# Question #19618

Nov 2, 2016

Many answers possible, but one such answer is $y = - 3 {x}^{3} + \frac{3}{2} {x}^{2} + \frac{5}{2} x + 2$.

#### Explanation:

A cubic polynomial is of the form $y = A {x}^{3} + B {x}^{2} + C x + D$. 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 :

$2 B = 3$

$B = \frac{3}{2}$

Resubstitute:

$A + C = - \frac{1}{2}$

$- A + - C = \frac{1}{2}$

$- \left(A + C\right) = \frac{1}{2}$

$A + C = - \frac{1}{2}$

So, all values of A and C that add to $- \frac{1}{2}$ will work. Let's take $A = - 3$ and $C = \frac{5}{2}$.

So, the function is $y = - 3 {x}^{3} + \frac{3}{2} {x}^{2} + \frac{5}{2} x + 2$.

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

Hopefully this helps!