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.

use the function to predict the number of births for month 8. there will be about births.

Explanation:

Step1: Set up the system of equations

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

Step2: Solve the system of equations

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

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

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

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

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

So \(f(x)=5x^{2}-42x + 174\)

Step3: Predict the number of births for month 8

Substitute \(x = 8\) into \(f(x)=5x^{2}-42x + 174\)
\(f(8)=5(8)^{2}-42(8)+174\)
\(f(8)=5\times64-336 + 174\)
\(f(8)=320-336 + 174\)
\(f(8)=158\)

Answer:

\(f(x)=5x^{2}-42x + 174\); \(158\)