SOLUTION: 16 = if x=1 then y=2; if x=2 then y=4; if x=3 then y=7; if x=4 then y=11; if x=5 then y=16; What will be the formula to solve for Y if X is given?

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Question 424932: 16 = if x=1 then y=2;
if x=2 then y=4;
if x=3 then y=7;
if x=4 then y=11;
if x=5 then y=16;
What will be the formula to solve for Y if X is given?

Answer by htmentor(1343) (Show Source):
You can put this solution on YOUR website!
Assume the formula is a quadratic of the form
The following (x,y) points lie on the curve:
x y
1 2
2 4
3 7
4 11
5 16
Now we need to find the coefficients, a, b and c.
Using the first 3 points, we can generate a system of 3 equations with 3 unknowns:
(1) 2 = a + b + c
(2) 4 = 4a + 2b + c
(3) 7 = 9a + 3b + c
Solve for c in (1): 2 - a - b = c
Substitute into (2) and (3):
4 = 4a + 2b + (2 - a - b) (4)
7 = 9a + 3b + (2 - a - b) (5)
Now solve for b in terms of a in (4): 4 = 3a + b + 2 -> b = 2 - 3a
Substitute this value for b into (5): 7 = 9a + 3(2 - 3a) + 2 - a - (2 - 3a)
Solving for a gives a gives 7 = 6 + 2a -> a = 1/2
From (4), 4 = 4(1/2) + 2b + 2 - 1/2 - b -> 4 = 7/2 + b -> b = 1/2
Using (1), 2 = 1/2 + 1/2 + c -> c = 1
So, the formula is
Check using points 4 and 5:
11 = 1/2(4)^2 + 1/2(4) + 1 = 8 + 2 + 1 = 11
16 = 1/2(5)^2 + 1/2(5) + 1 = 25/2 + 5/2 + 1 = 16