Question

Question #19618

Answer 1

Many answers possible, but one such answer is `y = -3x^3 + 3/2x^2 + 5/2x + 2`.

Explanation:

A cubic polynomial is of the form `y = Ax^3 + Bx^2 + Cx + D`. Knowing the input/output of the function, we can write a system of equations in `4` variables.

`A + B + C + D = 3`

`D = 2`

`-A + B - C + D = 4`

We instantly know that `D = 2`.

So, `A + B + C = 1` and `-A + B - C = 2`. By elimination, we have that :

`2B = 3`

`B = 3/2`

Resubstitute:

`A + C = -1/2`

`-A + -C = 1/2`

`-(A + C) = 1/2`

`A + C = -1/2`

So, all values of A and C that add to `-1/2` will work. Let's take `A = -3` and `C = 5/2`.

So, the function is `y = -3x^3 + 3/2x^2 + 5/2x + 2`.

Checking, you will find that the function passes through the points labelled above.

Hopefully this helps!