#y(x) = ax^3 + bx^2 + cx + d#
We can see that #y(0) = -4#:
#y(x) = a(0)^3 + b(0)^2 + c(0) + d = -4#
#d = -4#
#y(x) = ax^3 + bx^2 + cx - 4#
We can see that #y(-2) = 0#
#0 = a(-2)^3 + b(-2)^2 + c(-2) - 4#
#-8a + 4b - 2c = 4#
We can see that #y(2) = 0#
#8a + 4b + 2c = 4#
We are given the point #(5, 31.5)#
#y(5) = a(5)^3 + b(5)^2 + c(5) - 4 = -31.5#
#125a + 25b + 5c = -27.5#
#25a + 5b + c = -5.5#
#50a + 10b + 2c = -11#
Write the 3 equations as an augmented matrix:
#[
(-8,4, -2, |, 4),
(8,4, 2, |, 4),
(50,10,2,|,-11)
]#
#[
(-8,4, -2, |, 4),
(0,8, 0, |, 8),
(50,10,2,|,-11)
]#
#[
(-8,4, -2, |, 4),
(0,1, 0, |, 1),
(50,10,2,|,-11)
]#
#[
(-8,0, -2, |, 0),
(0,1, 0, |, 1),
(50,0,2,|,-21)
]#
#[
(-8,0, -2, |, 0),
(0,1, 0, |, 1),
(42,0,0,|,-21)
]#
#[
(-8,0, -2, |, 0),
(0,1, 0, |, 1),
(1,0,0,|,-0.5)
]#
#[
(0,0, -2, |, -4),
(0,1, 0, |, 1),
(1,0,0,|,-0.5)
]#
#[
(0,0, 1, |, 2),
(0,1, 0, |, 1),
(1,0,0,|,-0.5)
]#
#a = -1/2, b = 1, and c = 2#
The equation is:
#y = -1/2x^3 + x^2 + 2x - 4#
