Abiologist 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

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

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