Question

use cubic regression to find a function that fits the following points. (-1, 9) (0, 6) (1, 5) (2, 18) ?x³ + x² + x +

Explanation:

Step1: Set up the cubic function

Let the cubic function be $y = ax^{3}+bx^{2}+cx + d$.

Step2: Substitute the points

For point $(-1,9)$: $-a + b - c + d=9$.
For point $(0,6)$: $d = 6$.
For point $(1,5)$: $a + b + c + d=5$.
For point $(2,18)$: $8a+4b + 2c + d=18$.

Step3: Substitute $d = 6$ into other equations

$-a + b - c+6 = 9$ simplifies to $-a + b - c=3$.
$a + b + c+6 = 5$ simplifies to $a + b + c=-1$.
$8a+4b + 2c+6 = 18$ simplifies to $8a+4b + 2c = 12$.

Step4: Add the first - two new equations

$(-a + b - c)+(a + b + c)=3+( - 1)$ gives $2b = 2$, so $b = 1$.

Step5: Substitute $b = 1$ into the equations

$-a + 1 - c=3$ becomes $-a - c=2$.
$a + 1 + c=-1$ becomes $a + c=-2$.
$8a+4\times1 + 2c = 12$ becomes $8a+2c = 8$.

Step6: Solve the system of equations for $a$ and $c$

From $-a - c=2$ we have $c=-a - 2$.
Substitute $c=-a - 2$ into $8a+2c = 8$:
$8a+2(-a - 2)=8$.
$8a-2a-4 = 8$.
$6a=12$, so $a = 2$.
If $a = 2$, then $c=-a - 2=-4$.

Answer:

$2x^{3}+1x^{2}-4x + 6$