Precalculus Examples

${(x - 4)}^{2} = - 12 (y - 1)$

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

Rewrite the equation as $- 12 (y - 1) = {(x - 4)}^{2}$ .

$- 12 (y - 1) = {(x - 4)}^{2}$

Step 1.1.2

Divide each term in $- 12 (y - 1) = {(x - 4)}^{2}$ by $- 12$ and simplify.

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Step 1.1.2.1

Divide each term in $- 12 (y - 1) = {(x - 4)}^{2}$ by $- 12$ .

$\frac{- 12 (y - 1)}{- 12} = \frac{{(x - 4)}^{2}}{- 12}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 12$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 12 (y - 1)}{- 12} = \frac{{(x - 4)}^{2}}{- 12}$

Step 1.1.2.2.1.2

Divide $y - 1$ by $1$ .

$y - 1 = \frac{{(x - 4)}^{2}}{- 12}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$y - 1 = - \frac{{(x - 4)}^{2}}{12}$

Step 1.1.3

Add $1$ to both sides of the equation.

$y = - \frac{{(x - 4)}^{2}}{12} + 1$

Step 1.1.4

Reorder terms.

$y = - \frac{1}{12} \cdot {(x - 4)}^{2} + 1$

Step 1.2

Complete the square for $- \frac{1}{12} \cdot {(x - 4)}^{2} + 1$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$- \frac{1}{12} \cdot ((x - 4) (x - 4)) + 1$

Step 1.2.1.1.2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$- \frac{1}{12} \cdot (x (x - 4) - 4 (x - 4)) + 1$

Step 1.2.1.1.2.2

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot - 4 - 4 (x - 4)) + 1$

Step 1.2.1.1.2.3

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) + 1$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$- \frac{1}{12} \cdot (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) + 1$

Step 1.2.1.1.3.1.2

Move $- 4$ to the left of $x$ .

$- \frac{1}{12} \cdot (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) + 1$

Step 1.2.1.1.3.1.3

Multiply $- 4$ by $- 4$ .

$- \frac{1}{12} \cdot (x^{2} - 4 x - 4 x + 16) + 1$

Step 1.2.1.1.3.2

Subtract $4 x$ from $- 4 x$ .

$- \frac{1}{12} \cdot (x^{2} - 8 x + 16) + 1$

Step 1.2.1.1.4

Apply the distributive property.

$- \frac{1}{12} x^{2} - \frac{1}{12} (- 8 x) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $x^{2}$ and $\frac{1}{12}$ .

$- \frac{x^{2}}{12} - \frac{1}{12} (- 8 x) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.2

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.2.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$- \frac{x^{2}}{12} + \frac{- 1}{12} (- 8 x) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.2.2

Factor $4$ out of $12$ .

$- \frac{x^{2}}{12} + \frac{- 1}{4 (3)} (- 8 x) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.2.3

Factor $4$ out of $- 8 x$ .

$- \frac{x^{2}}{12} + \frac{- 1}{4 (3)} (4 (- 2 x)) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.2.4

Cancel the common factor.

$- \frac{x^{2}}{12} + \frac{- 1}{4 \cdot 3} (4 (- 2 x)) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.2.5

Rewrite the expression.

$- \frac{x^{2}}{12} + \frac{- 1}{3} (- 2 x) - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.3

Combine $- 2$ and $\frac{- 1}{3}$ .

$- \frac{x^{2}}{12} + \frac{- 2 \cdot - 1}{3} x - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.4

Multiply $- 2$ by $- 1$ .

$- \frac{x^{2}}{12} + \frac{2}{3} x - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.5

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

$- \frac{x^{2}}{12} + \frac{2 x}{3} - \frac{1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.6

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.6.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{12} \cdot 16 + 1$

Step 1.2.1.1.5.6.2

Factor $4$ out of $12$ .

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{4 (3)} \cdot 16 + 1$

Step 1.2.1.1.5.6.3

Factor $4$ out of $16$ .

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{4 \cdot 3} \cdot (4 \cdot 4) + 1$

Step 1.2.1.1.5.6.4

Cancel the common factor.

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{4 \cdot 3} \cdot (4 \cdot 4) + 1$

Step 1.2.1.1.5.6.5

Rewrite the expression.

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{3} \cdot 4 + 1$

Step 1.2.1.1.5.7

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

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1 \cdot 4}{3} + 1$

Step 1.2.1.1.5.8

Multiply $- 1$ by $4$ .

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 4}{3} + 1$

Step 1.2.1.1.6

Move the negative in front of the fraction.

