Question

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

Explanation:

Step1: Identify x-intercepts

The graph crosses the x-axis at $x=-1$, and touches (double root) at $x=4$. So the factors are $(x+1)$ and $(x-4)^2$. The function has the form $y=a(x+1)(x-4)^2$.

Step2: Use y-intercept to find a

The y-intercept is $(0,-8)$. Substitute $x=0$, $y=-8$ into the function:
$$-8=a(0+1)(0-4)^2$$
$$-8=a(1)(16)$$
$$a=\frac{-8}{16}=-\frac{1}{2}$$

Step3: Expand the function (optional, or keep factored)

Substitute $a=-\frac{1}{2}$ into the factored form:
$$y=-\frac{1}{2}(x+1)(x-4)^2$$
To expand fully:
First expand $(x-4)^2=x^2-8x+16$
Then multiply by $(x+1)$:
$$(x+1)(x^2-8x+16)=x^3-8x^2+16x+x^2-8x+16=x^3-7x^2+8x+16$$
Multiply by $-\frac{1}{2}$:
$$y=-\frac{1}{2}x^3+\frac{7}{2}x^2-4x-8$$

Answer:

$y=-\frac{1}{2}(x+1)(x-4)^2$ (or expanded form $y=-\frac{1}{2}x^3+\frac{7}{2}x^2-4x-8$)