To determine the correct quadratic equation for the given function [tex]\( C(t) \)[/tex], we start by examining the table of values for [tex]\( t \)[/tex] and [tex]\( C(t) \)[/tex] provided:
[tex]\[
\begin{array}{|c|c|}
\hline
t & C(t) \\
\hline
-2 & 1 \\
\hline
-1 & 4 \\
\hline
0 & 5 \\
\hline
1 & 4 \\
\hline
2 & 1 \\
\hline
\end{array}
\][/tex]
We are given four possible equations to consider:
1. [tex]\( C(t) = -(t-5)^2 \)[/tex]
2. [tex]\( C(t) = (t-5)^2 \)[/tex]
3. [tex]\( C(t) = -t^2 + 5 \)[/tex]
4. [tex]\( C(t) = t^2 + 5 \)[/tex]
Let's check these equations one by one against the given data points.
### Option 1: [tex]\( C(t) = -(t-5)^2 \)[/tex]
- For [tex]\( t = -2 \)[/tex]:
[tex]\[
C(-2) = -(-2-5)^2 = -(-7)^2 = -49
\][/tex]
This does not match the given [tex]\( C(t) = 1 \)[/tex].
Thus, this equation is not correct.
### Option 2: [tex]\( C(t) = (t-5)^2 \)[/tex]
- For [tex]\( t = -2 \)[/tex]:
[tex]\[
C(-2) = (-2-5)^2 = (-7)^2 = 49
\][/tex]
This does not match the given [tex]\( C(t) = 1 \)[/tex].
Thus, this equation is not correct.
### Option 3: [tex]\( C(t) = -t^2 + 5 \)[/tex]
- For [tex]\( t = -2 \)[/tex]:
[tex]\[
C(-2) = -(-2)^2 + 5 = -4 + 5 = 1
\][/tex]
This matches the given value.
- For [tex]\( t = -1 \)[/tex]:
[tex]\[
C(-1) = -(-1)^2 + 5 = -1 + 5 = 4
\][/tex]
This matches the given value.
- For [tex]\( t = 0 \)[/tex]:
[tex]\[
C(0) = -0^2 + 5 = 0 + 5 = 5
\][/tex]
This matches the given value.
- For [tex]\( t = 1 \)[/tex]:
[tex]\[
C(1) = -(1)^2 + 5 = -1 + 5 = 4
\][/tex]
This matches the given value.
- For [tex]\( t = 2 \)[/tex]:
[tex]\[
C(2) = -(2)^2 + 5 = -4 + 5 = 1
\][/tex]
This matches the given value.
Since all the provided data points fit this equation, [tex]\( C(t) = -t^2 + 5 \)[/tex] is the correct equation.
### Option 4: [tex]\( C(t) = t^2 + 5 \)[/tex]
- For [tex]\( t = -2 \)[/tex]:
[tex]\[
C(-2) = (-2)^2 + 5 = 4 + 5 = 9
\][/tex]
This does not match the given [tex]\( C(t) = 1 \)[/tex].
Thus, this equation is not correct.
Therefore, the correct quadratic function that fits the given data is:
[tex]\[
C(t) = -t^2 + 5
\][/tex]
