Question

Use cubic regression to find a function that fits the following points.

Answer 1

Answer:

The cubic regression that fits the points $(x,y) =(-2,-16)$, $(x,y) = (1,5)$, $(x,y) = (3, 59)$ and $(x,y) =(6,440)$ is $y =2\cdot x^{3}+x+2$.

Step-by-step explanation:

A cubic polynomial is a polynomial that has the following form:

$y = a\cdot x^{3}+b\cdot x^{2}+c\cdot x + d$ (1)

Where:

$x$ - Independent variable, dimensionless.

$y$ - Dependent variable, dimensionless.

$a$, $b$, $c$, $d$ - Coefficients, dimensionless.

We construct the following system of equations to determine the coefficients of the cubic regression:

$(x,y) =(-2,-16)$

$-8\cdot a +4\cdot b -2\cdot c + d = -16$ (1)

$(x,y) = (1,5)$

$a+b+c+d= 5$ (2)

$(x,y) = (3, 59)$

$27\cdot a + 9\cdot b + 3\cdot c + d = 59$ (3)

$(x,y) =(6,440)$

$216\cdot a + 36\cdot b + 6\cdot c + d = 440$ (4)

The solution of the system of linear equations is:

$a = 2$, $b = 0$, $c = 1$, $d = 2$

The cubic regression that fits the points $(x,y) =(-2,-16)$, $(x,y) = (1,5)$, $(x,y) = (3, 59)$ and $(x,y) =(6,440)$ is $y =2\cdot x^{3}+x+2$.