Examples

$(1, - 2)$ , $(3, 6)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(1, - 2)$ as the vertex $(h, k)$ and $(3, 6)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$6 = a {(3 - (1))}^{2} - 2$

Step 2

Using $6 = a {(3 - (1))}^{2} - 2$ to solve for $a$ , $a = 2$ .

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Step 2.1

Rewrite the equation as $a {(3 - (1))}^{2} - 2 = 6$ .

$a {(3 - (1))}^{2} - 2 = 6$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $1$ .

$a {(3 - 1)}^{2} - 2 = 6$

Step 2.2.2

Subtract $1$ from $3$ .

$a \cdot 2^{2} - 2 = 6$

Step 2.2.3

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

$a \cdot 4 - 2 = 6$

Step 2.2.4

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

$4 a - 2 = 6$

Step 2.3

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

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Step 2.3.1

Add $2$ to both sides of the equation.

$4 a = 6 + 2$

Step 2.3.2

Add $6$ and $2$ .

$4 a = 8$

Step 2.4

Divide each term in $4 a = 8$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 a = 8$ by $4$ .

$\frac{4 a}{4} = \frac{8}{4}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 a}{4} = \frac{8}{4}$

Step 2.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{8}{4}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $8$ by $4$ .

$a = 2$

Step 3

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

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

Step 4

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

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Step 4.1

Remove parentheses.

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

Step 4.2

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

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

Step 4.3

Remove parentheses.

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

Step 4.4

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

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $- 1$ by $1$ .

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

Step 4.4.1.2

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

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

Step 4.4.1.3

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

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Step 4.4.1.3.1

Apply the distributive property.

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

Step 4.4.1.3.2

Apply the distributive property.

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

Step 4.4.1.3.3

Apply the distributive property.

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

Step 4.4.1.4

Simplify and combine like terms.

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Step 4.4.1.4.1

Simplify each term.

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Step 4.4.1.4.1.1

Multiply $x$ by $x$ .

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

Step 4.4.1.4.1.2

Move $- 1$ to the left of $x$ .

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

Step 4.4.1.4.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.4.1.4.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 4.4.1.4.1.5

Multiply $- 1$ by $- 1$ .

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

Step 4.4.1.4.2

Subtract $x$ from $- x$ .

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

Step 4.4.1.5

Apply the distributive property.

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

Step 4.4.1.6

Simplify.

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Step 4.4.1.6.1

Multiply $- 2$ by $2$ .

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

Step 4.4.1.6.2

Multiply $2$ by $1$ .

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

Step 4.4.2

Combine the opposite terms in $2 x^{2} - 4 x + 2 - 2$ .

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Step 4.4.2.1

Subtract $2$ from $2$ .

$y = 2 x^{2} - 4 x + 0$

Step 4.4.2.2

Add $2 x^{2} - 4 x$ and $0$ .

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

Step 5

The standard form and vertex form are as follows.

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

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

Step 6

Simplify the standard form.

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

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

Step 7

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