To determine a cubic function of the form [tex]\( f(x) = ax^3 + bx^2 + cx + d \)[/tex] that fits the given points, we must utilize a method called cubic regression. Let's break down the steps and the resulting cubic function.
### Step-by-Step Solution:
1. Given Points:
The points we have are [tex]\((-2, -16)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((3, 59)\)[/tex], and [tex]\((6, 440)\)[/tex].
2. Dataset Representation:
- [tex]\( x \)[/tex] values are: [tex]\( [-2, 1, 3, 6] \)[/tex]
- [tex]\( y \)[/tex] values are: [tex]\( [-16, 5, 59, 440] \)[/tex]
3. Cubic Regression:
This process involves fitting a cubic polynomial of the form [tex]\( f(x) = ax^3 + bx^2 + cx + d \)[/tex] to the given data points to minimize the difference between the observed values and the values predicted by the polynomial.
4. Result of Regression:
After performing the cubic regression on the given data points, we obtain the following coefficients for the cubic polynomial:
- [tex]\( a \approx 1.9999999999999976 \)[/tex]
- [tex]\( b \approx 1.9911055729190188 \times 10^{-14} \)[/tex] (which is very close to 0 and can be considered negligible)
- [tex]\( c \approx 0.9999999999999597 \)[/tex]
- [tex]\( d \approx 2.0000000000000657 \)[/tex]
### Constructing the Cubic Function:
Given these coefficients, the cubic function that best fits the given data points is:
[tex]\[ f(x) = 2 \cdot x^3 + 0 \cdot x^2 + 1 \cdot x + 2 \][/tex]
Or, simplifying the negligible coefficient:
[tex]\[ f(x) = 2x^3 + x + 2 \][/tex]
### Final Answer:
The cubic polynomial that fits the given points [tex]\((-2, -16)\)[/tex], [tex]\((1, 5)\)[/tex], [tex]\((3, 59)\)[/tex], and [tex]\((6, 440)\)[/tex] is:
[tex]\[ f(x) = 2x^3 + x + 2 \][/tex]
