To determine which quadratic function is represented by the table, we need to evaluate each function at the given [tex]\( x \)[/tex] values and compare the results to the [tex]\( f(x) \)[/tex] values in the table.
The given [tex]\( x \)[/tex] values are [tex]\(-2, -1, 0, 1, 2\)[/tex] and the corresponding [tex]\( f(x) \)[/tex] values are [tex]\( 21, 10, 5, 6, 13 \)[/tex].
Let's evaluate each function one by one:
Option 1: [tex]\( f(x) = 3x^2 + 2x - 5 \)[/tex]
[tex]\[
\begin{aligned}
& f(-2) = 3(-2)^2 + 2(-2) - 5 = 3 \cdot 4 - 4 - 5 = 12 - 4 - 5 = 3 \\
& f(-1) = 3(-1)^2 + 2(-1) - 5 = 3 \cdot 1 - 2 - 5 = 3 - 2 - 5 = -4 \\
& f(0) = 3(0)^2 + 2(0) - 5 = 0 + 0 - 5 = -5 \\
& f(1) = 3(1)^2 + 2(1) - 5 = 3 \cdot 1 + 2 - 5 = 3 + 2 - 5 = 0 \\
& f(2) = 3(2)^2 + 2(2) - 5 = 3 \cdot 4 + 4 - 5 = 12 + 4 - 5 = 11 \\
\end{aligned}
\][/tex]
The values do not match the table values. Therefore, Option 1 is incorrect.
Option 2: [tex]\( f(x) = 3x^2 - 2x + 5 \)[/tex]
[tex]\[
\begin{aligned}
& f(-2) = 3(-2)^2 - 2(-2) + 5 = 3 \cdot 4 + 4 + 5 = 12 + 4 + 5 = 21 \\
& f(-1) = 3(-1)^2 - 2(-1) + 5 = 3 \cdot 1 + 2 + 5 = 3 + 2 + 5 = 10 \\
& f(0) = 3(0)^2 - 2(0) + 5 = 0 + 0 + 5 = 5 \\
& f(1) = 3(1)^2 - 2(1) + 5 = 3 \cdot 1 - 2 + 5 = 3 - 2 + 5 = 6 \\
& f(2) = 3(2)^2 - 2(2) + 5 = 3 \cdot 4 - 4 + 5 = 12 - 4 + 5 = 13 \\
\end{aligned}
\][/tex]
The values match the table values perfectly. Therefore, Option 2 is correct.
Option 3: [tex]\( f(x) = 2x^2 + 3x - 5 \)[/tex]
[tex]\[
\begin{aligned}
& f(-2) = 2(-2)^2 + 3(-2) - 5 = 2 \cdot 4 - 6 - 5 = 8 - 6 - 5 = -3 \\
& f(-1) = 2(-1)^2 + 3(-1) - 5 = 2 \cdot 1 - 3 - 5 = 2 - 3 - 5 = -6 \\
& f(0) = 2(0)^2 + 3(0) - 5 = 0 + 0 - 5 = -5 \\
& f(1) = 2(1)^2 + 3(1) - 5 = 2 \cdot 1 + 3 - 5 = 2 + 3 - 5 = 0 \\
& f(2) = 2(2)^2 + 3(2) - 5 = 2 \cdot 4 + 6 - 5 = 8 + 6 - 5 = 9 \\
\end{aligned}
\][/tex]
The values do not match the table values. Therefore, Option 3 is incorrect.
Option 4: [tex]\( f(x) = 2x^2 - 2x + 5 \)[/tex]
[tex]\[
\begin{aligned}
& f(-2) = 2(-2)^2 - 2(-2) + 5 = 2 \cdot 4 + 4 + 5 = 8 + 4 + 5 = 17 \\
& f(-1) = 2(-1)^2 - 2(-1) + 5 = 2 \cdot 1 + 2 + 5 = 2 + 2 + 5 = 9 \\
& f(0) = 2(0)^2 - 2(0) + 5 = 0 + 0 + 5 = 5 \\
& f(1) = 2(1)^2 - 2(1) + 5 = 2 \cdot 1 - 2 + 5 = 2 - 2 + 5 = 5 \\
& f(2) = 2(2)^2 - 2(2) + 5 = 2 \cdot 4 - 4 + 5 = 8 - 4 + 5 = 9 \\
\end{aligned}
\][/tex]
The values do not match the table values. Therefore, Option 4 is incorrect.
Thus, the quadratic function represented by the table is:
[tex]\[ f(x) = 3x^2 - 2x + 5 \][/tex]
