Answer:
y = x^2 +10x -9
Explanation:
The quadratic regression functions of a spreadsheet or graphing calculator can do this easily. Input the given points and ask for the equation.
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If you must do this by hand, you can write and solve three equations in the unknown coefficients of the quadratic.
y = ax^2 +bx +c
Filling in the given point values, these equations are ...
-33 = a(-4)^2 +b(-4) +c = 16a -4b +c
2 = a(1)^2 +b(1) +c = a +b +c
162 = a(9)^2 +b(9) +c = 81a +9b +c
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There are algorithms available for solving general systems of three linear equations in three unknowns. It is often faster to take advantage of any relationships that show up between the coefficients. Of course, calculators and web sites are available that will solve these equations for you.
To solve these, we can subtract the second equation from the other two:
15a -5b = -35
80a +8b = 160
Removing common factors from each of these, we get ...
3a -b = -7
10a +b = 20
Adding these two equations gives ...
13a = 13
a = 1
Substituting into either of the two-variable equations, we find ...
b = 3a+7 = 10
And using these values in the second of the original equations, we can find c:
1 + 10 + c = 2
c = -9 . . . . . . . .subtract 11
Now, we know the quadratic can be written as ...
y = x^2 +10x -9