Precalculus Examples

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

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Remove parentheses.

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

Step 1.1.2

Remove parentheses.

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

Step 1.1.3

Simplify $- \frac{1}{3} \cdot ((x + 1) (x - 5))$ .

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Step 1.1.3.1

Expand $(x + 1) (x - 5)$ using the FOIL Method.

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Step 1.1.3.1.1

Apply the distributive property.

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

Step 1.1.3.1.2

Apply the distributive property.

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

Step 1.1.3.1.3

Apply the distributive property.

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

Step 1.1.3.2

Simplify and combine like terms.

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Step 1.1.3.2.1

Simplify each term.

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Step 1.1.3.2.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.2.1.2

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

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

Step 1.1.3.2.1.3

Multiply $x$ by $1$ .

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

Step 1.1.3.2.1.4

Multiply $- 5$ by $1$ .

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

Step 1.1.3.2.2

Add $- 5 x$ and $x$ .

$y = - \frac{1}{3} \cdot (x^{2} - 4 x - 5)$

Step 1.1.3.3

Apply the distributive property.

$y = - \frac{1}{3} x^{2} - \frac{1}{3} (- 4 x) - \frac{1}{3} \cdot - 5$

Step 1.1.3.4

Simplify.

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Step 1.1.3.4.1

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

$y = - \frac{x^{2}}{3} - \frac{1}{3} (- 4 x) - \frac{1}{3} \cdot - 5$

Step 1.1.3.4.2

Multiply $- \frac{1}{3} (- 4 x)$ .

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Step 1.1.3.4.2.1

Multiply $- 4$ by $- 1$ .

$y = - \frac{x^{2}}{3} + 4 (\frac{1}{3}) x - \frac{1}{3} \cdot - 5$

Step 1.1.3.4.2.2

Combine $4$ and $\frac{1}{3}$ .

$y = - \frac{x^{2}}{3} + \frac{4}{3} x - \frac{1}{3} \cdot - 5$

Step 1.1.3.4.2.3

Combine $\frac{4}{3}$ and $x$ .

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} - \frac{1}{3} \cdot - 5$

Step 1.1.3.4.3

Multiply $- \frac{1}{3} \cdot - 5$ .

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Step 1.1.3.4.3.1

Multiply $- 5$ by $- 1$ .

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + 5 (\frac{1}{3})$

Step 1.1.3.4.3.2

Combine $5$ and $\frac{1}{3}$ .

$y = - \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{5}{3}$

Step 1.2

Complete the square for $- \frac{x^{2}}{3} + \frac{4 x}{3} + \frac{5}{3}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

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

$b = \frac{4}{3}$

$c = \frac{5}{3}$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{4}{3}}{2 (- \frac{1}{3})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{4}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 (2)}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 1.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 2}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 1.2.3.2.2.3

Rewrite the expression.

$d = \frac{2}{3} \cdot \frac{1}{- \frac{1}{3}}$

Step 1.2.3.2.3

Multiply $\frac{2}{3}$ by $\frac{1}{- \frac{1}{3}}$ .

$d = \frac{2}{3 (- \frac{1}{3})}$

Step 1.2.3.2.4

Multiply $- 1$ by $3$ .

$d = \frac{2}{- 3 (\frac{1}{3})}$

Step 1.2.3.2.5

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

$d = \frac{2}{\frac{- 3}{3}}$

Step 1.2.3.2.6

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

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Step 1.2.3.2.6.1

Factor $3$ out of $- 3$ .

$d = \frac{2}{\frac{3 \cdot - 1}{3}}$

Step 1.2.3.2.6.2

Cancel the common factors.

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Step 1.2.3.2.6.2.1

Factor $3$ out of $3$ .

$d = \frac{2}{\frac{3 \cdot - 1}{3 (1)}}$

Step 1.2.3.2.6.2.2

Cancel the common factor.

$d = \frac{2}{\frac{3 \cdot - 1}{3 \cdot 1}}$

Step 1.2.3.2.6.2.3

Rewrite the expression.

$d = \frac{2}{\frac{- 1}{1}}$

Step 1.2.3.2.6.2.4

Divide $- 1$ by $1$ .

$d = \frac{2}{- 1}$

Step 1.2.3.2.7

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 1.2.3.2.8

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{5}{3} - \frac{{(\frac{4}{3})}^{2}}{4 (- \frac{1}{3})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{4}{3}$ .

$e = \frac{5}{3} - \frac{\frac{4^{2}}{3^{2}}}{4 (- \frac{1}{3})}$

Step 1.2.4.2.1.1.2

Raise $4$ to the power of $2$ .

$e = \frac{5}{3} - \frac{\frac{16}{3^{2}}}{4 (- \frac{1}{3})}$

Step 1.2.4.2.1.1.3

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

$e = \frac{5}{3} - \frac{\frac{16}{9}}{4 (- \frac{1}{3})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{5}{3} - \frac{\frac{16}{9}}{- 4 (\frac{1}{3})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{3}$ .

$e = \frac{5}{3} - \frac{\frac{16}{9}}{\frac{- 4}{3}}$

Step 1.2.4.2.1.3

Move the negative in front of the fraction.

$e = \frac{5}{3} - \frac{\frac{16}{9}}{- \frac{4}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{5}{3} - (\frac{16}{9} (- \frac{3}{4}))$

Step 1.2.4.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.4.2.1.5.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$e = \frac{5}{3} - (\frac{16}{9} \cdot \frac{- 3}{4})$

Step 1.2.4.2.1.5.2

Factor $4$ out of $16$ .

$e = \frac{5}{3} - (\frac{4 (4)}{9} \cdot \frac{- 3}{4})$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = \frac{5}{3} - (\frac{4 \cdot 4}{9} \cdot \frac{- 3}{4})$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = \frac{5}{3} - (\frac{4}{9} \cdot - 3)$

Step 1.2.4.2.1.6

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.6.1

Factor $3$ out of $9$ .

$e = \frac{5}{3} - (\frac{4}{3 (3)} \cdot - 3)$

Step 1.2.4.2.1.6.2

Factor $3$ out of $- 3$ .

$e = \frac{5}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.3

Cancel the common factor.

$e = \frac{5}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.4

Rewrite the expression.

$e = \frac{5}{3} - (\frac{4}{3} \cdot - 1)$

Step 1.2.4.2.1.7

Combine $\frac{4}{3}$ and $- 1$ .

$e = \frac{5}{3} - \frac{4 \cdot - 1}{3}$

Step 1.2.4.2.1.8

Multiply $4$ by $- 1$ .

$e = \frac{5}{3} - \frac{- 4}{3}$

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

$e = \frac{5}{3} - - \frac{4}{3}$

Step 1.2.4.2.1.10

Multiply $- - \frac{4}{3}$ .

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Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = \frac{5}{3} + 1 (\frac{4}{3})$

Step 1.2.4.2.1.10.2

Multiply $\frac{4}{3}$ by $1$ .

$e = \frac{5}{3} + \frac{4}{3}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{5 + 4}{3}$

Step 1.2.4.2.3

Add $5$ and $4$ .

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

Step 1.2.4.2.4

Divide $9$ by $3$ .

$e = 3$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{3} {(x - 2)}^{2} + 3$ .

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

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

$h = 2$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(2, 3)$

Step 4