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Write the equation in vertex form for the parabola with vertex $(0,-2)$ and focus $(0,-8)$ . $\newline$ Simplify any fractions. $\newline$ ______

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Write equations of parabolas in vertex form using properties

Full solution

Q. Write the equation in vertex form for the parabola with vertex

(0,-2)

and focus

(0,-8)

.

\newline

Simplify any fractions.

\newline

______

Identify Parabola Orientation: Vertex: $(0,-2)$ $\newline$ Focus: $(0,-8)$ $\newline$ 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: Vertex: $(0,-2)$ $\newline$ Focus: $(0,-8)$ $\newline$ Determine if the parabola opens upward or downward. $\newline$ Since the focus is below the vertex, the parabola opens downward.
Calculate Distance for 'a': Vertex: $(0,-2)$ $\newline$ Focus: $(0,-8)$ $\newline$ Calculate the distance between vertex and focus to find the value of ' $a$ '. $\newline$ Distance: $|-8 - (-2)| = |-8 + 2| = 6$
Calculate 'a' Value: Distance between the vertex and focus: $6$ $\newline$ The value of 'a' is negative because the parabola opens downward. $\newline$ $a = -\frac{1}{4p}$ , where $p$ is the distance from the vertex to the focus. $\newline$ $a = -\frac{1}{4\times 6}$ $\newline$ $a = -\frac{1}{24}$
Substitute Values in Equation: We found: $\newline$ $a = -\frac{1}{24}$ $\newline$ Vertex $(h,k) = (0,-2)$ $\newline$ Substitute $-\frac{1}{24}$ for $a$ , $0$ for $h$ and $-2$ for $k$ in the vertex form equation. $\newline$ $y = -\frac{1}{24}(x-0)^2-2$ $\newline$ $y = -\frac{1}{24} x^2 - 2$

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