How to derive a cubic equation $ax^3+bx^2+cx+d =y$ from $x$ and $y$.

Question

Please let me show how to derive a cubic equation form $ax^3+bx^2+cx+d =y$ by using a set of $x$ and $y$ data. Simply the outline of the cubic equation derivation... Thank you

Answer 1

HINT. If all your points are exactly along a cubic function, what you want is to find the values of parameters (a,b,c,d). Then, take four points, since you have four unknowns, are write the corresponding equations
y1 = a x1 ^3 + b x1^2 + c x1 + d
y2 = a x2 ^3 + b x2^2 + c x2 + d
y3 = a x3 ^3 + b x3^2 + c x3 + d
y4 = a x4 ^3 + b x4^2 + c x4 + d
I suppose that you can see the matrix form.
If you prefer, you can easily find the solutions : eliminate "d" by substracting y1 from y2, y2 from y3, y3 from y4. Then, you now have three equations and three unknowns left (a,b,c). Repeat the process twice and you will end with the value of "a". Now compute backward "b", then "c", then "d".
If your points do not perfectly align on the cubic, then the problem is "polynomial regression" as mentioned by Zafer Sernikli.

Answer 2

Hint: Search for $3^{rd}$ order polynomial curve fitting. There you'll have your answer.