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A parabola opening up or down has vertex $(0,5)$ and passes through $(-8,-3)$ . Write its equation in vertex form. $\newline$ Simplify any fractions.

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Write a quadratic function from its vertex and another point

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Q. A parabola opening up or down has vertex

(0,5)

and passes through

(-8,-3)

. Write its equation in vertex form.

\newline

Simplify any fractions.

Vertex Form: What is the vertex form of the parabola? $\newline$ The vertex form of a parabola is given by the equation $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.
Vertex Equation: What is the equation of a parabola with a vertex at $(0, 5)$ ? $\newline$ Substitute $0$ for $h$ and $5$ for $k$ in the vertex form. $\newline$ $y = a(x - 0)^2 + 5$ $\newline$ $y = ax^2 + 5$
Find Value of $a$ : Use the point $(-8, -3)$ to find the value of $a$ . Replace the variables with $(-8, -3)$ in the equation. Substitute $-8$ for $x$ and $-3$ for $y$ . $-3 = a(-8)^2 + 5$ $-3 = 64a + 5$
Solve for a: Solve for a. $\newline$ $-3 = 64a + 5$ $\newline$ Subtract $5$ from both sides. $\newline$ $-3 - 5 = 64a$ $\newline$ $-8 = 64a$ $\newline$ Divide both sides by $64$ . $\newline$ $-\frac{8}{64} = a$ $\newline$ $-\frac{1}{8} = a$
Write Parabola Equation: Write the equation of the parabola with $a = -\frac{1}{8}$ . Substitute $-\frac{1}{8}$ for $a$ in the equation $y = ax^2 + 5$ . $y = \left(-\frac{1}{8}\right)x^2 + 5$ Vertex form of the parabola: $y = -\left(\frac{1}{8}\right)x^2 + 5$

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