The problem asks to perform cubic regression to find the coefficients of a cubic polynomial of the form:

$f(x) = ax^3 + bx^2 + cx + d$

which fits the points: $(-3, -28),\ (0, -4),\ (4, 168),\ (-5, -174).$

Steps to Solve:

1. Substitute the points into the cubic polynomial:
   Each point gives a linear equation in terms of $a, b, c, d$. For example:

   • At $(-3, -28)$:
     $a(-3)^3 + b(-3)^2 + c(-3) + d = -28$ Simplifies to:
     $-27a + 9b - 3c + d = -28$.
   • Repeat this process for the other points.

2. Form a system of equations:
   Use the same substitution for all 4 points to form a system of linear equations.

3. Solve the system:
   Solve the equations simultaneously to determine $a$, $b$, $c$, and $d$.

Let me calculate the solution for you. sympy import symbols, Eq, solve

Define the variables

a, b, c, d = symbols('a b c d')

Define the equations based on the given points

eq1 = Eq(a*(-3)*3 + b(-3)2 + c(-3) + d, -28) # Point (-3, -28) eq2 = Eq(a(0)*3 + b(0)2 + c(0) + d, -4) # Point (0, -4) eq3 = Eq(a(4)*3 + b(4)2 + c(4) + d, 168) # Point (4, 168) eq4 = Eq(a(-5)*3 + b(-5)*2 + c(-5) + d, -174) # Point (-5, -174)

Solve the system of equations

solution = solve([eq1, eq2, eq3, eq4], (a, b, c, d)) solutionThe coefficients of the cubic polynomial are:

$a = 2, \quad b = 3, \quad c = -1, \quad d = -4$

Thus, the cubic function is:

$f(x) = 2x^3 + 3x^2 - x - 4$

Let me know if you want a breakdown of the calculations or have any questions!

Related Questions:

1. What is the difference between linear and cubic regression?
2. How do you determine the degree of a polynomial needed for regression?
3. How do changes in one point affect the coefficients in polynomial regression?
4. Can polynomial regression always fit any set of points exactly?
5. What methods are available to solve systems of linear equations?

Tip:

Always verify the polynomial by substituting the points back into the equation to ensure accuracy!