Answer:
[tex]p(x)=-\frac{1}{50}x^2+2x+269[/tex]
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:
[tex]y=ax^2+bx+c[/tex] 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:
[tex]269=a(0)^2+b(0)+c[/tex] which simplifies to
[tex]269=c[/tex] 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:
[tex]287=a(10)^2+b(10)+269[/tex] simplifies to
287 = 100a + 10b + 269 and
18 = 100a + 10b
Now doing the same with the first coordinate:
[tex]247=a(-10)^2+b(-10)+269[/tex] 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:
[tex]100(-\frac{1}{50})+10b=18[/tex] simplifies to
-2 + 10b = 18 and
10b = 20 so
b = 2
Now we have our quadratic:
[tex]p(x)=-\frac{1}{50}x^2+2x+269[/tex]
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.
