Precalculus Examples

${(y + 4)}^{2} = 12 (x - 5)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Rewrite the equation as $12 (x - 5) = {(y + 4)}^{2}$ .

$12 (x - 5) = {(y + 4)}^{2}$

Step 1.1.2

Divide each term in $12 (x - 5) = {(y + 4)}^{2}$ by $12$ and simplify.

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Step 1.1.2.1

Divide each term in $12 (x - 5) = {(y + 4)}^{2}$ by $12$ .

$\frac{12 (x - 5)}{12} = \frac{{(y + 4)}^{2}}{12}$

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.

$\frac{12 (x - 5)}{12} = \frac{{(y + 4)}^{2}}{12}$

Step 1.1.2.2.1.2

Divide $x - 5$ by $1$ .

$x - 5 = \frac{{(y + 4)}^{2}}{12}$

Step 1.1.3

Add $5$ to both sides of the equation.

$x = \frac{{(y + 4)}^{2}}{12} + 5$

Step 1.1.4

Reorder terms.

$x = \frac{1}{12} \cdot {(y + 4)}^{2} + 5$

Step 1.2

Complete the square for $\frac{1}{12} \cdot {(y + 4)}^{2} + 5$ .

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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 ${(y + 4)}^{2}$ as $(y + 4) (y + 4)$ .

$\frac{1}{12} \cdot ((y + 4) (y + 4)) + 5$

Step 1.2.1.1.2

Expand $(y + 4) (y + 4)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{12} \cdot (y (y + 4) + 4 (y + 4)) + 5$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot 4 + 4 (y + 4)) + 5$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4) + 5$

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 $y$ by $y$ .

$\frac{1}{12} \cdot (y^{2} + y \cdot 4 + 4 y + 4 \cdot 4) + 5$

Step 1.2.1.1.3.1.2

Move $4$ to the left of $y$ .

$\frac{1}{12} \cdot (y^{2} + 4 \cdot y + 4 y + 4 \cdot 4) + 5$

Step 1.2.1.1.3.1.3

Multiply $4$ by $4$ .

$\frac{1}{12} \cdot (y^{2} + 4 y + 4 y + 16) + 5$

Step 1.2.1.1.3.2

Add $4 y$ and $4 y$ .

$\frac{1}{12} \cdot (y^{2} + 8 y + 16) + 5$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{12} y^{2} + \frac{1}{12} (8 y) + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

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

$\frac{y^{2}}{12} + \frac{1}{12} (8 y) + \frac{1}{12} \cdot 16 + 5$

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

Factor $4$ out of $12$ .

$\frac{y^{2}}{12} + \frac{1}{4 (3)} (8 y) + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.2.2

Factor $4$ out of $8 y$ .

$\frac{y^{2}}{12} + \frac{1}{4 (3)} (4 (2 y)) + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{y^{2}}{12} + \frac{1}{4 \cdot 3} (4 (2 y)) + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{y^{2}}{12} + \frac{1}{3} (2 y) + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.3

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

$\frac{y^{2}}{12} + \frac{2}{3} y + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.4

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

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{1}{12} \cdot 16 + 5$

Step 1.2.1.1.5.5

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.5.1

Factor $4$ out of $12$ .

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{1}{4 (3)} \cdot 16 + 5$

Step 1.2.1.1.5.5.2

Factor $4$ out of $16$ .

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{1}{4 \cdot 3} \cdot (4 \cdot 4) + 5$

Step 1.2.1.1.5.5.3

Cancel the common factor.

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{1}{4 \cdot 3} \cdot (4 \cdot 4) + 5$

Step 1.2.1.1.5.5.4

Rewrite the expression.

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{1}{3} \cdot 4 + 5$

Step 1.2.1.1.5.6

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

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{4}{3} + 5$

Step 1.2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{4}{3} + 5 \cdot \frac{3}{3}$

Step 1.2.1.3

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

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{4}{3} + \frac{5 \cdot 3}{3}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{4 + 5 \cdot 3}{3}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $5$ by $3$ .

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{4 + 15}{3}$

Step 1.2.1.5.2

Add $4$ and $15$ .

$\frac{y^{2}}{12} + \frac{2 y}{3} + \frac{19}{3}$

Step 1.2.2

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

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

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

$c = \frac{19}{3}$

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

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

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

Step 1.2.4.2.5

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

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Step 1.2.4.2.5.1

Factor $3$ out of $3$ .

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

Step 1.2.4.2.5.2

Cancel the common factors.

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Step 1.2.4.2.5.2.1

Factor $3$ out of $12$ .

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

Step 1.2.4.2.5.2.2

Cancel the common factor.

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

Step 1.2.4.2.5.2.3

Rewrite the expression.

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

Step 1.2.4.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 4$

Step 1.2.4.2.7

Multiply $4$ 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 $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{19}{3} - \frac{{(\frac{2}{3})}^{2}}{4 (\frac{1}{12})}$

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 $\frac{2}{3}$ .

$e = \frac{19}{3} - \frac{\frac{2^{2}}{3^{2}}}{4 (\frac{1}{12})}$

Step 1.2.5.2.1.1.2

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

$e = \frac{19}{3} - \frac{\frac{4}{3^{2}}}{4 (\frac{1}{12})}$

Step 1.2.5.2.1.1.3

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

$e = \frac{19}{3} - \frac{\frac{4}{9}}{4 (\frac{1}{12})}$

Step 1.2.5.2.1.2

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

$e = \frac{19}{3} - \frac{\frac{4}{9}}{\frac{4}{12}}$

Step 1.2.5.2.1.3

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

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{19}{3} - \frac{\frac{4}{9}}{\frac{4 (1)}{12}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $4$ out of $12$ .

$e = \frac{19}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = \frac{19}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = \frac{19}{3} - \frac{\frac{4}{9}}{\frac{1}{3}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{19}{3} - (\frac{4}{9} \cdot 3)$

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 $9$ .

$e = \frac{19}{3} - (\frac{4}{3 (3)} \cdot 3)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{19}{3} - (\frac{4}{3 \cdot 3} \cdot 3)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{19}{3} - \frac{4}{3}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{19 - 4}{3}$

Step 1.2.5.2.3

Subtract $4$ from $19$ .

$e = \frac{15}{3}$

Step 1.2.5.2.4

Divide $15$ by $3$ .

$e = 5$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(y + 4)}^{2} + 5$ .

$\frac{1}{12} {(y + 4)}^{2} + 5$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{12} \cdot {(y + 4)}^{2} + 5$

Step 2

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

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

$h = 5$

$k = - 4$

Step 3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(5, - 4)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{12}}$

Step 5.3

Simplify.

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Step 5.3.1

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

$\frac{1}{\frac{4}{12}}$

Step 5.3.2

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

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Step 5.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{12}}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $4$ out of $12$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 5.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{3}}$

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 3$

Step 5.3.4

Multiply $3$ by $1$ .

$3$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(8, - 4)$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = - 4$

Step 8