
[tex]\[ \text{C. } T_{<-2, 2\rangle} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{C. } T_{<-2, 2\rangle}} \][/tex]
