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: Assume 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

Substituting $d = 6$ into $-a + b - c + d=9$ gives $-a + b - c=3$.
Substituting $d = 6$ into $a + b + c + d=5$ gives $a + b + c=-1$.
Substituting $d = 6$ into $8a+4b + 2c + d=18$ gives $8a+4b + 2c=12$.

Step4: Add the first - two new equations

$(-a + b - c)+(a + b + c)=3+( - 1)$, which simplifies to $2b = 2$, so $b = 1$.

Step5: Substitute $b = 1$ into equations

Substituting $b = 1$ into $-a + b - c=3$ gives $-a - c=2$.
Substituting $b = 1$ into $8a+4b + 2c=12$ gives $8a+2c=8$.

Step6: Multiply $-a - c=2$ by 2

$2(-a - c)=2\times2$, so $-2a-2c = 4$.

Step7: Add the new equations

$(8a + 2c)+(-2a-2c)=8 + 4$, which simplifies to $6a=12$, so $a = 2$.

Step8: Find $c$

Substitute $a = 2$ into $-a - c=2$, we get $-2 - c=2$, so $c=-4$.

Answer:

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