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Write the equation in vertex form for the parabola with vertex $(0,2)$ and directrix $y = 5$ . $\newline$ 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

(0,2)

and directrix

y = 5

.

\newline

Simplify any fractions.

\newline

______

Identify Orientation: Identify the orientation of the parabola. $\newline$ Since the directrix is horizontal $y=5$ , the parabola is vertical.
Determine Direction: Determine the direction the parabola opens. $\newline$ The vertex $(0,2)$ is below the directrix $y=5$ , so the parabola opens downward.
Calculate Distance: Calculate the distance between the vertex and the directrix. $\newline$ Distance = $|2 - 5| = 3$ .
Find Value of a: Find the value of $a$ using the distance. $\newline$ The distance is equal to $\frac{1}{4a}$ for a vertical parabola that opens downward, so $a = -\frac{1}{4\times 3}$ .
Simplify Value of a: Simplify the value of a. $\newline$ $a = -\frac{1}{12}$ .
Write Equation: Write the equation in vertex form. $\newline$ Vertex form is $y = a(x-h)^2 + k$ . $\newline$ Substitute $a = -\frac{1}{12}$ , $h = 0$ , and $k = 2$ . $\newline$ $y = -\frac{1}{12}(x-0)^2 + 2$ .
Simplify Equation: Simplify the equation. $y = -\frac{1}{12}x^2 + 2$ .

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