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Question

Write a function to model the data.

x​	−1​	0​	1​	2​	3​
y​	0​	−6​	−6​	0​	12​

The function is y=​ .

Answer 1

-1 and 2 are both zeros of the function.

Therefore f(x) = a(x + 1)(x - 2).

Uses any other point to determine value of a.

Answer 2

You can fit the quartic polynomical: ax⁴+bx³+cx²+dx+e=y

You then can plug the values in for x and y and get 5 simultaneous equations for the unknown constants.

-1: a-b+c-d+e = 0

0: e=-6

1: a+b+c+d+e = -6 or a+b+c+d = 0

2: 16a+8b+4c+2d+e =0

3: 81a+27b+9c+3d+e = 12

You can solve this to get {a,b,c,d,e} = {0,0,3,3,-6}

You therefore could have tried a quadratic of the form a(x+1)(x-2) = y using the zeros and having one arbitrary constant. You can use the x = 0 point to solve for a (= 3). and get the same answer. The other two points would now have to fit (which they do)