Algebra help needed

Question

The table below shows the profit, p(x), in dollars from selling x items

X 1 2 3 6 9 14

Y 90.6 106.4 112.4 71.6 -57.4 -468.4

Using your quadratic regression, estimate the profit from selling 12 items? 

Answer 1

I ran this quadratic regression in R (code inserted below), and found that it is matched up 100% with the data. This leads me to believe that you should be able to just do this using linear algebra. I've set up the table using ax² + bx + c = y (where we know what the x values are based on the table).

a b c y

1 1 1 90.6

4 2 1 106.4

9 3 1 112.4

36 6 1 71.6

81 9 1 -57.4

196 14 1 -468.4

Feel free to use any 3 lines of this matrix to solve for a, b, and c. Then your formula would be

p(12) = a(144) + b(12) + c.

I got this from R.

x=c(1,2,3,6,9,14)

y=c(90.6,106.4,112.4,71.6,-57.4,-468.4)

x2=x^2

quadreg=lm(y~x+x2)

This generates the model was y = -4.9x² + 30.5x + 6.5. This gives a prediction p(12) = -333.1