determine an equation for the pictured graph. write your answer in factored form. remember to start with…

Explanation:

Step1: Identify the x - intercepts

The x - intercepts (roots) of the polynomial function are the points where the graph crosses or touches the x - axis. From the graph, the x - intercepts are (x=-2), (x = 0), and (x=2). So the factors are ((x + 2)), (x), and ((x - 2)). So (f(x)=a(x + 2)x(x - 2)).

Step2: Find the value of (a)

We can use another point on the graph to find the value of (a). Let's use the point ((1,4)). Substitute (x = 1) and (y=f(1)=4) into (f(x)=a(x + 2)x(x - 2)). [ \begin{align*} 4&=a(1 + 2)\times1\times(1 - 2)\ 4&=a\times3\times1\times(-1)\ 4&=- 3a\ a&=-\frac{4}{3} \end{align*} ]

Step3: Write the equation

Substitute (a =-\frac{4}{3}) into (f(x)=a(x + 2)x(x - 2)). The equation of the function is (y=-\frac{4}{3}x(x + 2)(x - 2)).

Answer:

(y =-\frac{4}{3}x(x + 2)(x - 2))