To determine the correct quadratic function [tex]\( C(t) \)[/tex] that fits the given data points, we will evaluate each provided function option against the data in the table.
Let’s review each given function:
1. [tex]\( C(t) = -(t-4)^2 \)[/tex]
2. [tex]\( C(t) = (t-4)^2 \)[/tex]
3. [tex]\( C(t) = -t^2 + 4 \)[/tex]
4. [tex]\( C(t) = t^2 + 4 \)[/tex]
We'll compare each function with the given data points:
[tex]\[
\begin{array}{|c|c|}
\hline
t & C(t) \\
\hline
2 & 4 \\
\hline
3 & 1 \\
\hline
4 & 0 \\
\hline
5 & 1 \\
\hline
6 & 4 \\
\hline
\end{array}
\][/tex]
### Checking the functions
1. For [tex]\( C(t) = -(t-4)^2 \)[/tex]:
- [tex]\( C(2) = -(2-4)^2 = -(2)^2 = -4 \)[/tex]
- [tex]\( C(3) = -(3-4)^2 = -(1)^2 = -1 \)[/tex]
- [tex]\( C(4) = -(4-4)^2 = -(0)^2 = 0 \)[/tex]
- [tex]\( C(5) = -(5-4)^2 = -(1)^2 = -1 \)[/tex]
- [tex]\( C(6) = -(6-4)^2 = -(2)^2 = -4 \)[/tex]
Calculated values: [tex]\([-4, -1, 0, -1, -4]\)[/tex]
2. For [tex]\( C(t) = (t-4)^2 \)[/tex]:
- [tex]\( C(2) = (2-4)^2 = (2)^2 = 4 \)[/tex]
- [tex]\( C(3) = (3-4)^2 = (1)^2 = 1 \)[/tex]
- [tex]\( C(4) = (4-4)^2 = (0)^2 = 0 \)[/tex]
- [tex]\( C(5) = (5-4)^2 = (1)^2 = 1 \)[/tex]
- [tex]\( C(6) = (6-4)^2 = (2)^2 = 4 \)[/tex]
Calculated values: [tex]\([4, 1, 0, 1, 4]\)[/tex]
3. For [tex]\( C(t) = -t^2 + 4 \)[/tex]:
- [tex]\( C(2) = -2^2 + 4 = -4 + 4 = 0 \)[/tex]
- [tex]\( C(3) = -3^2 + 4 = -9 + 4 = -5 \)[/tex]
- [tex]\( C(4) = -4^2 + 4 = -16 + 4 = -12 \)[/tex]
- [tex]\( C(5) = -5^2 + 4 = -25 + 4 = -21 \)[/tex]
- [tex]\( C(6) = -6^2 + 4 = -36 + 4 = -32 \)[/tex]
Calculated values: [tex]\([0, -5, -12, -21, -32]\)[/tex]
4. For [tex]\( C(t) = t^2 + 4 \)[/tex]:
- [tex]\( C(2) = 2^2 + 4 = 4 + 4 = 8 \)[/tex]
- [tex]\( C(3) = 3^2 + 4 = 9 + 4 = 13 \)[/tex]
- [tex]\( C(4) = 4^2 + 4 = 16 + 4 = 20 \)[/tex]
- [tex]\( C(5) = 5^2 + 4 = 25 + 4 = 29 \)[/tex]
- [tex]\( C(6) = 6^2 + 4 = 36 + 4 = 40 \)[/tex]
Calculated values: [tex]\([8, 13, 20, 29, 40]\)[/tex]
### Matching Values
- The only function whose calculated values exactly match the given data points [tex]\([4, 1, 0, 1, 4]\)[/tex] is [tex]\( C(t) = (t-4)^2 \)[/tex].
### Conclusion
The correct equation of [tex]\( C(t) \)[/tex] is:
[tex]\[ C(t) = (t-4)^2 \][/tex]
