Find the Parabola Through (-2,-20) with Vertex (0,-12) (-2,-20) , (0,-12)

$(- 2, - 20)$ , $(0, - 12)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, - 12)$ as the vertex $(h, k)$ and $(- 2, - 20)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 20 = a {(- 2 - (0))}^{2} - 12$

Step 2

Using $- 20 = a {(- 2 - (0))}^{2} - 12$ to solve for $a$ , $a = - 2$ .

Rewrite the equation as $a {(- 2 - (0))}^{2} - 12 = - 20$ .

$a {(- 2 - (0))}^{2} - 12 = - 20$

Simplify each term.

Subtract $0$ from $- 2$ .

$a {(- 2)}^{2} - 12 = - 20$

Raise $- 2$ to the power of $2$ .

$a \cdot 4 - 12 = - 20$

Move $4$ to the left of $a$ .

$4 a - 12 = - 20$

Move all terms not containing $a$ to the right side of the equation.

Add $12$ to both sides of the equation.

$4 a = - 20 + 12$

Add $- 20$ and $12$ .

$4 a = - 8$

Divide each term in $4 a = - 8$ by $4$ and simplify.

Divide each term in $4 a = - 8$ by $4$ .

$\frac{4 a}{4} = \frac{- 8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{- 8}{4}$

Divide $a$ by $1$ .

$a = \frac{- 8}{4}$

Simplify the right side.

Divide $- 8$ by $4$ .

$a = - 2$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(0, - 12)$ and $a = - 2$ is $y = (- 2) {(x - (0))}^{2} - 12$ .

$y = (- 2) {(x - (0))}^{2} - 12$

Step 4

Solve $y = (- 2) {(x - (0))}^{2} - 12$ for $y$ .

Remove parentheses.

$y = (- 2) {(x - (0))}^{2} - 12$

Multiply $- 2$ by ${(x - (0))}^{2}$ .

$y = - 2 {(x - (0))}^{2} - 12$

Remove parentheses.

$y = (- 2) {(x - (0))}^{2} - 12$

Subtract $0$ from $x$ .

$y = - 2 x^{2} - 12$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 2 x^{2} - 12$

Vertex Form: $y = (- 2) {(x - (0))}^{2} - 12$

Step 6

Simplify the standard form.

Standard Form: $y = - 2 x^{2} - 12$

Vertex Form: $y = - 2 x^{2} - 12$

Step 7

Find the Parabola Through (-2,-20) with Vertex (0,-12) (-2,-20) , (0,-12)

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