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A parabola opening up or down has vertex $(0,-7)$ and passes through $(-12,11)$ . 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,-7)

and passes through

(-12,11)

. Write its equation in vertex form.

\newline

Simplify any fractions.

Vertex form of parabola: What is the vertex form of the parabola? $\newline$ Vertex form of a parabola: $y = a(x - h)^2 + k$
Equation with vertex: What is the equation of a parabola with a vertex at $(0, -7)$ ? $\newline$ Substitute $0$ for $h$ and $-7$ for $k$ in vertex form. $\newline$ $y = a(x - 0)^2 + (-7)$ $\newline$ $y = ax^2 - 7$
Substitute values: $y = ax^2 - 7$ $\newline$ Replace the variables with $(-12, 11)$ in the equation. $\newline$ Substitute $-12$ for $x$ and $11$ for $y$ . $\newline$ $11 = a(-12)^2 - 7$ $\newline$ $11 = 144a - 7$
Solve for $a$ : $11 = 144a - 7$ $\newline$ Solve for $a$ . $\newline$ $11 + 7 = 144a$ $\newline$ $18 = 144a$ $\newline$ $\frac{18}{144} = a$ $\newline$ $\frac{1}{8} = a$
Equation with $a=\frac{1}{8}$ : $y = ax^2 - 7$ $\newline$ What is the equation of the parabola if $a = \frac{1}{8}$ ? $\newline$ Substitute $\frac{1}{8}$ for $a$ . $\newline$ $y = (\frac{1}{8})x^2 - 7$ $\newline$ Vertex form of the parabola: $y = (\frac{1}{8})x^2 - 7$

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Solve by completing the square.

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