Algebra Examples

$(- 5, - 1)$ , $(- 2, 2)$

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(- 5, - 1)$ as the vertex $(h, k)$ and $(- 2, 2)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

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

Using $2 = a {(- 2 - (- 5))}^{2} - 1$ to solve for $a$ , $a = \frac{1}{3}$ .

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Rewrite the equation as $a {(- 2 - (- 5))}^{2} - 1 = 2$ .

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

Simplify each term.

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Multiply $- 1$ by $- 5$ .

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

Add $- 2$ and $5$ .

$a \cdot 3^{2} - 1 = 2$

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

$a \cdot 9 - 1 = 2$

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

$9 a - 1 = 2$

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

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Add $1$ to both sides of the equation.

$9 a = 2 + 1$

Add $2$ and $1$ .

$9 a = 3$

Divide each term in $9 a = 3$ by $9$ and simplify.

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Divide each term in $9 a = 3$ by $9$ .

$\frac{9 a}{9} = \frac{3}{9}$

Simplify the left side.

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Cancel the common factor of $9$ .

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Cancel the common factor.

$\frac{9 a}{9} = \frac{3}{9}$

Divide $a$ by $1$ .

$a = \frac{3}{9}$

Simplify the right side.

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Cancel the common factor of $3$ and $9$ .

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Factor $3$ out of $3$ .

$a = \frac{3 (1)}{9}$

Cancel the common factors.

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Factor $3$ out of $9$ .

$a = \frac{3 \cdot 1}{3 \cdot 3}$

Cancel the common factor.

$a = \frac{3 \cdot 1}{3 \cdot 3}$

Rewrite the expression.

$a = \frac{1}{3}$

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(- 5, - 1)$ and $a = \frac{1}{3}$ is $y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$ .

$y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$

Solve $y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$ for $y$ .

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Remove parentheses.

$y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$

Multiply $\frac{1}{3}$ by ${(x - (- 5))}^{2}$ .

$y = \frac{1}{3} \cdot {(x - (- 5))}^{2} - 1$

Remove parentheses.

$y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$

Simplify $(\frac{1}{3}) {(x - (- 5))}^{2} - 1$ .

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Simplify each term.

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Multiply $- 1$ by $- 5$ .

$y = \frac{1}{3} {(x + 5)}^{2} - 1$

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$y = \frac{1}{3} ((x + 5) (x + 5)) - 1$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Apply the distributive property.

$y = \frac{1}{3} (x (x + 5) + 5 (x + 5)) - 1$

Apply the distributive property.

$y = \frac{1}{3} (x \cdot x + x \cdot 5 + 5 (x + 5)) - 1$

Apply the distributive property.

$y = \frac{1}{3} (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$y = \frac{1}{3} (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 1$

Move $5$ to the left of $x$ .

$y = \frac{1}{3} (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 1$

Multiply $5$ by $5$ .

$y = \frac{1}{3} (x^{2} + 5 x + 5 x + 25) - 1$

Add $5 x$ and $5 x$ .

$y = \frac{1}{3} (x^{2} + 10 x + 25) - 1$

Apply the distributive property.

$y = \frac{1}{3} x^{2} + \frac{1}{3} (10 x) + \frac{1}{3} \cdot 25 - 1$

Simplify.

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Combine $\frac{1}{3}$ and $x^{2}$ .

$y = \frac{x^{2}}{3} + \frac{1}{3} (10 x) + \frac{1}{3} \cdot 25 - 1$

Multiply $\frac{1}{3} (10 x)$ .

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Combine $10$ and $\frac{1}{3}$ .

$y = \frac{x^{2}}{3} + \frac{10}{3} x + \frac{1}{3} \cdot 25 - 1$

Combine $\frac{10}{3}$ and $x$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{1}{3} \cdot 25 - 1$

Combine $\frac{1}{3}$ and $25$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{25}{3} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{25}{3} - 1 \cdot \frac{3}{3}$

Combine $- 1$ and $\frac{3}{3}$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{25}{3} + \frac{- 1 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{25 - 1 \cdot 3}{3}$

Simplify the numerator.

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Multiply $- 1$ by $3$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{25 - 3}{3}$

Subtract $3$ from $25$ .

$y = \frac{x^{2}}{3} + \frac{10 x}{3} + \frac{22}{3}$

The standard form and vertex form are as follows.

Standard Form: $y = \frac{1}{3} x^{2} + \frac{10}{3} x + \frac{22}{3}$

Vertex Form: $y = (\frac{1}{3}) {(x - (- 5))}^{2} - 1$

Simplify the standard form.

Standard Form: $y = \frac{1}{3} x^{2} + \frac{10}{3} x + \frac{22}{3}$

Vertex Form: $y = \frac{1}{3} {(x + 5)}^{2} - 1$