Suppose f(x) is a quadratic function such that f(1) = -24, f(4) = 0, and f(7) = 50. Determine the value of f(-1).
Let the function be of the form
ax^2 + bx + c
We have this system of equations
a(1)^2 + b(1) + c = -24 ⇒ a + b + c = -24 (1)
a(4)^2 + b(4) + c = 0 ⇒ 16a + 4b + c = 0 (2)
a(7)^2 + b(7) + c = 50 ⇒ 49a + 7b + c = 50 (3)
Multiply ( 1) by -1 and add to (2) ⇒ 15a + 3b = 24 ⇒ 5a + b = 8 (4)
Multiply (1) by -1 and add to (3) ⇒ 48a + 6b = 74 ⇒ 24a + 3b = 37 (5)
Multiply (4) by -3 and add to (5) ⇒ 9a = 13 ⇒ a = 13/9
And 5 (13/9) + b = 8
65/9 + b = 72/9
b = 72/9 - 65/9 = 7/9
And (13/9) + (7/9) + c = -24
(20/9) + c = -24
c = -24 - 20/9 = -236/9
The function is (13/9)x^2 + (7/9) x - 236/9
And f(-1) = (13/9) - (7/9) - 236/9 = -230 / 9