determine the equation for the cubic function shown below. you may leave your answer in factored form. the…

Explanation:

Step1: Identify the roots and multiplicity

The graph touches the (x -)axis at (x = 0) (with a "bounce", so multiplicity (2)) and crosses the (x -)axis at (x = 4) (multiplicity (1)). So the factored form of a cubic function is (y=a(x - 0)^{2}(x - 4)=ax^{2}(x - 4)).

Step2: Determine the leading coefficient (a)

As (x\to+\infty), (y\to-\infty). For the function (y = ax^{2}(x - 4)=a(x^{3}-4x^{2})), when (x\to+\infty), the leading term is (ax^{3}). Since (x^{3}\to+\infty) as (x\to+\infty) and (y\to-\infty) as (x\to+\infty), then (a=- 1).

Answer:

(y=-x^{2}(x - 4))