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
percent 11.5 8.7 6.7 5.5 4.4 6.5 8.4 10.5 12.6
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 (40,4.4) as the vertex and (20,6.7) 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

Vertex $(h,k)=(40,4.4)$, so $h=40$, $k=4.4$. Substitute into vertex form:
$f(x)=a(x-40)^2 + 4.4$

Step2: Substitute given point (20,6.7)

Plug $x=20$, $f(x)=6.7$ into the equation:
$6.7 = a(20-40)^2 + 4.4$

Step3: Solve for $a$

First calculate $(20-40)^2=(-20)^2=400$. Rearrange to isolate $a$:
$6.7 - 4.4 = 400a$
$2.3 = 400a$
$a=\frac{2.3}{400}=0.00575$
Round to 4 decimal places: $a=0.0058$

Step4: Write final function

Substitute $a$, $h$, $k$ back into vertex form:
$f(x)=0.0058(x-40)^2 + 4.4$

Answer:

$f(x)=0.0058(x-40)^2 + 4.4$