Question

Can someone help me with this question?
A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lakes population of waterfowl on each of the next six weeks. Find a quadratic function that models the data as a function of x, the number of weeks. Use the model to estimate the number of waterfowl at the lake on week 8.
week 0 1 2 3 4 5 6
population 635 644 719 860 1,067 1,340 1,769

Answer 1

First of all, you want to create a table


now you know that a quadratic takes the standard form
\[ax^{2}+bx+c\]

now if we let x be the number of weeks, and y be the population
then we know that when x=0, substitute it into the standard form of a quadratic, and we are left with

a(0)^2 +b(0)+c=c
so we know that from the table, if x=0, we get 635
this will be your constant c

Answer 2

now to figure out a, of the quadratic, we will find the rate of the rate at which it increases, this means
find the difference from one population size, to the next for example
then the difference of that difference


the "rate of the rate" is supposed to be the same throughout, which makes me think that the question has either made a mistake or it is an outlier, Anyways
this helps us because, we want to divide the rate of the rate by 2 to get a

so 66 is our rate of the rate, so 66/2=33

so, so far we have
ax^2 +bx + c= 33x^2 +bx +635

so now, all we have to do is substitute the x values into our unfinished quadratic

let x=0, let x=1, let x=2, etc

so we get

x=0, 635
x=1, 33+b + 635= 644
x=2, 33(2)^2 + 2b + 635=719
x=3, 33(3)^2 +3x +635=860
carry on for all values of x
so we have

644-668=b
b=-24

we also have

132 +2b +635=719
719-765=2b
b= -24

we also have 297+3b+635=860
860-932=3b
b=-24