Find the cubic function that models the data in the table below.

Question

                                                                                                       
x
minus2
minus1
0
1
2
3
4
y
negative 36
negative 7
0
negative 3
negative 4
9
48
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Part 1
yequals
  
enter your response here
​(Simplify your answer. Do not factor. Use integers or decimals for any numbers in the expression. Round to three decimal places as​ needed.)

Answer 1

To find the cubic function that models the given data, we can express the cubic function in the standard form:

\[ y = ax^3 + bx^2 + cx + d \]

where \(a\), \(b\), \(c\), and \(d\) are coefficients we need to determine. Given the points from the table:

\[ \begin{align*} (-2, -36) \ (-1, -7) \ (0, 0) \ (1, -3) \ (2, -4) \ (3, 9) \ (4, 48) \ \end{align*} \]

Let's set up a system of equations using these points.

Substituting each point into the cubic equation:

1. For \((-2, -36)\): \[-36 = a(-8) + b(4) + c(-2) + d\] \[ -8a + 4b - 2c + d = -36 \quad \text{(1)}\]

2. For \((-1, -7)\): \[-7 = a(-1) + b(1) + c(0) + d\] \[ -a + b + d = -7 \quad \text{(2)}\]

3. For \((0, 0)\): \[0 = d\] \[d = 0 \quad \text{(3)}\]

4. For \((1, -3)\): \[-3 = a(1) + b(1) + c(1) + d\] \[a + b + c + d = -3\] Substituting \(d = 0\) gives us: \[a + b + c = -3 \quad \text{(4)}\]

5. For \((2, -4)\): \[-4 = a(8) + b(4) + c(2) + d\] \[8a + 4b + 2c = -4 \quad \text{(5)}\]

6. For \((3, 9)\): \[9 = a(27) + b(9) + c(3) + d\] \[27a + 9b + 3c = 9 \quad \text{(6)}\]

7. For \((4, 48)\): \[48 = a(64) + b(16) + c(4) + d\] \[64a + 16b + 4c = 48 \quad \text{(7)}\]

Using equation (3): From equation (3), we have \(d = 0\).

Revising equations: Substituting \(d = 0\) into equations (1), (2), (4), (5), (6), and (7):

• (1): \( -8a + 4b - 2c = -36\)
• (2): \( -a + b = -7\)
• (4): \( a + b + c = -3\)
• (5): \( 8a + 4b + 2c = -4\)
• (6): \( 27a + 9b + 3c = 9\)
• (7): \( 64a + 16b + 4c = 48\)

We can express these equations and solve for \(a\), \(b\), and \(c\).

From equation (2): \[ b = -7 + a \]

Substituting \(b\) into equation (4): \[ a + (-7 + a) + c = -3 \] \[ 2a + c - 7 = -3 \] \[ 2a + c = 4 \quad \text{(8)}\]

Now substituting for \(b\) in equations (1), (5), (6) and (7):

1. \( -8a + 4(-7 + a) - 2c = -36 \) \[-8a -28 + 4a - 2c = -36\] \[-4a - 2c = -8\] \[2a + c = 4 \quad \text{(already derived)}\]

2. Substituting \(b\) into: \( 8a + 4(-7 + a) + 2c = -4 \) \[ 8a - 28 + 4a + 2c = -4 \] \[ 12a + 2c = 24 \] \[ 6a + c = 12 \quad \text{(9)}\]

Solve (8) and (9): From (8): \[ c = 4 - 2a \]

Substituting into (9): \[ 6a + (4 - 2a) = 12 \] \[ 4a = 8 \] \[ a = 2 \]

Substituting \(a = 2\) back into (8): \[ 2(2) + c = 4 \] \[ 4 + c = 4 \] \[ c = 0 \]

Now substituting back to find \(b\): \[ b = -7 + a = -7 + 2 = -5 \]

So, we have:

• \(a = 2\)
• \(b = -5\)
• \(c = 0\)
• \(d = 0\)

Thus, the cubic function is: \[ y = 2x^3 - 5x^2 \]

So the final answer for the cubic function is: \[ \boxed{2x^3 - 5x^2} \]