Find the Parabola Through (0,13) with Vertex (-3,-14) (-3,-14) , (0,13)

$(- 3, - 14)$ , $(0, 13)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 3, - 14)$ as the vertex $(h, k)$ and $(0, 13)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$13 = a {(0 - (- 3))}^{2} - 14$

Step 2

Using $13 = a {(0 - (- 3))}^{2} - 14$ to solve for $a$ , $a = 3$ .

Rewrite the equation as $a {(0 - (- 3))}^{2} - 14 = 13$ .

$a {(0 - (- 3))}^{2} - 14 = 13$

Simplify $a {(0 - (- 3))}^{2} - 14$ .

Subtract $- 3$ from $0$ .

$a {(- (- 3))}^{2} - 14 = 13$

Simplify each term.

Multiply $- 1$ by $- 3$ .

$a \cdot 3^{2} - 14 = 13$

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

$a \cdot 9 - 14 = 13$

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

$9 a - 14 = 13$

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

Add $14$ to both sides of the equation.

$9 a = 13 + 14$

Add $13$ and $14$ .

$9 a = 27$

Divide each term in $9 a = 27$ by $9$ and simplify.

Divide each term in $9 a = 27$ by $9$ .

$\frac{9 a}{9} = \frac{27}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 a}{9} = \frac{27}{9}$

Divide $a$ by $1$ .

$a = \frac{27}{9}$

Simplify the right side.

Divide $27$ by $9$ .

$a = 3$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 3, - 14)$ and $a = 3$ is $y = (3) {(x - (- 3))}^{2} - 14$ .

$y = (3) {(x - (- 3))}^{2} - 14$

Step 4

Solve $y = (3) {(x - (- 3))}^{2} - 14$ for $y$ .

Remove parentheses.

$y = (3) {(x - (- 3))}^{2} - 14$

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

$y = 3 {(x - (- 3))}^{2} - 14$

Remove parentheses.

$y = (3) {(x - (- 3))}^{2} - 14$

Simplify $(3) {(x - (- 3))}^{2} - 14$ .

Simplify each term.

Multiply $- 1$ by $- 3$ .

$y = 3 {(x + 3)}^{2} - 14$

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

$y = 3 ((x + 3) (x + 3)) - 14$

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

Apply the distributive property.

$y = 3 (x (x + 3) + 3 (x + 3)) - 14$

Apply the distributive property.

$y = 3 (x \cdot x + x \cdot 3 + 3 (x + 3)) - 14$

Apply the distributive property.

$y = 3 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) - 14$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = 3 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) - 14$

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

$y = 3 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) - 14$

Multiply $3$ by $3$ .

$y = 3 (x^{2} + 3 x + 3 x + 9) - 14$

Add $3 x$ and $3 x$ .

$y = 3 (x^{2} + 6 x + 9) - 14$

Apply the distributive property.

$y = 3 x^{2} + 3 (6 x) + 3 \cdot 9 - 14$

Simplify.

Multiply $6$ by $3$ .

$y = 3 x^{2} + 18 x + 3 \cdot 9 - 14$

Multiply $3$ by $9$ .

$y = 3 x^{2} + 18 x + 27 - 14$

Subtract $14$ from $27$ .

$y = 3 x^{2} + 18 x + 13$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = 3 x^{2} + 18 x + 13$

Vertex Form: $y = (3) {(x - (- 3))}^{2} - 14$

Step 6

Simplify the standard form.

Standard Form: $y = 3 x^{2} + 18 x + 13$

Vertex Form: $y = 3 {(x + 3)}^{2} - 14$

Step 7

Find the Parabola Through (0,13) with Vertex (-3,-14) (-3,-14) , (0,13)

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