write a function in any form that would match the graph shown below.

Explanation:

Step1: Identify x - intercepts

The graph has x - intercepts at (x=- 4) and (x = 2). A polynomial function with these x - intercepts can be written in factored form (y=a(x + 4)(x - 2)).

Step2: Find the value of (a)

The graph passes through the point ((0,8)). Substitute (x = 0) and (y=8) into (y=a(x + 4)(x - 2)): [ \begin{align*} 8&=a(0 + 4)(0 - 2)\ 8&=a(4)(-2)\ 8&=-8a\ a&=-1 \end{align*} ]

Step3: Write the function

Substitute (a=-1) into (y=a(x + 4)(x - 2)), we get (y=-(x + 4)(x - 2)=-x^{2}-2x + 8).

Answer:

(y=-(x + 4)(x - 2)) (factored form) or (y=-x^{2}-2x + 8) (standard form)