Question

a new hospital tracked the number of births during its first 6 months of operation, as shown in the table.

month	number of births
2	110
3	93
4	70
5	65
6	77

find a quadratic function that models the data. round numerical values to the nearest whole number.
f(x) = x²+ x +
use the function to predict the number of births for month 8. there will be about births.

Explanation:

Step1: Set up the quadratic function

A quadratic function is of the form $f(x)=ax^{2}+bx + c$. We have the points $(1,137),(2,110),(3,93)$ etc. Substitute the first three points into the function:
When $x = 1$, $f(1)=a\times1^{2}+b\times1 + c=a + b + c=137$.
When $x = 2$, $f(2)=a\times2^{2}+b\times2 + c=4a+2b + c=110$.
When $x = 3$, $f(3)=a\times3^{2}+b\times3 + c=9a+3b + c=93$.

Step2: Subtract equations to eliminate $c$

Subtract the first - equation from the second:
$(4a + 2b + c)-(a + b + c)=110 - 137$
$4a+2b + c - a - b - c=-27$
$3a + b=-27$.
Subtract the second - equation from the third:
$(9a + 3b + c)-(4a+2b + c)=93 - 110$
$9a+3b + c - 4a - 2b - c=-17$
$5a + b=-17$.

Step3: Solve the system of equations for $a$ and $b$

Subtract the equation $3a + b=-27$ from $5a + b=-17$:
$(5a + b)-(3a + b)=-17+27$
$5a + b - 3a - b = 10$
$2a=10$
$a = 5$.
Substitute $a = 5$ into $3a + b=-27$:
$3\times5 + b=-27$
$15 + b=-27$
$b=-42$.

Step4: Solve for $c$

Substitute $a = 5$ and $b=-42$ into $a + b + c=137$:
$5-42 + c=137$
$-37 + c=137$
$c = 174$.
So the quadratic function is $f(x)=5x^{2}-42x + 174$.

Step5: Predict the number of births for month 8

Substitute $x = 8$ into $f(x)=5x^{2}-42x + 174$:
$f(8)=5\times8^{2}-42\times8 + 174$
$=5\times64-336 + 174$
$=320-336 + 174$
$=158$.

Answer:

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