write a quadratic function $f(x)$ for the given graph. write the equation in standard form.

Explanation:

Step1: Identify x-intercepts

The graph crosses the x-axis at $x=-2$ and $x=6$. So the factored form is $f(x)=a(x+2)(x-6)$.

Step2: Use y-intercept to find a

The y-intercept is $(0,6)$. Substitute $x=0, f(x)=6$: $6=a(0+2)(0-6)$ $6=a(2)(-6)$ $6=-12a$ $a=\frac{6}{-12}=-\frac{1}{2}$

Step3: Expand to standard form

First expand the factored form with $a=-\frac{1}{2}$: $f(x)=-\frac{1}{2}(x+2)(x-6)$ Multiply $(x+2)(x-6)=x^2-6x+2x-12=x^2-4x-12$ Then distribute $-\frac{1}{2}$: $f(x)=-\frac{1}{2}x^2 + 2x + 6$

Answer:

$f(x)=-\frac{1}{2}x^2 + 2x + 6$