Question

finding and using a quadratic regression model 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.

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

Explanation:

Step1: Define quadratic model form

The quadratic function is $f(x)=ax^2+bx+c$, where $x$ is the month, $f(x)$ is the number of births, and $a,b,c$ are constants to solve for.

Step2: Set up system of equations

Substitute 3 data points into the model:

• For $x=1, f(x)=137$: $a(1)^2+b(1)+c=137 \implies a+b+c=137$
• For $x=2, f(x)=110$: $a(2)^2+b(2)+c=110 \implies 4a+2b+c=110$
• For $x=3, f(x)=93$: $a(3)^2+b(3)+c=93 \implies 9a+3b+c=93$

Step3: Solve the system

Subtract first equation from second:
$(4a+2b+c)-(a+b+c)=110-137 \implies 3a+b=-27$
Subtract second equation from third:
$(9a+3b+c)-(4a+2b+c)=93-110 \implies 5a+b=-17$
Subtract the new first result from the new second result:
$(5a+b)-(3a+b)=-17-(-27) \implies 2a=10 \implies a=5$
Substitute $a=5$ into $3a+b=-27$:
$3(5)+b=-27 \implies 15+b=-27 \implies b=-42$
Substitute $a=5,b=-42$ into $a+b+c=137$:
$5-42+c=137 \implies -37+c=137 \implies c=174$

Step4: Finalize the quadratic model

Round values to whole numbers (already whole): $f(x)=5x^2-42x+174$

Step5: Predict births for month 8

Substitute $x=8$ into the model:
$f(8)=5(8)^2-42(8)+174=5(64)-336+174=320-336+174$

Answer:

$f(x)=5x^2 - 42x + 174$
There will be about 158 births