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 $x=-3$ and $x=8$. So the factors are $(x+3)$ and $(x-8)$. We can write the quadratic function as $y=a(x+3)(x-8)$.

Step2: Use y-intercept to find a

The y-intercept is $(0,-24)$. Substitute $x=0$, $y=-24$ into the function:
$$-24=a(0+3)(0-8)$$
Simplify the right side:
$$-24=a(3)(-8) \implies -24=-24a$$
Solve for $a$:
$$a=\frac{-24}{-24}=1$$

Step3: Write the final function

Substitute $a=1$ back into the factored form:
$$y=(x+3)(x-8)$$
(This can also be expanded to standard form: $y=x^2-5x-24$)

Answer:

$y=(x+3)(x-8)$ (or $y=x^2-5x-24$)