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A parabola opening up or down has vertex $(0,0)$ and passes through $(-4,4)$ . Write its equation in vertex form. $\newline$ Simplify any fractions.

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Write a quadratic function from its vertex and another point

Full solution

Q. A parabola opening up or down has vertex

(0,0)

and passes through

(-4,4)

. Write its equation in vertex form.

\newline

Simplify any fractions.

Vertex Form: What is the vertex form of the parabola? $\newline$ Vertex form of a parabola: $y = a(x - h)^2 + k$
Equation at $(0,0)$ : What is the equation of a parabola with a vertex at $(0, 0)$ ? $\newline$ Substitute $0$ for $h$ and $0$ for $k$ in vertex form. $\newline$ $y = a(x - 0)^2 + 0$ $\newline$ $y = ax^2$
Substitute $(-4,4)$ : $y = ax^2$ $\newline$ Replace the variables with $(-4, 4)$ in the equation. $\newline$ Substitute $-4$ for $x$ and $4$ for $y$ . $\newline$ $4 = a(-4)^2$ $\newline$ $4 = 16a$
Solve for $a$ : $4 = 16a$ $\newline$ Solve for $a$ . $\newline$ $4 = 16a$ $\newline$ $\frac{4}{16} = a$ $\newline$ $\frac{1}{4} = a$
Equation with $a=\frac{1}{4}$ : $y = ax^2$ $\newline$ What is the equation of the parabola if $a = \frac{1}{4}$ ? $\newline$ Substitute $\frac{1}{4}$ for $a$ . $\newline$ $y = (\frac{1}{4})x^2$ $\newline$ Vertex form of the parabola: $y = (\frac{1}{4})x^2$

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Solve by completing the square.

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