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The table represents a quadratic function \( C(t) \).

[tex]\[
\begin{array}{|c|c|}
\hline
t & C(t) \\
\hline
-2 & 7 \\
\hline
-1 & 4 \\
\hline
0 & 3 \\
\hline
1 & 4 \\
\hline
2 & 7 \\
\hline
\end{array}
\][/tex]

What is the equation of \( C(t) \)?

A. \( C(t) = -(t-3)^2 \)

B. \( C(t) = (t-3)^2 \)

C. \( C(t) = -t^2 + 3 \)

D. [tex]\( C(t) = t^2 + 3 \)[/tex]


Sagot :

Given the data points \((-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7)\), we are to determine the quadratic function \(C(t)\) that fits these points.

### Step-by-Step Solution:

1. Identify the general form of a quadratic function:
[tex]\[ C(t) = at^2 + bt + c \][/tex]

2. Find the coefficients \(a\), \(b\), and \(c\):
- Using the given data points, we can fit a quadratic polynomial \(C(t) = at^2 + bt + c\).

3. Determine the coefficients using curve fitting (in practice, this can be solved through methods like the least squares method, but we use the already provided result):
- The coefficients \(a\), \(b\), and \(c\) for the quadratic function best fitting the given data points are:
[tex]\[ a = 1.0000000000000002, \quad b = 0.0, \quad c = 3.0 \][/tex]

4. Form the equation:
- Substitute \(a\), \(b\), and \(c\) into the general quadratic form:
[tex]\[ C(t) = 1.0000000000000002 t^2 + 0.0 t + 3.0 \][/tex]
- Simplify the equation:
[tex]\[ C(t) = t^2 + 3 \][/tex]

5. Verify the function with the given options:
- Among the provided options, we need to match the quadratic function we derived:
[tex]\[ \text{Option 1: } C(t) = -(t-3)^2 \][/tex]
[tex]\[ \text{Option 2: } C(t) = (t-3)^2 \][/tex]
[tex]\[ \text{Option 3: } C(t) = -t^2 + 3 \][/tex]
[tex]\[ \text{Option 4: } C(t) = t^2 + 3 \][/tex]

- The derived function \(C(t) = t^2 + 3\) matches Option 4.

### Conclusion:

The equation of \(C(t)\) that fits the given data points is:
[tex]\[ C(t) = t^2 + 3 \][/tex]