Describe the translation.

[tex]\[ y = (x + 3)^2 + 4 \rightarrow y = (x + 1)^2 + 6 \][/tex]

A. [tex]\[ T_{\ \textless \ -2, -2\ \textgreater \ } \][/tex]

B. [tex]\[ T_{\ \textless \ 2, -2\ \textgreater \ } \][/tex]

C. [tex]\[ T_{\ \textless \ -2, 2\ \textgreater \ } \][/tex]

D. [tex]\[ T_{\ \textless \ 2, 2\ \textgreater \ } \][/tex]



Answer :

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]