The curve that describes the cubic function passes through
the given point, if and only if the coordinates of the points verify the expression of
the function.
We'll write the cubic
function:
f(x) = ax^3 + bx^2 + cx +
d
The point (-3, 0) belongs to the graph of cubic function
if and only if:
f(-3) =
0
f(-3) = a*(-3)^3 + b*(-3)^2 + c*(-3) +
d
f(-3) = -27a + 9b - 3c +
d
-27a + 9b - 3c + d = 0
(1)
The point (-1, 10) belongs to the graph of cubic
function if and only if:
f(-1) =
10
f(-1) = a*(-1)^3 + b*(-1)^2 + c*(-1) +
d
f(-1) = -a + b - c + d
-a +
b - c + d = 10 (2)
The point (0,0) belongs to the graph of
cubic function if and only if:
f(0) =
0
d = 0
The point (4, 0)
belongs to the graph of cubic function if and only if:
f(4)
= 0
a*(4)^3 + b*(4)^2 + 4c + d =
0
64a + 16b + 4c = 0 (3)
We'll
form the system from the equtaions (1),(2),(3):
-27a + 9b -
3c = 0
-a + b - c = 10
64a +
16b + 4c = 0
We'll multiply (2) by (-3) and we'll add to
(1):
3a - 3b + 3c - 27a + 9b - 3c =
30
We'll combine and eliminate like
terms:
-24a + 6b = 30
We'll
divide by 6:
-4a + b = 5
(4)
We'll multiply (2) by 4 and we'll add to
(3):
-4a + 4b - 4c + 64a + 16b + 4c =
40
We'll combine and eliminate like
terms:
60a + 20b = 40
We'll
divide by 20:
3a + b = 2
(5)
We'll multiply (5) by (-1) and we'll add it to
(4):
-3a - b - 4a + b = 5 -
2
-7a = 3
a =
-3/7
We'll substitute a in
(5):
-9/7 + b = 2
b = 2 +
9/7
b =
23/7
We'll substitute a and b in
(2):
-a + b - c = 10
3/7 +
23/7 - c = 10
26/7 - c = 10
c
= 26/7 - 10
c =
-44/7
The cubic function
is:
f(x) = (-3/7)x^3 +
(23/7)x^2 - (44/7)x
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