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Write the equation in vertex form for the parabola with vertex $(0,-3)$ and focus $(0,4)$ . $\newline$ 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,-3)

and focus

(0,4)

.

\newline

Simplify any fractions.

\newline

______

Identify Parabola Orientation: Vertex: $(0,-3)$ $\newline$ Focus: $(0,4)$ $\newline$ Identify whether the parabola is vertical or horizontal. $\newline$ Since the $x$ -coordinates of the vertex and focus are the same, the parabola is vertical.
Vertex Form Explanation: Vertex form of a vertical parabola: $y = a(x-h)^2+k$ Here, $(h,k)$ is the vertex.
Determine Parabola Direction: Determine if the parabola opens upward or downward. $\newline$ The focus $(0,4)$ is above the vertex $(0,-3)$ , so the parabola opens upward.
Calculate Distance for 'a': Calculate the distance between the vertex and focus to find the value of 'a'. $\newline$ Distance: $|4 - (-3)| = 7$ $\newline$ The distance is the same as $\frac{1}{4a}$ , so $7 = \frac{1}{4a}$ .
Solve for 'a': Solve for 'a'. $\newline$ $\frac{1}{4a} = 7$ $\newline$ $a = \frac{1}{4\times7}$ $\newline$ $a = \frac{1}{28}$
Substitute Values into Equation: Substitute the values of $a$ , $h$ , and $k$ into the vertex form equation. $y = a(x-h)^2+k$ $y = \frac{1}{28}(x-0)^2+(-3)$ $y = \frac{1}{28}x^2 - 3$

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Question

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The given equation is graphed in the

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Question

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