Question

find the equation of the quadratic function g whose graph is shown below.
g(x) =
(-3, -3)
(-2, -6)
try again

Explanation:

Step1: Identify vertex form

The vertex form of a quadratic function is $g(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex.

Step2: Plug in vertex coordinates

The vertex is $(-2,-6)$, so substitute $h=-2$, $k=-6$:
$g(x)=a(x+2)^2-6$

Step3: Solve for $a$ using given point

Use the point $(-3,-3)$: substitute $x=-3$, $g(x)=-3$
$-3=a(-3+2)^2-6$
$-3=a(1)^2-6$
$-3=a-6$
$a=-3+6=3$

Step4: Expand to standard form (optional, but complete)

Substitute $a=3$ into vertex form:
$g(x)=3(x+2)^2-6$
Expand: $g(x)=3(x^2+4x+4)-6=3x^2+12x+12-6=3x^2+12x+6$

Answer:

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