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

Explanation:

Step1: Identify the vertex - form of a quadratic function

The vertex - form of a quadratic function is $f(x)=a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola. From the graph, the vertex of the parabola is $(3,- 40)$. So, $h = 3$ and $k=-40$, and the function becomes $f(x)=a(x - 3)^2-40$.

Step2: Find the value of $a$

We can use another point on the graph. Let's use the $y$ - intercept. When $x = 0$, from the graph $y=-10$. Substitute $x = 0$ and $y=-10$ into $f(x)=a(x - 3)^2-40$: [ \begin{align*} -10&=a(0 - 3)^2-40\ -10&=9a-40\ 9a&=30\ a&=\frac{10}{3} \end{align*} ]

Step3: Write the function

Substitute $a=\frac{10}{3}$ into $f(x)=a(x - 3)^2-40$. The function is $f(x)=\frac{10}{3}(x - 3)^2-40$. Expanding it: [ \begin{align*} f(x)&=\frac{10}{3}(x^{2}-6x + 9)-40\ &=\frac{10}{3}x^{2}-20x+30 - 40\ &=\frac{10}{3}x^{2}-20x - 10 \end{align*} ]

Answer:

$f(x)=\frac{10}{3}(x - 3)^2-40$ (vertex - form) or $f(x)=\frac{10}{3}x^{2}-20x - 10$ (standard - form)