Find the Parabola Through (0,-3) with Vertex (-3,-12) (-3,-12) , (0,-3)

$(- 3, - 12)$ , $(0, - 3)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 3, - 12)$ as the vertex $(h, k)$ and $(0, - 3)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 3 = a {(0 - (- 3))}^{2} - 12$

Step 2

Using $- 3 = a {(0 - (- 3))}^{2} - 12$ to solve for $a$ , $a = 1$ .

Rewrite the equation as $a {(0 - (- 3))}^{2} - 12 = - 3$ .

$a {(0 - (- 3))}^{2} - 12 = - 3$

Simplify $a {(0 - (- 3))}^{2} - 12$ .

Subtract $- 3$ from $0$ .

$a {(- (- 3))}^{2} - 12 = - 3$

Simplify each term.

Multiply $- 1$ by $- 3$ .

$a \cdot 3^{2} - 12 = - 3$

Raise $3$ to the power of $2$ .

$a \cdot 9 - 12 = - 3$

Move $9$ to the left of $a$ .

$9 a - 12 = - 3$

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

Add $12$ to both sides of the equation.

$9 a = - 3 + 12$

Add $- 3$ and $12$ .

$9 a = 9$

Divide each term in $9 a = 9$ by $9$ and simplify.

Divide each term in $9 a = 9$ by $9$ .

$\frac{9 a}{9} = \frac{9}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} = \frac{9}{9}$

Divide $a$ by $1$ .

$a = \frac{9}{9}$

Simplify the right side.

Divide $9$ by $9$ .

$a = 1$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 3, - 12)$ and $a = 1$ is $y = (1) {(x - (- 3))}^{2} - 12$ .

$y = (1) {(x - (- 3))}^{2} - 12$

Step 4

Solve $y = (1) {(x - (- 3))}^{2} - 12$ for $y$ .

Remove parentheses.

$y = (1) {(x - (- 3))}^{2} - 12$

Multiply $1$ by ${(x - (- 3))}^{2}$ .

$y = 1 {(x - (- 3))}^{2} - 12$

Remove parentheses.

$y = (1) {(x - (- 3))}^{2} - 12$

Simplify $(1) {(x - (- 3))}^{2} - 12$ .

Simplify each term.

Multiply ${(x - (- 3))}^{2}$ by $1$ .

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

Multiply $- 1$ by $- 3$ .

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

Rewrite ${(x + 3)}^{2}$ as $(x + 3) (x + 3)$ .

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

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$y = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $3$ to the left of $x$ .

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

Multiply $3$ by $3$ .

$y = x^{2} + 3 x + 3 x + 9 - 12$

Add $3 x$ and $3 x$ .

$y = x^{2} + 6 x + 9 - 12$

Subtract $12$ from $9$ .

$y = x^{2} + 6 x - 3$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = x^{2} + 6 x - 3$

Vertex Form: $y = (1) {(x - (- 3))}^{2} - 12$

Step 6

Simplify the standard form.

Standard Form: $y = x^{2} + 6 x - 3$

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

Step 7

Find the Parabola Through (0,-3) with Vertex (-3,-12) (-3,-12) , (0,-3)

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