Question

solution
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 $x=-1$ (touching, so double root) and $x=4$ (crossing, single root). So the factored form is $y=a(x+1)^2(x-4)$.

Step2: Solve for a using y-intercept

The y-intercept is $(0,-2)$. Substitute $x=0, y=-2$:
$-2=a(0+1)^2(0-4)$
$-2=a(1)(-4)$
$a=\frac{-2}{-4}=\frac{1}{2}$

Step3: Write the final function

Substitute $a=\frac{1}{2}$ into the factored form.

Answer:

$y=\frac{1}{2}(x+1)^2(x-4)$