To determine the translation from [tex]\( y = (x+3)^2 + 4 \)[/tex] to [tex]\( y = (x+1)^2 + 6 \)[/tex], we need to look at the horizontal and vertical shifts.
1. Horizontal Shift:
- Original equation: [tex]\( y = (x+3)^2 + 4 \)[/tex]
- The term [tex]\( x+3 \)[/tex] implies that the vertex is shifted 3 units to the left of the origin, so the horizontal component [tex]\( h \)[/tex] is -3.
- Translated equation: [tex]\( y = (x+1)^2 + 6 \)[/tex]
- The term [tex]\( x+1 \)[/tex] implies that the vertex is shifted 1 unit to the left of the origin, so the new horizontal component [tex]\( h \)[/tex] is -1.
The horizontal shift is the difference between the new horizontal component and the original horizontal component:
[tex]\[
\Delta h = -1 - (-3) = -1 + 3 = 2
\][/tex]
Therefore, the horizontal shift is 2 units to the right.
2. Vertical Shift:
- Original equation: [tex]\( y = (x+3)^2 + 4 \)[/tex]
- The constant term [tex]\( +4 \)[/tex] indicates the vertical position of the vertex is 4 units above the origin, so the vertical component [tex]\( k \)[/tex] is 4.
- Translated equation: [tex]\( y = (x+1)^2 + 6 \)[/tex]
- The constant term [tex]\( +6 \)[/tex] indicates the new vertical position of the vertex is 6 units above the origin, so the new vertical component [tex]\( k \)[/tex] is 6.
The vertical shift is the difference between the new vertical component and the original vertical component:
[tex]\[
\Delta k = 6 - 4 = 2
\][/tex]
Therefore, the vertical shift is 2 units upwards.
Combining these shifts, the translation [tex]\( T \)[/tex] is described by the vector [tex]\( T_{<\Delta h, \Delta k>} = T_{<2, 2>} \)[/tex].
Thus, the correct option is:
D. [tex]\( T_{<2, 2>} \)[/tex]
