To determine the correct equation of [tex]\( f(x) \)[/tex], we will compare the given data in the table with each of the provided functions:
1. Function [tex]\( f(x) = (x+5)^2 - 2 \)[/tex]:
- Calculate [tex]\( f(-8) \)[/tex]:
[tex]\[ (-8 + 5)^2 - 2 = (-3)^2 - 2 = 9 - 2 = 7 \][/tex]
This does not match the table value of 13.
2. Function [tex]\( f(x) = (x+4)^2 - 3 \)[/tex]:
- Calculate [tex]\( f(-8) \)[/tex] and compare to other values:
[tex]\[ f(-8) = (-8 + 4)^2 - 3 = (-4)^2 - 3 = 16 - 3 = 13 \][/tex]
[tex]\[ f(-7) = (-7 + 4)^2 - 3 = (-3)^2 - 3 = 9 - 3 = 6 \][/tex]
[tex]\[ f(-6) = (-6 + 4)^2 - 3 = (-2)^2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ f(-5) = (-5 + 4)^2 - 3 = (-1)^2 - 3 = 1 - 3 = -2 \][/tex]
[tex]\[ f(-4) = (-4 + 4)^2 - 3 = (0)^2 - 3 = 0 - 3 = -3 \][/tex]
[tex]\[ f(-3) = (-3 + 4)^2 - 3 = (1)^2 - 3 = 1 - 3 = -2 \][/tex]
[tex]\[ f(-2) = (-2 + 4)^2 - 3 = (2)^2 - 3 = 4 - 3 = 1 \][/tex]
[tex]\[ f(-1) = (-1 + 4)^2 - 3 = (3)^2 - 3 = 9 - 3 = 6 \][/tex]
[tex]\[ f(0) = (0 + 4)^2 - 3 = (4)^2 - 3 = 16 - 3 = 13 \][/tex]
All values match the table, indicating a potential solution.
3. Function [tex]\( f(x) = (x-4)^2 - 3 \)[/tex]:
- Calculate [tex]\( f(-8) \)[/tex]:
[tex]\[ (-8 - 4)^2 - 3 = (-12)^2 - 3 = 144 - 3 = 141 \][/tex]
This does not match the table value of 13.
4. Function [tex]\( f(x) = (x-5)^2 - 2 \)[/tex]:
- Calculate [tex]\( f(-8) \)[/tex]:
[tex]\[ (-8 - 5)^2 - 2 = (-13)^2 - 2 = 169 - 2 = 167 \][/tex]
This does not match the table value of 13.
Based on the calculations, the function that fits all the given values in the table is [tex]\( f(x) = (x+4)^2 - 3 \)[/tex].
Thus, the correct equation for [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = (x+4)^2 - 3 \][/tex]
