Find the Parabola Through (0,-12) with Vertex (-2,-20) (-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 $(- 2, - 20)$ as the vertex $(h, k)$ and $(0, - 12)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

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

Step 2

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

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

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

Simplify $a {(0 - (- 2))}^{2} - 20$ .

Subtract $- 2$ from $0$ .

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

Simplify each term.

Multiply $- 1$ by $- 2$ .

$a \cdot 2^{2} - 20 = - 12$

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

$a \cdot 4 - 20 = - 12$

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

$4 a - 20 = - 12$

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

Add $20$ to both sides of the equation.

$4 a = - 12 + 20$

Add $- 12$ and $20$ .

$4 a = 8$

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 $(- 2, - 20)$ and $a = 2$ is $y = (2) {(x - (- 2))}^{2} - 20$ .

$y = (2) {(x - (- 2))}^{2} - 20$

Step 4

Solve $y = (2) {(x - (- 2))}^{2} - 20$ for $y$ .

Remove parentheses.

$y = (2) {(x - (- 2))}^{2} - 20$

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

$y = 2 {(x - (- 2))}^{2} - 20$

Remove parentheses.

$y = (2) {(x - (- 2))}^{2} - 20$

Simplify $(2) {(x - (- 2))}^{2} - 20$ .

Simplify each term.

Multiply $- 1$ by $- 2$ .

$y = 2 {(x + 2)}^{2} - 20$

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$y = 2 ((x + 2) (x + 2)) - 20$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$y = 2 (x (x + 2) + 2 (x + 2)) - 20$

Apply the distributive property.

$y = 2 (x \cdot x + x \cdot 2 + 2 (x + 2)) - 20$

Apply the distributive property.

$y = 2 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 20$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = 2 (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 20$

Move $2$ to the left of $x$ .

$y = 2 (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 20$

Multiply $2$ by $2$ .

$y = 2 (x^{2} + 2 x + 2 x + 4) - 20$

Add $2 x$ and $2 x$ .

$y = 2 (x^{2} + 4 x + 4) - 20$

Apply the distributive property.

$y = 2 x^{2} + 2 (4 x) + 2 \cdot 4 - 20$

Simplify.

Multiply $4$ by $2$ .

$y = 2 x^{2} + 8 x + 2 \cdot 4 - 20$

Multiply $2$ by $4$ .

$y = 2 x^{2} + 8 x + 8 - 20$

Subtract $20$ from $8$ .

$y = 2 x^{2} + 8 x - 12$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 2 x^{2} + 8 x - 12$

Vertex Form: $y = (2) {(x - (- 2))}^{2} - 20$

Step 6

Simplify the standard form.

Standard Form: $y = 2 x^{2} + 8 x - 12$

Vertex Form: $y = 2 {(x + 2)}^{2} - 20$

Step 7

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

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