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A parabola opening up or down has vertex $(0,3)$ and passes through $(-8,19)$ . 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

(0,3)

and passes through

(-8,19)

. 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.
Equation at $(0, 3)$ : What is the equation of a parabola with a vertex at $(0, 3)$ ? $\newline$ Substitute $0$ for $h$ and $3$ for $k$ in the vertex form. $\newline$ $y = a(x - 0)^2 + 3$ $\newline$ $y = ax^2 + 3$
Use Point $(-8, 19)$ : Use the point $(-8, 19)$ to find the value of $a$ . Replace the variables with $(-8, 19)$ in the equation. Substitute $-8$ for $x$ and $19$ for $y$ . $19 = a(-8)^2 + 3$ $19 = 64a + 3$
Solve for 'a': Solve for 'a'. $\newline$ $19 = 64a + 3$ $\newline$ Subtract $3$ from both sides. $\newline$ $16 = 64a$ $\newline$ Divide both sides by $64$ . $\newline$ $\frac{16}{64} = a$ $\newline$ $\frac{1}{4} = a$
Write Equation: Write the equation of the parabola with the value of 'a' found. $\newline$ Substitute $\frac{1}{4}$ for $a$ in the equation $y = ax^2 + 3$ . $\newline$ $y = \left(\frac{1}{4}\right)x^2 + 3$ $\newline$ Vertex form of the parabola: $y = \left(\frac{1}{4}\right)x^2 + 3$

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