Question

select one:
a. $y = \frac{2}{27}(x + 3)(x + 1)(x - 3)^2$

b. $y = \frac{2}{9}(x + 3)(x + 1)(x - 3)$

c. $y = -\frac{2}{9}(x + 3)(x + 1)(x - 3)$

d. $y = 2(x + 3)(x + 1)(x - 3)$

Explanation:

Step1: Identify x-intercepts

The graph crosses x-axis at $x=-3$, $x=-1$, $x=3$. So the factors are $(x+3)$, $(x+1)$, $(x-3)$.

Step2: Check end behavior

As $x\to+\infty$, $y\to-\infty$, so leading coefficient is negative.

Step3: Solve for leading coefficient a

Use y-intercept $(0,2)$:
Substitute $x=0$, $y=2$ into $y=a(x+3)(x+1)(x-3)$:
$2 = a(0+3)(0+1)(0-3)$
$2 = a(3)(1)(-3)$
$2 = -9a$
$a = -\frac{2}{9}$

Step4: Form the equation

Substitute $a$ into the factored form: $y=-\frac{2}{9}(x+3)(x+1)(x-3)$

Answer:

C. $y = -\frac{2}{9}(x + 3)(x + 1)(x - 3)$