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Question
Find function for X values 1, 3, 5, 7
Y values 3, 6, 11,18
Construct a difference table.
It looks like
1 3
3 6 3/2
5 11 5/2 1/4
7 18 7/2 1/4 0
The 1st column (1,3,5,7) is the X values.
The 2nd column (3,6,11,18) is the Y values.
The 3rd column (3/2, 5/2, 7/2) is found by taking the difference in the successive Y values over the difference in the successive X values. Thus, (6-3)/(3-1) = 3/2, (11-6)/(5-3) = 5/2, etc.
The 4th column (1/4, 1/4) is found by taking the successive elements of the 3rd column over the x value in that row and the X value 2 rows before. Thus, (5/2 - 3/2)/(5-1) = (2/2)/4 = 1/4 and
(7/2 - 5/2)/(7-3) = (2/2)/4 = 1/4.
This gives the polynomial [(1/4)(x-3) + 3/2](x-1) + 3.
That reduces to [(x-3+6)/4](x-1) + 3.
That is the same as (x+3)(x-1)/4 + 12/4.
That multiplies out to (x^2 + 2x - 3 + 12)/4.
Reducing the last term gives (x^2 + 2x + 9)/4.
It should be noted that the minimum value is at x = -1 and the value is 2. The function is an upward sloping parabola.
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Scott A Wilson
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I earned high honors in both my BS degree and MS degree from Oregon State.
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