Question

Find the cubic function that models the data in the table below.
                                                                                                       
x
minus2
minus1
0
1
2
3
4
y
negative 36
negative 7
0
negative 3
negative 4
9
48
Question content area bottom
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. \( -8a + 4(-7 + a) - 2c = -36 \)
\[-8a -28 + 4a - 2c = -36\]
\[-4a - 2c = -8\]
\[2a + c = 4 \quad \text{(already derived)}\]

5. 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:
\[
\boxed{2x^3 - 5x^2}
\]