Find the Parabola Through (-3,-14) with Vertex (0,13) (-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 $(0, 13)$ as the vertex $(h, k)$ and $(- 3, - 14)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

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

Step 2

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

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

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

Simplify each term.

Subtract $0$ from $- 3$ .

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

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

$a \cdot 9 + 13 = - 14$

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

$9 a + 13 = - 14$

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

Subtract $13$ from both sides of the equation.

$9 a = - 14 - 13$

Subtract $13$ from $- 14$ .

$9 a = - 27$

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 $(0, 13)$ and $a = - 3$ is $y = (- 3) {(x - (0))}^{2} + 13$ .

$y = (- 3) {(x - (0))}^{2} + 13$

Step 4

Solve $y = (- 3) {(x - (0))}^{2} + 13$ for $y$ .

Remove parentheses.

$y = (- 3) {(x - (0))}^{2} + 13$

Multiply $- 3$ by ${(x - (0))}^{2}$ .

$y = - 3 {(x - (0))}^{2} + 13$

Remove parentheses.

$y = (- 3) {(x - (0))}^{2} + 13$

Subtract $0$ from $x$ .

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

Step 5

The standard form and vertex form are as follows.

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

Vertex Form: $y = (- 3) {(x - (0))}^{2} + 13$

Step 6

Simplify the standard form.

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

Vertex Form: $y = - 3 x^{2} + 13$

Step 7

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

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