Write the cubic function whose graph is shown.

Question

(-2,0) (-1,2) (0,0) (1,0) <-- Coordinates/Points

Write the cubic function whose graph is shown.

f(x)=?

Answer 1

A cubic function can be written as

y = a(x - x₁)(x - x₂)(x - x₃),

where x₁, x₂, and x₃ are the zeros, or where the function hits the x-axis (y = 0), and a determines how a function is stretched, and how the cubic will behave at very high and low values of x. This is almost the same as a quadratic, except one more root is added.

We are given 3 coordinates with a y = 0, and the x-coordinates of these are -2, 0, and 1. This follows the definition of a "zero" in the above paragraph, so substitute these into the cubic function and we get

y = a(x - 0)(x + 2)(x - 1) = a*x(x + 2)(x - 1).

To determine the value of a, plug in the last remaining coordinate (-1, 2) into the equation:

2 = a(-1)(-1 + 2)(-1 - 1) = a(-1)(1)(-2) = 2a

⇒ a = 2/2 = 1

So, our final function is

y = x(x + 2)(x - 1).

If you want to write this in standard form, expand the above equation to get

y = x³ + x² - 2x (Both forms are correct).