Answer:
f(x) = x^3 - 3x^2 - 6x + 8
Explanation:
The graph has a local maximum and a local minimum, so it is a 3rd degree polynomial equation.
f(x) = ax^3 + bx^2 + cx + d
x = 0
f(0) = a(0^3) + b(0^2) + c(0) + d = 8
d = 8
f(x) = ax^3 + bx^2 + cx + 8
x = -2
f(-2) = a(-2)^3 + b(-2)^2 + c(-2) + 8 = 0
-8a + 4b - 2c = -8
4a - 2b + c = 4 Eq. 1
x = 1
f(1) = a + b + c + 8 = 0
a + b + c = -8 Eq. 2
x = 4
f(4) = a(4^3) + b(4^2) + c(4) + 8 = 0
64a + 16b + 4c = -8
16a + 4b + c = -2 Eq. 3
Equations 1, 2 and 3 form a system of equation in three variables, a, b, c.
4a - 2b + c = 4
a + b + c = -8
16a + 4b + c = -2
2 × Eq. 2 + Eq. 1
6a + 3c = -12
-4 × Eq.1 2 + Eq. 3
12a - 3c = 30
18a = 18
a = 1
6(1) + 3c = -12
3c = -18
c = -6
a + b + c = -8
1 + b - 6 = -8
b = -3
a = 1; b = -3; c = -6; d = 8
Answer:
f(x) = x^3 - 3x^2 - 6x + 8