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

Explanation:

Step1: Set up the system of equations

Let the quadratic function be $f(x)=ax^{2}+bx + c$. Substitute the points $(x,y)$ from the table: For $x = 1,y=137$: $a(1)^{2}+b(1)+c=137$, so $a + b + c=137$. For $x = 2,y = 110$: $a(2)^{2}+b(2)+c=110$, so $4a+2b + c=110$. For $x = 3,y = 93$: $a(3)^{2}+b(3)+c=93$, so $9a+3b + c=93$.

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 new system of equations for $a$ and $b$

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

Step4: Solve for $c$

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

Step5: Predict the number of births for month 8

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

Answer:

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