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A parabola opening up or down has vertex $(0,1)$ and passes through $(-8,9)$ . 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

(0,1)

and passes through

(-8,9)

. Write its equation in vertex form.

\newline

Simplify any fractions.

Vertex Form Explanation: 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 with Vertex: What is the equation of a parabola with a vertex at $(0, 1)$ ? $\newline$ Substitute $0$ for $h$ and $1$ for $k$ in the vertex form. $\newline$ $y = a(x - 0)^2 + 1$ $\newline$ $y = ax^2 + 1$
Use Point to Find 'a': Use the point $(-8, 9)$ to find the value of 'a'. $\newline$ Replace the variables with $(-8, 9)$ in the equation. $\newline$ Substitute $-8$ for $x$ and $9$ for $y$ . $\newline$ $9 = a(-8)^2 + 1$ $\newline$ $9 = 64a + 1$
Solve for 'a': Solve for 'a'. $\newline$ $9 = 64a + 1$ $\newline$ Subtract $1$ from both sides. $\newline$ $8 = 64a$ $\newline$ Divide both sides by $64$ . $\newline$ $\frac{8}{64} = a$ $\newline$ $\frac{1}{8} = a$
Write Equation with 'a': Write the equation of the parabola with the value of 'a'. $\newline$ Substitute $\frac{1}{8}$ for $a$ in the equation $y = ax^2 + 1$ . $\newline$ $y = \left(\frac{1}{8}\right)x^2 + 1$ $\newline$ Vertex form of the parabola: $y = \left(\frac{1}{8}\right)x^2 + 1$

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