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Sagot :
To determine which equation describes the function modeled in the given table, we should compare the values generated by each function option with the values provided in the table.
Given the table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline $f(x)$ & 6 & 0 & -2 & 0 & 6 & 16 & 30 \\ \hline \end{tabular} \][/tex]
Let's evaluate each option:
### Option A: [tex]\( y = 2(x-1)^2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2-1)^2 = 2(-3)^2 = 18 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1-1)^2 = 2(-2)^2 = 8 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0-1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1-1)^2 = 2(0)^2 = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2-1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3-1)^2 = 2(2)^2 = 8 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4-1)^2 = 2(3)^2 = 18 \)[/tex]
The values [tex]\(6, 0, -2, 0, 6, 16, 30\)[/tex] do not match, so this equation is incorrect.
### Option B: [tex]\( y = 2x^2 - 2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2)^2 - 2 = 8 - 2 = 6 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1)^2 - 2 = 2 - 2 = 0 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0)^2 - 2 = 0 - 2 = -2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1)^2 - 2 = 2 - 2 = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2)^2 - 2 = 8 - 2 = 6 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3)^2 - 2 = 18 - 2 = 16 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4)^2 - 2 = 32 - 2 = 30 \)[/tex]
The values match perfectly, so this equation is correct.
### Option C: [tex]\( y = (2x-1)(x-1) \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = (2(-2)-1)((-2)-1) = (-5)(-3) = 15 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = (2(-1)-1)((-1)-1) = (-3)(-2) = 6 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = (2(0)-1)(0-1) = (-1)(-1) = 1 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = (2(1)-1)(1-1) = (1)(0) = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = (2(2)-1)(2-1) = (3)(1) = 3 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = (2(3)-1)(3-1) = (5)(2) = 10 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = (2(4)-1)(4-1) = (7)(3) = 21 \)[/tex]
The values [tex]\(15, 6, 1, 0, 3, 10, 21\)[/tex] do not match, so this equation is incorrect.
### Option D: [tex]\( y = 2(x+1)^2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2+1)^2 = 2(-1)^2 = 2 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1+1)^2 = 2(0)^2 = 0 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0+1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1+1)^2 = 2(2)^2 = 8 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2+1)^2 = 2(3)^2 = 18 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3+1)^2 = 2(4)^2 = 32 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4+1)^2 = 2(5)^2 = 50 \)[/tex]
The values [tex]\(2, 0, 2, 8, 18, 32, 50\)[/tex] do not match, so this equation is incorrect.
Therefore, the correct equation that describes the function modeled in the given table is:
B. [tex]\( y = 2x^2 - 2 \)[/tex]
Given the table:
[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline $f(x)$ & 6 & 0 & -2 & 0 & 6 & 16 & 30 \\ \hline \end{tabular} \][/tex]
Let's evaluate each option:
### Option A: [tex]\( y = 2(x-1)^2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2-1)^2 = 2(-3)^2 = 18 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1-1)^2 = 2(-2)^2 = 8 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0-1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1-1)^2 = 2(0)^2 = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2-1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3-1)^2 = 2(2)^2 = 8 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4-1)^2 = 2(3)^2 = 18 \)[/tex]
The values [tex]\(6, 0, -2, 0, 6, 16, 30\)[/tex] do not match, so this equation is incorrect.
### Option B: [tex]\( y = 2x^2 - 2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2)^2 - 2 = 8 - 2 = 6 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1)^2 - 2 = 2 - 2 = 0 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0)^2 - 2 = 0 - 2 = -2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1)^2 - 2 = 2 - 2 = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2)^2 - 2 = 8 - 2 = 6 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3)^2 - 2 = 18 - 2 = 16 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4)^2 - 2 = 32 - 2 = 30 \)[/tex]
The values match perfectly, so this equation is correct.
### Option C: [tex]\( y = (2x-1)(x-1) \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = (2(-2)-1)((-2)-1) = (-5)(-3) = 15 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = (2(-1)-1)((-1)-1) = (-3)(-2) = 6 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = (2(0)-1)(0-1) = (-1)(-1) = 1 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = (2(1)-1)(1-1) = (1)(0) = 0 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = (2(2)-1)(2-1) = (3)(1) = 3 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = (2(3)-1)(3-1) = (5)(2) = 10 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = (2(4)-1)(4-1) = (7)(3) = 21 \)[/tex]
The values [tex]\(15, 6, 1, 0, 3, 10, 21\)[/tex] do not match, so this equation is incorrect.
### Option D: [tex]\( y = 2(x+1)^2 \)[/tex]
Calculating [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( x = -2 \)[/tex]: [tex]\( y = 2(-2+1)^2 = 2(-1)^2 = 2 \)[/tex]
- [tex]\( x = -1 \)[/tex]: [tex]\( y = 2(-1+1)^2 = 2(0)^2 = 0 \)[/tex]
- [tex]\( x = 0 \)[/tex]: [tex]\( y = 2(0+1)^2 = 2(1)^2 = 2 \)[/tex]
- [tex]\( x = 1 \)[/tex]: [tex]\( y = 2(1+1)^2 = 2(2)^2 = 8 \)[/tex]
- [tex]\( x = 2 \)[/tex]: [tex]\( y = 2(2+1)^2 = 2(3)^2 = 18 \)[/tex]
- [tex]\( x = 3 \)[/tex]: [tex]\( y = 2(3+1)^2 = 2(4)^2 = 32 \)[/tex]
- [tex]\( x = 4 \)[/tex]: [tex]\( y = 2(4+1)^2 = 2(5)^2 = 50 \)[/tex]
The values [tex]\(2, 0, 2, 8, 18, 32, 50\)[/tex] do not match, so this equation is incorrect.
Therefore, the correct equation that describes the function modeled in the given table is:
B. [tex]\( y = 2x^2 - 2 \)[/tex]
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