Question

write the equation in vertex form for the parabola with vertex (0,0) and focus (0, -9). simplify any fractions.

Explanation:

Step1: Determine the direction of the parabola

Since the vertex is \((0,0)\) and the focus is \((0, - 9)\), the parabola opens downwards.

Step2: Find the value of \(p\)

The distance between the vertex \((0,0)\) and the focus \((0,-9)\) is \(|p|\). Using the formula for the distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) which is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1 = 0,y_1 = 0,x_2 = 0,y_2=-9\), so \(d=\sqrt{(0 - 0)^2+(-9 - 0)^2}=9\). Since the parabola opens downwards, \(p=-9\).

Step3: Write the vertex - form equation of the parabola

The vertex - form equation of a parabola is \(x^{2}=4py\) (for parabolas that open up or down with vertex \((h,k)=(0,0)\)). Substitute \(p = - 9\) into the equation:

$$x^{2}=4(-9)y$$

$$x^{2}=-36y$$

$$y=-\frac{1}{36}x^{2}$$

Answer:

\(y =-\frac{1}{36}x^{2}\)