Question

A biologist took a count of the number of migrating waterfowl at a particular lake, and recounted the lake's population of waterfowl on each of the next six weeks.
Week 0 1 2 3 4 5 6
Population 660 668 738 870 1064 1320 1638

Estimate the number of waterfowl at the lake on week 7

Answer 1

Quadratic function ( in standard form ):

y = a x² + b x + c

x = 0:

635 = 0 + 0 + c

c = 635

x = 1:

644 = a + b + 635

a + b = 9

x = 2:

719 = 4 a + 2 b + 635

4 a + 2 b = 84

and now we have a system:

a + b = 9   /  * ( - 2 )

4 a + 2 b = 84

-------------------------

- 2 a + 2 b = - 18

+

 4 a + 2 b  = 84

-------------------------

  2 a = 66

 a = 33;   b = - 24;

 Function is:

 f ( x ) = 33 x² - 24 x + 635

f ( 8 ) = 33 * 64 - 24 * 8 + 635 = 2,112 - 192 + 635 = 2,555

 Answer: The number of waterfowl on week 8 would be 2,555.