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A parabola has a vertex at $(-4,-2)$ and goes through the point $(-8,30)$ . Determine its equation in vertex form and rewrite equation in standard form.

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Write a quadratic function from its vertex and another point

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Q. A parabola has a vertex at

(-4,-2)

and goes through the point

(-8,30)

. Determine its equation in vertex form and rewrite equation in standard form.

Identify vertex values: Identify the vertex values $h$ and $k$ from the given vertex $(-4, -2)$ . $h = -4$ , $k = -2$
Write vertex form equation: Write the vertex form equation using the values of $h$ and $k$ . $\newline$ Vertex form: $y = a(x - h)^2 + k$ $\newline$ Substitute $h = -4$ and $k = -2$ : $\newline$ $y = a(x + 4)^2 - 2$
Find value of $a$ : Use the point $(-8, 30)$ to find the value of $a$ . Substitute $x = -8$ and $y = 30$ into the equation: $30 = a(-8 + 4)^2 - 2$ $30 = a(-4)^2 - 2$ $30 = 16a - 2$ $32 = 16a$ $a = \frac{32}{16}$ $a = 2$
Substitute value of $a$ : Substitute the value of $a$ back into the vertex form equation. $y = 2(x + 4)^2 - 2$
Expand to standard form: Expand the vertex form to standard form. $\newline$ $y = 2(x^2 + 8x + 16) - 2$ $\newline$ $y = 2x^2 + 16x + 32 - 2$ $\newline$ $y = 2x^2 + 16x + 30$

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