Question

the graph shows a parabola and its focus. write the equation of the parabola in vertex form.

Explanation:

Step1: Identify vertex of parabola

The vertex $(h,k)$ is at $(0,0)$ (where the parabola intersects the origin).

Step2: Identify focus of parabola

The focus is at $(0,-4)$. For a downward-opening parabola, the focus is $(h, k-p)$, so $k-p = -4$. Since $k=0$, solve for $p$:
$0-p = -4 \implies p=4$

Step3: Use vertex form formula

The vertex form of a downward-opening parabola is $y = \frac{-1}{4p}(x-h)^2 + k$. Substitute $h=0$, $k=0$, $p=4$:
$y = \frac{-1}{4(4)}(x-0)^2 + 0$

Step4: Simplify the equation

$y = \frac{-1}{16}x^2$

Answer:

$y = -\frac{1}{16}x^2$