Given the table of values for the function [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 48 \\
\hline
-1 & 50 \\
\hline
0 & 48 \\
\hline
1 & 42 \\
\hline
2 & 32 \\
\hline
3 & 18 \\
\hline
4 & 0 \\
\hline
\end{array}
\][/tex]
We need to determine which of the given quadratic equations corresponds to the data in the table. The candidate equations are:
1. [tex]\( f(x) = -2x^2 - 4x + 48 \)[/tex]
2. [tex]\( f(x) = 2x^2 + 4x - 48 \)[/tex]
3. [tex]\( f(x) = x^2 + 2x - 24 \)[/tex]
4. [tex]\( f(x) = -x^2 - 2x + 24 \)[/tex]
We will match the values of [tex]\(x\)[/tex] and [tex]\(f(x)\)[/tex] from the table against each equation to find the correct one.
### Equation 1: [tex]\( f(x) = -2x^2 - 4x + 48 \)[/tex]
- For [tex]\(x = -2\)[/tex]:
[tex]\[
f(-2) = -2(-2)^2 - 4(-2) + 48 = -2(4) + 8 + 48 = -8 + 8 + 48 = 48
\][/tex]
- For [tex]\(x = -1\)[/tex]:
[tex]\[
f(-1) = -2(-1)^2 - 4(-1) + 48 = -2(1) + 4 + 48 = -2 + 4 + 48 = 50
\][/tex]
- For [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = -2(0)^2 - 4(0) + 48 = 0 + 0 + 48 = 48
\][/tex]
- For [tex]\(x = 1\)[/tex]:
[tex]\[
f(1) = -2(1)^2 - 4(1) + 48 = -2(1) - 4 + 48 = -2 - 4 + 48 = 42
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = -2(2)^2 - 4(2) + 48 = -2(4) - 8 + 48 = -8 - 8 + 48 = 32
\][/tex]
- For [tex]\(x = 3\)[/tex]:
[tex]\[
f(3) = -2(3)^2 - 4(3) + 48 = -2(9) - 12 + 48 = -18 - 12 + 48 = 18
\][/tex]
- For [tex]\(x = 4\)[/tex]:
[tex]\[
f(4) = -2(4)^2 - 4(4) + 48 = -2(16) - 16 + 48 = -32 - 16 + 48 = 0
\][/tex]
All the calculations match the given values in the table of values. Therefore, the equation that represents the given function [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = -2x^2 - 4x + 48
\][/tex]
