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Write the equation in vertex form for the parabola with vertex $(0,-2)$ and directrix $y = -9$ . $\newline$ 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 = -9

.

\newline

Simplify any fractions.

\newline

______

Vertex and Directrix Information: Vertex: $(0,-2)$ $\newline$ Directrix: $y = -9$ $\newline$ Since the directrix is horizontal, the parabola is vertical.
Vertex Form of Parabola: Vertex form of a vertical parabola: $y = a(x-h)^2 + k$
Direction of Parabola: The vertex is above the directrix, so the parabola opens upwards.
Distance Calculation: Distance between vertex and directrix: $|-2 - (-9)| = 7$
Calculation of 'a': The value of $a$ is $\frac{1}{4p}$ , where $p$ is the distance from the vertex to the focus (which is the same as the distance to the directrix). $\newline$ $a = \frac{1}{4\times7}$ $\newline$ $a = \frac{1}{28}$
Substitution into Vertex Form: Substitute $a = \frac{1}{28}$ , $h = 0$ , and $k = -2$ into the vertex form equation. $\newline$ $y = \frac{1}{28}(x - 0)^2 - 2$ $\newline$ $y = \frac{1}{28}x^2 - 2$

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