Question

A country's census lists the population of the country as 247 million in 1990, 269 million in 2000, and 287 million in 2010. Fit a second-degree polynomial passing through these three points. (Let the year 2000 be x = 0 and let p(x) represent the population in millions.)

Answer 1

Answer:

$p(x)=-\frac{1}{50}x^2+2x+269$

Step-by-step explanation:

Each of those sets of data can be written as a coordinate.  In order:

(-10, 247)  (0, 269)   (10, 287)

We need to fill in the equation:

$y=ax^2+bx+c$ so we use those equations to give us the a, b, and c we are looking for.  Start with the coordinate that has an x value of 0.  Filling in:

$269=a(0)^2+b(0)+c$ which simplifies to

$269=c$  We can use that c value now in the other 2 equations to solve for a and b.  We will use the third equation now:

$287=a(10)^2+b(10)+269$ simplifies to

287 = 100a + 10b + 269 and

18 = 100a + 10b

Now doing the same with the first coordinate:

$247=a(-10)^2+b(-10)+269$ simplifies to

247 = 100a - 10b + 269 and

-22 = 100a - 10b

Solve the 2 bold equations by elimination/addition method:

    18   =    100a   +   10b

   -22  =    100a   -    10b

The b's eliminate each other automatically, leaving us with:

-4 = 100a so

a = -1/50

Filling back in with that value of a:

$100(-\frac{1}{50})+10b=18$ simplifies to

-2 + 10b = 18 and

10b = 20 so

b = 2

Now we have our quadratic:

$p(x)=-\frac{1}{50}x^2+2x+269$

Thi i an upside down parabola that tells you the population will continue to grow to the ma value of the vertex (yr, population), then will decline after the vertex value.