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

Can you say "successive difference"?

644 - 635 = 9
719 - 644 = 75
860 - 719 = 141
1067 - 860 = 207
1340 - 1067 = 273
1679 - 1340 = 429

Doesn't look like much, does it? Do it again!

75 - 9 = 66
141 - 75 = 66
207 - 141 = 66
273 - 207 = 66
429 - 273 = 156

Those first four are very suspicious. Seems like a quadratic equation would be a great model. Then that last one is a little annoying.

Answer 2

the answer should look like p(x)=33x^2_24x+635; either 2,555 or 2,084 waterfowl. Just not sure how to get to this

Answer 3

Yup. There's that quadratic equation. What tools have you for solving systems of equations?

Answer 4

y=ax^2+bxc

Answer 5

"+c", I hope. What after that? Least Squares? Generalized Matrix Inverse? Hard Core solution of multiple equations? You'll have to provide a clue what sort of material you are working on.

Answer 6

would you use 0 1 2 for the three different equations

Answer 7

That's why I asked. A quadratic equation has only three parameters. With seven (7) data points, it's just not a natural fit for a simple substitution and system-solve. Are yousure youhaven't been studying "Least Squares"?

Answer 8

On the other hand, the Least Squares solution doesn't match the lovely "66" was saw in teh 2nd Diferences. I'm torn.

Answer 9

y=a(0)^2+b(0)+c
y=a(1)^2+b(1)+c
y=a(2)^2+b(2)+c

Answer 10

Okay, if you're going to do that, youshoudl get the rest of it right and then solve away!

635=a(0)^2+b(0)+c
644=a(1)^2+b(1)+c
719=a(2)^2+b(2)+c

Let's see what you get.

Answer 11

BTW, that's a horrible solution, but it does see to be what is wanted. It completely ignores that last values and it is only by coincidence that it includes the 4th, 5th, and 6th values. Oh, well. It is hard to write a good question.

Answer 12

644=a(1)^2+b(1)+635
644=a+b+635
a+b=9
719=a(2)^2+b(2)+635
719=a+b+635
a+b=84

Answer 13

719=4a+2b+635
4a+2b=84
2a+b=42

Answer 14

What happened to these?
c = 635
a+b=9
a+b=84

Solve that. :-)