$- \frac{x^{2}}{12} + \frac{2 x}{3} - \frac{4}{3} + 1$

Step 1.2.1.2

Write $1$ as a fraction with a common denominator.

$- \frac{x^{2}}{12} + \frac{2 x}{3} - \frac{4}{3} + \frac{3}{3}$

Step 1.2.1.3

Combine the numerators over the common denominator.

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 4 + 3}{3}$

Step 1.2.1.4

Add $- 4$ and $3$ .

$- \frac{x^{2}}{12} + \frac{2 x}{3} + \frac{- 1}{3}$

Step 1.2.1.5

Move the negative in front of the fraction.

$- \frac{x^{2}}{12} + \frac{2 x}{3} - \frac{1}{3}$

Step 1.2.2

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

$a = - \frac{1}{12}$

$b = \frac{2}{3}$

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

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{\frac{2}{3}}{2 (- \frac{1}{12})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{2}{3} \cdot \frac{1}{2 (- \frac{1}{12})}$

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Cancel the common factor.

$d = \frac{2}{3} \cdot \frac{1}{2 (- \frac{1}{12})}$

Step 1.2.4.2.2.2

Rewrite the expression.

$d = \frac{1}{3} \cdot \frac{1}{- \frac{1}{12}}$

Step 1.2.4.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{- \frac{1}{12}}$ .

$d = \frac{1}{3 (- \frac{1}{12})}$

Step 1.2.4.2.4

Multiply $- 1$ by $3$ .

$d = \frac{1}{- 3 (\frac{1}{12})}$

Step 1.2.4.2.5

Combine $- 3$ and $\frac{1}{12}$ .

$d = \frac{1}{\frac{- 3}{12}}$

Step 1.2.4.2.6

Cancel the common factor of $- 3$ and $12$ .

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Step 1.2.4.2.6.1

Factor $3$ out of $- 3$ .

$d = \frac{1}{\frac{3 (- 1)}{12}}$

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $3$ out of $12$ .

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

Step 1.2.4.2.6.2.2

Cancel the common factor.

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

Step 1.2.4.2.6.2.3

Rewrite the expression.

$d = \frac{1}{\frac{- 1}{4}}$

Step 1.2.4.2.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.8

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

$d = \frac{4}{- 1}$

Step 1.2.4.2.9

Divide $4$ by $- 1$ .

$d = - 4$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{1}{3} - \frac{{(\frac{2}{3})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $\frac{2}{3}$ .

$e = - \frac{1}{3} - \frac{\frac{2^{2}}{3^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.2

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

$e = - \frac{1}{3} - \frac{\frac{4}{3^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.3

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

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.2

Simplify the denominator.

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{- 4 (\frac{1}{12})}$

Step 1.2.5.2.1.2.2

Combine $- 4$ and $\frac{1}{12}$ .

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{\frac{- 4}{12}}$

Step 1.2.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.5.2.1.3.1

Cancel the common factor of $- 4$ and $12$ .

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Step 1.2.5.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{\frac{4 (- 1)}{12}}$

Step 1.2.5.2.1.3.1.2

Cancel the common factors.

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Step 1.2.5.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{\frac{- 1}{3}}$

Step 1.2.5.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{1}{3} - \frac{\frac{4}{9}}{- \frac{1}{3}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.5.2.1.5

Cancel the common factor of $3$ .

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Step 1.2.5.2.1.5.1

Factor $3$ out of $9$ .

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

Step 1.2.5.2.1.5.2

Factor $3$ out of $- 1 \cdot 3$ .

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

Step 1.2.5.2.1.5.3

Cancel the common factor.

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

Step 1.2.5.2.1.5.4

Rewrite the expression.

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

Step 1.2.5.2.1.6

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

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

Step 1.2.5.2.1.7

Multiply $4$ by $- 1$ .

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

Step 1.2.5.2.1.8

Move the negative in front of the fraction.

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

Step 1.2.5.2.1.9

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

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Step 1.2.5.2.1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.2.1.9.2

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

$e = - \frac{1}{3} + \frac{4}{3}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 1 + 4}{3}$

Step 1.2.5.2.3

Add $- 1$ and $4$ .

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

Step 1.2.5.2.4

Divide $3$ by $3$ .

$e = 1$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(x - 4)}^{2} + 1$ .

$- \frac{1}{12} {(x - 4)}^{2} + 1$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{12} \cdot {(x - 4)}^{2} + 1$

Step 2

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

$a = - \frac{1}{12}$

$h = 4$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(4, 1)$

Step 4