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Write the equation in vertex form for the parabola with vertex $(0,0)$ and directrix $y = 8$ . $\newline$ Simplify any fractions. $\newline$ ______

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Write equations of parabolas in vertex form using properties

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Q. Write the equation in vertex form for the parabola with vertex

(0,0)

and directrix

y = 8

.

\newline

Simplify any fractions.

\newline

______

Identify Orientation: Identify the orientation of the parabola based on the directrix. $\newline$ Since the directrix is $y = 8$ , the parabola is vertical.
Determine Opening Direction: Determine the direction the parabola opens. $\newline$ The vertex $(0,0)$ is below the directrix $y = 8$ , so the parabola opens upward.
Calculate Distance: Calculate the distance between the vertex and the directrix. $\newline$ The distance is $|0 - 8| = 8$ .
Find Value of a: Find the value of a using the distance. $\newline$ The distance is $8$ , so $8 = \frac{1}{4a}$ . $\newline$ Solving for a gives $a = \frac{1}{4\times 8} = \frac{1}{32}$ .
Write Vertex Form Equation: Write the equation in vertex form. $\newline$ Substitute $a = \frac{1}{32}$ and the vertex $(h,k) = (0,0)$ into the vertex form equation $y = a(x-h)^2+k$ . $\newline$ $y = \frac{1}{32}(x-0)^2+0$ .
Simplify Equation: Simplify the equation. $y = \frac{1}{32} x^2$ .

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