Use cubic regression to fit a curve through the four points given in the table:
x=-3  -1  1  3

y=-9  21  7  -15



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Use cubic regression to fit a curve through the points (-3,-9),(-1,21),(1,7), and (3,-15).

(1) If you are allowed to use technology (a graphing calculator, Excel spreadsheet, or web sites such as Wolfram Alpha) you find the answer to be `ax^3+bx^2+cx+d=y` where `a=3/4,b=(-13)/4,c=(-31)/4, d=(69)/4` with  `r^2=1` .

(2) To compute by hand, set up the following system of equations:




Solve this system -- you could use substitution, linear combinations, Gaussian elimination, etc...

Using linear combinations, subtract the second, third, and fourth equations from the first equation to get:


We use combinations on this system:

`2a+2c=-14` or `a+c=-7`
`-54a-6c=6` or `-9a-c=1`

Again using combinations we get `-8a=-6 => a=3/4` .

Using back substitution we get `c=-7-3/4=-31/4`


Then the answer is :`ax^3+bx^2+cx+d=y` where `ax^3+bx^2+cx+d=y` where `a=3/4,b=(-13)/4,c=(-31)/4,d=(69)/4```

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