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]
