Algebra Examples

$(- 3, 5)$ , $(0, - 14)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 3, 5)$ as the vertex $(h, k)$ and $(0, - 14)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$- 14 = a {(0 - (- 3))}^{2} + 5$

Step 2

Using $- 14 = a {(0 - (- 3))}^{2} + 5$ to solve for $a$ , $a = - \frac{19}{9}$ .

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Step 2.1

Rewrite the equation as $a {(0 - (- 3))}^{2} + 5 = - 14$ .

$a {(0 - (- 3))}^{2} + 5 = - 14$

Step 2.2

Simplify $a {(0 - (- 3))}^{2} + 5$ .

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Step 2.2.1

Subtract $- 3$ from $0$ .

$a {(- (- 3))}^{2} + 5 = - 14$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $- 1$ by $- 3$ .

$a \cdot 3^{2} + 5 = - 14$

Step 2.2.2.2

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

$a \cdot 9 + 5 = - 14$

Step 2.2.2.3

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

$9 a + 5 = - 14$

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 $5$ from both sides of the equation.

$9 a = - 14 - 5$

Step 2.3.2

Subtract $5$ from $- 14$ .

$9 a = - 19$

Step 2.4

Divide each term in $9 a = - 19$ by $9$ and simplify.

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Step 2.4.1

Divide each term in $9 a = - 19$ by $9$ .

$\frac{9 a}{9} = \frac{- 19}{9}$

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.

$\frac{9 a}{9} = \frac{- 19}{9}$

Step 2.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 19}{9}$

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.

$a = - \frac{19}{9}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 3, 5)$ and $a = - \frac{19}{9}$ is $y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$ .

$y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$

Step 4

Solve $y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$ for $y$ .

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Step 4.1

Remove parentheses.

$y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$

Step 4.2

Multiply $- \frac{19}{9}$ by ${(x - (- 3))}^{2}$ .

$y = - \frac{19}{9} \cdot {(x - (- 3))}^{2} + 5$

Step 4.3

Remove parentheses.

$y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$

Step 4.4

Simplify $(- \frac{19}{9}) {(x - (- 3))}^{2} + 5$ .

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $- 1$ by $- 3$ .

$y = - \frac{19}{9} {(x + 3)}^{2} + 5$

Step 4.4.1.2

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

$y = - \frac{19}{9} ((x + 3) (x + 3)) + 5$

Step 4.4.1.3

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

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Step 4.4.1.3.1

Apply the distributive property.

$y = - \frac{19}{9} (x (x + 3) + 3 (x + 3)) + 5$

Step 4.4.1.3.2

Apply the distributive property.

$y = - \frac{19}{9} (x \cdot x + x \cdot 3 + 3 (x + 3)) + 5$

Step 4.4.1.3.3

Apply the distributive property.

$y = - \frac{19}{9} (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 5$

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 $x$ .

$y = - \frac{19}{9} (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 5$

Step 4.4.1.4.1.2

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

$y = - \frac{19}{9} (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 5$

Step 4.4.1.4.1.3

Multiply $3$ by $3$ .

$y = - \frac{19}{9} (x^{2} + 3 x + 3 x + 9) + 5$

Step 4.4.1.4.2

Add $3 x$ and $3 x$ .

$y = - \frac{19}{9} (x^{2} + 6 x + 9) + 5$

Step 4.4.1.5

Apply the distributive property.

$y = - \frac{19}{9} x^{2} - \frac{19}{9} (6 x) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6

Simplify.

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Step 4.4.1.6.1

Combine $x^{2}$ and $\frac{19}{9}$ .

$y = - \frac{x^{2} \cdot 19}{9} - \frac{19}{9} (6 x) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.2

Cancel the common factor of $3$ .

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Step 4.4.1.6.2.1

Move the leading negative in $- \frac{19}{9}$ into the numerator.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 19}{9} (6 x) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.2.2

Factor $3$ out of $9$ .

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 19}{3 (3)} (6 x) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.2.3

Factor $3$ out of $6 x$ .

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 19}{3 (3)} (3 (2 x)) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.2.4

Cancel the common factor.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 19}{3 \cdot 3} (3 (2 x)) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.2.5

Rewrite the expression.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 19}{3} (2 x) - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.3

Combine $2$ and $\frac{- 19}{3}$ .

$y = - \frac{x^{2} \cdot 19}{9} + \frac{2 \cdot - 19}{3} x - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.4

Multiply $2$ by $- 19$ .

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 38}{3} x - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.5

Combine $\frac{- 38}{3}$ and $x$ .

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 38 x}{3} - \frac{19}{9} \cdot 9 + 5$

Step 4.4.1.6.6

Cancel the common factor of $9$ .

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Step 4.4.1.6.6.1

Move the leading negative in $- \frac{19}{9}$ into the numerator.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 38 x}{3} + \frac{- 19}{9} \cdot 9 + 5$

Step 4.4.1.6.6.2

Cancel the common factor.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 38 x}{3} + \frac{- 19}{9} \cdot 9 + 5$

Step 4.4.1.6.6.3

Rewrite the expression.

$y = - \frac{x^{2} \cdot 19}{9} + \frac{- 38 x}{3} - 19 + 5$

Step 4.4.1.7

Simplify each term.

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Step 4.4.1.7.1

Move $19$ to the left of $x^{2}$ .

$y = - \frac{19 \cdot x^{2}}{9} + \frac{- 38 x}{3} - 19 + 5$

Step 4.4.1.7.2

Move the negative in front of the fraction.

$y = - \frac{19 x^{2}}{9} - \frac{38 x}{3} - 19 + 5$

Step 4.4.2

Add $- 19$ and $5$ .

$y = - \frac{19 x^{2}}{9} - \frac{38 x}{3} - 14$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = - \frac{19}{9} x^{2} - \frac{38}{3} x - 14$

Vertex Form: $y = (- \frac{19}{9}) {(x - (- 3))}^{2} + 5$

Step 6

Simplify the standard form.

Standard Form: $y = - \frac{19}{9} x^{2} - \frac{38}{3} x - 14$

Vertex Form: $y = - \frac{19}{9} {(x + 3)}^{2} + 5$

Step 7