Examples

$(2, 3)$ ,

$(- 1, - 5)$

Step 1

The general equation of a parabola with vertex

$(h, k)$ is

$y = a {(x - h)}^{2} + k$ . In this case we have

$(2, 3)$ as the vertex

$(h, k)$ and

$(- 1, - 5)$ is a point

$(x, y)$ on the parabola. To find

$a$ , substitute the two points in

$y = a {(x - h)}^{2} + k$ .

$- 5 = a {(- 1 - (2))}^{2} + 3$

Step 2

Using

$- 5 = a {(- 1 - (2))}^{2} + 3$ to solve for

$a$ ,

$a = - \frac{8}{9}$ .

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Step 2.1

Rewrite the equation as

$a {(- 1 - (2))}^{2} + 3 = - 5$ .

$a {(- 1 - (2))}^{2} + 3 = - 5$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply

$- 1$ by

$2$ .

$a {(- 1 - 2)}^{2} + 3 = - 5$

Step 2.2.2

Subtract

$2$ from

$- 1$ .

$a {(- 3)}^{2} + 3 = - 5$

Step 2.2.3

Raise

$- 3$ to the power of

$2$ .

$a \cdot 9 + 3 = - 5$

Step 2.2.4

Move

$9$ to the left of

$a$ .

$9 a + 3 = - 5$

Step 2.3

Move all terms not containing

$a$ to the right side of the equation.

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Step 2.3.1

Subtract

$3$ from both sides of the equation.

$9 a = - 5 - 3$

Step 2.3.2

Subtract

$3$ from

$- 5$ .

$9 a = - 8$

Step 2.4

Divide each term in

$9 a = - 8$ by

$9$ and simplify.

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Step 2.4.1

Divide each term in

$9 a = - 8$ by

$9$ .

$\frac{9 a}{9} = \frac{- 8}{9}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of

$9$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{9 a}{9} = \frac{- 8}{9}$

Step 2.4.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{- 8}{9}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Move the negative in front of the fraction.

$a = - \frac{8}{9}$

Step 3

Using

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

$(2, 3)$ and

$a = - \frac{8}{9}$ is

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$ .

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$

Step 4

Solve

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$ for

$y$ .

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Step 4.1

Remove parentheses.

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$

Step 4.2

Multiply

$- \frac{8}{9}$ by

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

$y = - \frac{8}{9} \cdot {(x - (2))}^{2} + 3$

Step 4.3

Remove parentheses.

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$

Step 4.4

Simplify

$(- \frac{8}{9}) {(x - (2))}^{2} + 3$ .

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply

$- 1$ by

$2$ .

$y = - \frac{8}{9} {(x - 2)}^{2} + 3$

Step 4.4.1.2

Rewrite

${(x - 2)}^{2}$ as

$(x - 2) (x - 2)$ .

$y = - \frac{8}{9} ((x - 2) (x - 2)) + 3$

Step 4.4.1.3

Expand

$(x - 2) (x - 2)$ using the FOIL Method.

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Step 4.4.1.3.1

Apply the distributive property.

$y = - \frac{8}{9} (x (x - 2) - 2 (x - 2)) + 3$

Step 4.4.1.3.2

Apply the distributive property.

$y = - \frac{8}{9} (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 3$

Step 4.4.1.3.3

Apply the distributive property.

$y = - \frac{8}{9} (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 3$

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

$x$ .

$y = - \frac{8}{9} (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + 3$

Step 4.4.1.4.1.2

Move

$- 2$ to the left of

$x$ .

$y = - \frac{8}{9} (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + 3$

Step 4.4.1.4.1.3

Multiply

$- 2$ by

$- 2$ .

$y = - \frac{8}{9} (x^{2} - 2 x - 2 x + 4) + 3$

Step 4.4.1.4.2

Subtract

$2 x$ from

$- 2 x$ .

$y = - \frac{8}{9} (x^{2} - 4 x + 4) + 3$

Step 4.4.1.5

Apply the distributive property.

$y = - \frac{8}{9} x^{2} - \frac{8}{9} (- 4 x) - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6

Simplify.

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Step 4.4.1.6.1

Combine

$x^{2}$ and

$\frac{8}{9}$ .

$y = - \frac{x^{2} \cdot 8}{9} - \frac{8}{9} (- 4 x) - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6.2

Multiply

$- \frac{8}{9} (- 4 x)$ .

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Step 4.4.1.6.2.1

Multiply

$- 4$ by

$- 1$ .

$y = - \frac{x^{2} \cdot 8}{9} + 4 (\frac{8}{9}) x - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6.2.2

Combine

$4$ and

$\frac{8}{9}$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{4 \cdot 8}{9} x - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6.2.3

Multiply

$4$ by

$8$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{32}{9} x - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6.2.4

Combine

$\frac{32}{9}$ and

$x$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{32 x}{9} - \frac{8}{9} \cdot 4 + 3$

Step 4.4.1.6.3

Multiply

$- \frac{8}{9} \cdot 4$ .

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Step 4.4.1.6.3.1

Multiply

$4$ by

$- 1$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{32 x}{9} - 4 (\frac{8}{9}) + 3$

Step 4.4.1.6.3.2

Combine

$- 4$ and

$\frac{8}{9}$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{32 x}{9} + \frac{- 4 \cdot 8}{9} + 3$

Step 4.4.1.6.3.3

Multiply

$- 4$ by

$8$ .

$y = - \frac{x^{2} \cdot 8}{9} + \frac{32 x}{9} + \frac{- 32}{9} + 3$

Step 4.4.1.7

Simplify each term.

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Step 4.4.1.7.1

Move

$8$ to the left of

$x^{2}$ .

$y = - \frac{8 \cdot x^{2}}{9} + \frac{32 x}{9} + \frac{- 32}{9} + 3$

Step 4.4.1.7.2

Move the negative in front of the fraction.

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} - \frac{32}{9} + 3$

Step 4.4.2

To write

$3$ as a fraction with a common denominator, multiply by

$\frac{9}{9}$ .

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} - \frac{32}{9} + 3 \cdot \frac{9}{9}$

Step 4.4.3

Combine

$3$ and

$\frac{9}{9}$ .

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} - \frac{32}{9} + \frac{3 \cdot 9}{9}$

Step 4.4.4

Combine the numerators over the common denominator.

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} + \frac{- 32 + 3 \cdot 9}{9}$

Step 4.4.5

Simplify the numerator.

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Step 4.4.5.1

Multiply

$3$ by

$9$ .

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} + \frac{- 32 + 27}{9}$

Step 4.4.5.2

Add

$- 32$ and

$27$ .

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} + \frac{- 5}{9}$

Step 4.4.6

Move the negative in front of the fraction.

$y = - \frac{8 x^{2}}{9} + \frac{32 x}{9} - \frac{5}{9}$

Step 5

The standard form and vertex form are as follows.

Standard Form:

$y = - \frac{8}{9} x^{2} + \frac{32}{9} x - \frac{5}{9}$

Vertex Form:

$y = (- \frac{8}{9}) {(x - (2))}^{2} + 3$

Step 6

Simplify the standard form.

Standard Form:

$y = - \frac{8}{9} x^{2} + \frac{32}{9} x - \frac{5}{9}$

Vertex Form:

$y = - \frac{8}{9} {(x - 2)}^{2} + 3$

Step 7

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