Question

find the equation of the quadratic function g whose graph is shown below.
$g(x)=\square$

Explanation:

Step1: Identify vertex form

A quadratic function's vertex form is $g(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex. From the graph, the vertex is $(3,3)$, so substitute $h=3, k=3$:
$g(x)=a(x-3)^2+3$

Step2: Solve for coefficient $a$

Use the point $(6,12)$ on the graph. Substitute $x=6, g(x)=12$ into the equation:
$12=a(6-3)^2+3$
Simplify the equation:
$12=9a+3$
Subtract 3 from both sides:
$9=9a$
Divide by 9:
$a=1$

Step3: Expand to standard form (optional, or keep vertex form)

Substitute $a=1$ back into vertex form, then expand:
$g(x)=(x-3)^2+3 = x^2-6x+9+3 = x^2-6x+12$

Answer:

$g(x)=(x-3)^2+3$ or $g(x)=x^2-6x+12$