How to write an equation when given a cubic function graphed.

[GrannySmith]

Alright, so lets say you are given a graph. You know it is a cubic function, and you are told to write an equation for that cubic function. How would you go about doing so?

Do you just use the parent function, and then see what translations were made by using an x y table? I've had some difficulty doing this, and it seems very hard.

Any other easier methods?

 

[richardt]

If the function is a polynomial, you may wish to consider the factor theorem if your graph is explicit with regard to zeros.

Rich

 

[HallsofIvy]

Alright, so lets say you are given a graph. You know it is a cubic function, and you are told to write an equation for that cubic function. How would you go about doing so?

Do you just use the parent function, and then see what translations were made by using an x y table? I've had some difficulty doing this, and it seems very hard.

Any other easier methods?

A cubic function is if the form \(\displaystyle y= ax^3+ bx^2+ cx+ d\). Since you have four unknown coefficients, you will need four equations to find them. Choose any four points on your graph and substitute for x and y to get those four equations.

Equivalently, use "Lagrange' formula": the cubic function whose graph passes through \(\displaystyle (x_0, y_0)\), \(\displaystyle (x_1, y_1)\), \(\displaystyle (x_1, y_1)\), \(\displaystyle (x_2, y_2)\), and \(\displaystyle (x_3, y_3)\) is
\(\displaystyle y_0\frac{(x- x_1)(x- x_2)(x- x_3)}{(x_0- x_1)(x_0- x_2)(x_0- x_3)}+ y_1\frac{(x- x_0)(x- x_2)(x- x_3)}{(x_1- x_0)(x_1- x_2)(x_1- x_3)}+ y_2\frac{(x- x_0)(x- x_1)(x- x_3)}{(x_2- x_0)(x_2- x_1)(x_2- x_3)}+ y_3\frac{(x- x_0)(x- x_1)(x- x_2)}{(x_3- x_0)(x_3- x_1)(x_3- x_2)}\)