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Write the equation in vertex form for the parabola with vertex $(0,-7)$ 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,-7)

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 Opening Direction: Determine the direction the parabola opens. $\newline$ The vertex $(0,-7)$ is below the directrix $y = -5$ , so the parabola opens upwards.
Calculate Distance: Calculate the distance between the vertex and the directrix. $\newline$ Distance = $|-7 - (-5)| = |-7 + 5| = |2| = 2$ .
Find Value of $a$ : Find the value of $a$ using the distance. $\newline$ The distance is equal to $\frac{1}{4a}$ , so $2 = \frac{1}{4a}$ . $\newline$ Solving for $a$ gives $a = \frac{1}{4\times 2} = \frac{1}{8}$ .
Write Equation in Vertex Form: Write the equation in vertex form. $\newline$ The vertex form is $y = a(x-h)^2 + k$ . $\newline$ Substitute $a = \frac{1}{8}$ , $h = 0$ , and $k = -7$ . $\newline$ $y = \frac{1}{8}(x-0)^2 - 7$ .
Simplify Equation: Simplify the equation. $y = \frac{1}{8}x^2 - 7$ .

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