Find the Parabola Through (0,2) with Vertex (-1,4) (-1,4) , (0,2)

$(- 1, 4)$ , $(0, 2)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 1, 4)$ as the vertex $(h, k)$ and $(0, 2)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$2 = a {(0 - (- 1))}^{2} + 4$

Step 2

Using $2 = a {(0 - (- 1))}^{2} + 4$ to solve for $a$ , $a = - 2$ .

Rewrite the equation as $a {(0 - (- 1))}^{2} + 4 = 2$ .

$a {(0 - (- 1))}^{2} + 4 = 2$

Simplify $a {(0 - (- 1))}^{2} + 4$ .

Subtract $- 1$ from $0$ .

$a {(- (- 1))}^{2} + 4 = 2$

Simplify each term.

Multiply $- 1$ by $- 1$ .

$a \cdot 1^{2} + 4 = 2$

One to any power is one.

$a \cdot 1 + 4 = 2$

Multiply $a$ by $1$ .

$a + 4 = 2$

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

Subtract $4$ from both sides of the equation.

$a = 2 - 4$

Subtract $4$ from $2$ .

$a = - 2$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 1, 4)$ and $a = - 2$ is $y = (- 2) {(x - (- 1))}^{2} + 4$ .

$y = (- 2) {(x - (- 1))}^{2} + 4$

Step 4

Solve $y = (- 2) {(x - (- 1))}^{2} + 4$ for $y$ .

Remove parentheses.

$y = (- 2) {(x - (- 1))}^{2} + 4$

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

$y = - 2 {(x - (- 1))}^{2} + 4$

Remove parentheses.

$y = (- 2) {(x - (- 1))}^{2} + 4$

Simplify $(- 2) {(x - (- 1))}^{2} + 4$ .

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = - 2 {(x + 1)}^{2} + 4$

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$y = - 2 ((x + 1) (x + 1)) + 4$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = - 2 (x (x + 1) + 1 (x + 1)) + 4$

Apply the distributive property.

$y = - 2 (x \cdot x + x \cdot 1 + 1 (x + 1)) + 4$

Apply the distributive property.

$y = - 2 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - 2 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) + 4$

Multiply $x$ by $1$ .

$y = - 2 (x^{2} + x + 1 x + 1 \cdot 1) + 4$

Multiply $x$ by $1$ .

$y = - 2 (x^{2} + x + x + 1 \cdot 1) + 4$

Multiply $1$ by $1$ .

$y = - 2 (x^{2} + x + x + 1) + 4$

Add $x$ and $x$ .

$y = - 2 (x^{2} + 2 x + 1) + 4$

Apply the distributive property.

$y = - 2 x^{2} - 2 (2 x) - 2 \cdot 1 + 4$

Simplify.

Multiply $2$ by $- 2$ .

$y = - 2 x^{2} - 4 x - 2 \cdot 1 + 4$

Multiply $- 2$ by $1$ .

$y = - 2 x^{2} - 4 x - 2 + 4$

Add $- 2$ and $4$ .

$y = - 2 x^{2} - 4 x + 2$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - 2 x^{2} - 4 x + 2$

Vertex Form: $y = (- 2) {(x - (- 1))}^{2} + 4$

Step 6

Simplify the standard form.

Standard Form: $y = - 2 x^{2} - 4 x + 2$

Vertex Form: $y = - 2 {(x + 1)}^{2} + 4$

Step 7

Find the Parabola Through (0,2) with Vertex (-1,4) (-1,4) , (0,2)

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