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A parabola opening up or down has vertex $(0,3)$ and passes through $(-8,-13)$ . 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,-13)

. 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 $(0, 3)$ ? $\newline$ Substitute $0$ for $h$ and $3$ for $k$ in the vertex form. $\newline$ $y = a(x - 0)^2 + 3$ $\newline$ $y = ax^2 + 3$
Finding 'a' Value: Use the point $(-8, -13)$ to find the value of 'a'. $\newline$ Replace the variables with $(-8, -13)$ in the equation. $\newline$ Substitute $-8$ for $x$ and $-13$ for $y$ . $\newline$ $-13 = a(-8)^2 + 3$ $\newline$ $-13 = 64a + 3$
Solving for 'a': Solve for $a$ . $-13 = 64a + 3$ Subtract $3$ from both sides. $-16 = 64a$ Divide both sides by $64$ . $a = -\frac{16}{64}$ Simplify the fraction. $a = -\frac{1}{4}$
Final Parabola Equation: Write the equation of the parabola with the value of $a$ . $\newline$ Substitute $-\frac{1}{4}$ for $a$ in the equation $y = ax^2 + 3$ . $\newline$ $y = \left(-\frac{1}{4}\right)x^2 + 3$ $\newline$ The vertex form of the parabola is $y = -\left(\frac{1}{4}\right)x^2 + 3$ .

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