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Write the equation in vertex form for the parabola with vertex $(0,-7)$ and directrix $y = -4$ . $\newline$ 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 = -4

.

\newline

Simplify any fractions.

\newline

______

Identify orientation: Identify the orientation of the parabola. $\newline$ Since the directrix is horizontal $y = -4$ , the parabola is vertical.
Determine opening direction: Determine the direction the parabola opens. $\newline$ The vertex $(0,-7)$ is below the directrix $y = -4$ , so the parabola opens upward.
Calculate distance to directrix: Calculate the distance between the vertex and the directrix. $\newline$ Distance = $| -7 - (-4) | = |-7 + 4| = | -3 | = 3$
Find value of $a$ : Find the value of $a$ using the distance. $\newline$ The distance is equal to $\frac{1}{4a}$ , so $3 = \frac{1}{4a}$ . $\newline$ Solve for $a$ : $a = \frac{1}{4\times 3} = \frac{1}{12}$ .
Write equation in vertex form: Write the equation in vertex form. $\newline$ Vertex form for a vertical parabola is $y = a(x-h)^2 + k$ . $\newline$ Substitute $a = \frac{1}{12}$ , $h = 0$ , and $k = -7$ . $\newline$ $y = \frac{1}{12}(x - 0)^2 - 7$ $\newline$ $y = \frac{1}{12}x^2 - 7$

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