Question

a new hospital tracked the number of births during its first 6 months of operation, as shown in the table. find a quadratic function that models the data. round numerical values to the nearest whole number.
$f(x)=\square x^{2}+\square x+\square$
use the function to predict the number of births for month 8. there will be about \square births.

Explanation:

Step1: Set up the system of equations

Let the quadratic function be \(f(x)=ax^{2}+bx + c\).
Substitute \(x = 1,y=137\): \(a\times1^{2}+b\times1 + c=137\), so \(a + b + c=137\).
Substitute \(x = 2,y = 110\): \(a\times2^{2}+b\times2 + c=110\), so \(4a+2b + c=110\).
Substitute \(x = 3,y = 93\): \(a\times3^{2}+b\times3 + c=93\), so \(9a+3b + c=93\).

Step2: Solve the system of equations

Subtract the first equation from the second equation:
\((4a + 2b + c)-(a + b + c)=110 - 137\)
\(4a+2b + c-a - b - c=-27\)
\(3a + b=-27\) (Equation 1)

Subtract the second equation from the third equation:
\((9a+3b + c)-(4a + 2b + c)=93 - 110\)
\(9a+3b + c-4a - 2b - c=-17\)
\(5a + b=-17\) (Equation 2)

Subtract Equation 1 from Equation 2:
\((5a + b)-(3a + b)=-17-(-27)\)
\(5a + b-3a - b = 10\)
\(2a=10\), so \(a = 5\)

Substitute \(a = 5\) into Equation 1: \(3\times5 + b=-27\), \(15 + b=-27\), \(b=-42\)

Substitute \(a = 5\) and \(b=-42\) into \(a + b + c=137\): \(5-42 + c=137\), \(c=174\)

Step3: Predict for \(x = 8\)

The function is \(f(x)=5x^{2}-42x + 174\)
When \(x = 8\), \(f(8)=5\times8^{2}-42\times8 + 174\)
\(=5\times64-336 + 174\)
\(=320-336 + 174\)
\(=158\)

Answer:

The quadratic function is \(f(x)=5x^{2}-42x + 174\) and the number of births in month 8 is \(158\)