Find the Parabola Through (0,-36) with Vertex (-4,12) (-4,12) , (0,-36)

$(- 4, 12)$ , $(0, - 36)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 4, 12)$ as the vertex $(h, k)$ and $(0, - 36)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 36 = a {(0 - (- 4))}^{2} + 12$

Step 2

Using $- 36 = a {(0 - (- 4))}^{2} + 12$ to solve for $a$ , $a = - 3$ .

Rewrite the equation as $a {(0 - (- 4))}^{2} + 12 = - 36$ .

$a {(0 - (- 4))}^{2} + 12 = - 36$

Simplify $a {(0 - (- 4))}^{2} + 12$ .

Subtract $- 4$ from $0$ .

$a {(- (- 4))}^{2} + 12 = - 36$

Simplify each term.

Multiply $- 1$ by $- 4$ .

$a \cdot 4^{2} + 12 = - 36$

Raise $4$ to the power of $2$ .

$a \cdot 16 + 12 = - 36$

Move $16$ to the left of $a$ .

$16 a + 12 = - 36$

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

Subtract $12$ from both sides of the equation.

$16 a = - 36 - 12$

Subtract $12$ from $- 36$ .

$16 a = - 48$

Divide each term in $16 a = - 48$ by $16$ and simplify.

Divide each term in $16 a = - 48$ by $16$ .

$\frac{16 a}{16} = \frac{- 48}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 a}{16} = \frac{- 48}{16}$

Divide $a$ by $1$ .

$a = \frac{- 48}{16}$

Simplify the right side.

Divide $- 48$ by $16$ .

$a = - 3$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 4, 12)$ and $a = - 3$ is $y = (- 3) {(x - (- 4))}^{2} + 12$ .

$y = (- 3) {(x - (- 4))}^{2} + 12$

Step 4

Solve $y = (- 3) {(x - (- 4))}^{2} + 12$ for $y$ .

Remove parentheses.

$y = (- 3) {(x - (- 4))}^{2} + 12$

Multiply $- 3$ by ${(x - (- 4))}^{2}$ .

$y = - 3 {(x - (- 4))}^{2} + 12$

Remove parentheses.

$y = (- 3) {(x - (- 4))}^{2} + 12$

Simplify $(- 3) {(x - (- 4))}^{2} + 12$ .

Simplify each term.

Multiply $- 1$ by $- 4$ .

$y = - 3 {(x + 4)}^{2} + 12$

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$y = - 3 ((x + 4) (x + 4)) + 12$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$y = - 3 (x (x + 4) + 4 (x + 4)) + 12$

Apply the distributive property.

$y = - 3 (x \cdot x + x \cdot 4 + 4 (x + 4)) + 12$

Apply the distributive property.

$y = - 3 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - 3 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) + 12$

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

$y = - 3 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) + 12$

Multiply $4$ by $4$ .

$y = - 3 (x^{2} + 4 x + 4 x + 16) + 12$

Add $4 x$ and $4 x$ .

$y = - 3 (x^{2} + 8 x + 16) + 12$

Apply the distributive property.

$y = - 3 x^{2} - 3 (8 x) - 3 \cdot 16 + 12$

Simplify.

Multiply $8$ by $- 3$ .

$y = - 3 x^{2} - 24 x - 3 \cdot 16 + 12$

Multiply $- 3$ by $16$ .

$y = - 3 x^{2} - 24 x - 48 + 12$

Add $- 48$ and $12$ .

$y = - 3 x^{2} - 24 x - 36$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 3 x^{2} - 24 x - 36$

Vertex Form: $y = (- 3) {(x - (- 4))}^{2} + 12$

Step 6

Simplify the standard form.

Standard Form: $y = - 3 x^{2} - 24 x - 36$

Vertex Form: $y = - 3 {(x + 4)}^{2} + 12$

Step 7

Find the Parabola Through (0,-36) with Vertex (-4,12) (-4,12) , (0,-36)

Related Questions

Support LCMGCF.com by disabling your adblocker.