f(x) = 7x^2 -7x -3
We have 3 known points and 3 unknowns. So let's set up a series of equations to solve.
f(1) = -3;
a(1)^2 + b(1) + c = -3
a + b + c = -3
f(-2) = 39;
a(-2)^2 + b(-2) + c = 39
4a - 2b + c = 39
f(2) = 11;
a(2)^2 + b(2) + c = 11
4a + 2b + c = 11
Giving us the 3 equations to solve:
(1) a + b + c = -3
(2) 4a - 2b + c = 39
(3) 4a + 2b + c = 11
Let's subtract equation (2) from equation (3)
(4a + 2b + c = 11) - (4a - 2b + c = 39) = (4b = -28)
4b = -28
b = -7
Now let's add equations (2) and (3) together, then solve for c
(4a - 2b + c = 39) - (4a + 2b + c = 11) = (8a + 2c = 50)
8a + 2c = 50
4a + c = 25
c = 25 - 4a
Let's use the calculated value for b, and the equation for c and substitute into equation (1) above to calculate c.
a + b + c = -3
a + (-7) + (25 - 4a) = -3
a + (25 - 4a) = 4
-3a + 25 = 4
-3a = -21
a = 7
Finally, let's calculate c
c = 25 - 4a
c = 25 - 4*7
c = 25 - 28
c = -3
So the final desired equation is
f(x) = 7x^2 -7x -3
Let's verify with the known points.
x = 1
f(1) = 7(1)^2 - 7(1) - 3
f(1) = 7 - 7 - 3
f(1) = -3
x = -2
f(-2) = 7(-2)^2 - 7(-2) - 3
f(-2) = 7(4) + 14 - 3
f(-2) = 28 + 14 - 3
f(-2) = 39
x = 2
f(2) = 7(2^2) - 7(2) - 3
f(2) = 7(4) - 14 - 3
f(2) = 28 - 14 - 3
f(2) = 11
All three points are verified, so the function is f(x) = 7x^2 -7x -3
