State the function for the corresponding data set
{({1,2,3,4,5,6,7}{9,9.5,10.25,11.25 ,12.5,14,15.75}): }
domain range

Given:
x: [1      2         3        4      5    6        7]
y: [9  9.5  10.25  11.25  12.5  14  15.75]

Let the function which models the data be
y(x) = ax² + bx + c

Apply the function to the first three data points
  a +   b + c = 9           .(1)
4a + 2b + c = 9.5        (2)
9a + 3b + c = 10.25    (3)

Subtract (1) from (2).
3a + b = 0.5
or
b = 0.5 - 3a               (4)
Subtract (1) from (3).
8a + 2b = 1.25          (5)
Substitute (4) into (5).
8a + 2(0.5 - 3a) = 1.25
2a = 0.25
a  = 0.125
From (4),
b = 0.5-3*0.125 = 0.125
From (1),
c = 9 - 0.125 - 0.125 = 8.75

The function is
y = 8.75 + 0.125x + 0.125x²

Check that the function works for x=4,5,6,7.
From the calculator, obtain
y(4) = 11.25  Correct
y(5) = 12.5  Correct
y(6) = 14      Correct
y(7) = 15.75 Correct

The graph of the function is shown below.

 y = 8.75 + 0.125x + 0.125x²

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