# Cubic FunctionWrite a cubic function that passes through the points: (-3, 0), (-1, 10), (0,0), (4, 0).

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### 1 Answer

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