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A parabola opening up or down has vertex $(0,2)$ and passes through $(4,-2)$ . 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

(0,2)

and passes through

(4,-2)

. Write its equation in vertex form.

\newline

Simplify any fractions.

Identify vertex form: Identify the vertex form of a 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.
Plug vertex coordinates: Plug the vertex coordinates into the vertex form. $\newline$ Since the vertex is given as $(0, 2)$ , we substitute $h = 0$ and $k = 2$ into the vertex form equation. $\newline$ $y = a(x - 0)^2 + 2$ $\newline$ $y = ax^2 + 2$
Use point to find 'a': Use the point $(4, -2)$ to find the value of 'a'. $\newline$ We know the parabola passes through the point $(4, -2)$ , so we can substitute $x = 4$ and $y = -2$ into the equation to solve for 'a'. $\newline$ $-2 = a(4)^2 + 2$ $\newline$ $-2 = 16a + 2$
Solve for 'a': Solve for 'a'. $\newline$ Subtract $2$ from both sides of the equation to isolate the term with 'a'. $\newline$ $-2 - 2 = 16a$ $\newline$ $-4 = 16a$ $\newline$ Divide both sides by $16$ to solve for 'a'. $\newline$ $a = -4 / 16$ $\newline$ $a = -1 / 4$
Write final equation: Write the final equation of the parabola in vertex form. $\newline$ Now that we have the value of $a$ , we can write the equation of the parabola. $\newline$ $y = (-\frac{1}{4})x^2 + 2$ $\newline$ This is the equation of the parabola in vertex form.

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