Question

the table below lists the percent of a countrys population that is foreign-born for selected years.

year	1930	1940	1950	1960	1970	1980	1990	2000	2008

answer parts a through e.
a.
b.
c.
d.
0,80,10 by 0,15,3
(b) find a quadratic function $f(x)=a(x-h)^2 + k$ that models the data. use (40,4.3) as the vertex and (20,6.6) as the other point to determine a.
$f(x) = \square$
(use integers or decimals for any numbers in the expression. type an integer or decimal rounded to four decimal places as needed.)

Explanation:

Step1: Identify vertex values

The vertex form is $f(x)=a(x-h)^2+k$, with vertex $(h,k)=(40,4.3)$. Substitute $h=40$, $k=4.3$:
$f(x)=a(x-40)^2+4.3$

Step2: Substitute the point (20,6.6)

Plug $x=20$, $f(x)=6.6$ into the equation:
$6.6=a(20-40)^2+4.3$

Step3: Solve for $a$

First calculate $(20-40)^2=(-20)^2=400$:
$6.6=400a+4.3$
Subtract 4.3 from both sides:
$6.6-4.3=400a$
$2.3=400a$
Solve for $a$:
$a=\frac{2.3}{400}=0.00575$
Round to 4 decimal places: $a=0.0058$

Step4: Write the final function

Substitute $a=0.0058$, $h=40$, $k=4.3$ back into vertex form:
$f(x)=0.0058(x-40)^2+4.3$

Answer:

$f(x)=0.0058(x-40)^2+4.3$