Question

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

Explanation:

Step1: Identify x-intercepts

The graph crosses/touches the x-axis at $x=-3$ (double root, since it touches the axis), $x=1$, and $x=2$. So the factored form starts as $y=a(x+3)^2(x-1)(x-2)$.

Step2: Use y-intercept to find a

The y-intercept is $(0,-30)$. Substitute $x=0$, $y=-30$:
$-30=a(0+3)^2(0-1)(0-2)$
$-30=a(9)(-1)(-2)$
$-30=18a$
$a=\frac{-30}{18}=-\frac{5}{3}$

Step3: Write final function

Substitute $a$ back into the factored form.

Answer:

$y=-\frac{5}{3}(x+3)^2(x-1)(x-2)$