Precalculus Examples

${(y - 6)}^{2} = 12 (x + 2)$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Rewrite the equation as $12 (x + 2) = {(y - 6)}^{2}$ .

$12 (x + 2) = {(y - 6)}^{2}$

Step 1.1.2

Divide each term in $12 (x + 2) = {(y - 6)}^{2}$ by $12$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $12 (x + 2) = {(y - 6)}^{2}$ by $12$ .

$\frac{12 (x + 2)}{12} = \frac{{(y - 6)}^{2}}{12}$

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{12 (x + 2)}{12} = \frac{{(y - 6)}^{2}}{12}$

Step 1.1.2.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{{(y - 6)}^{2}}{12}$

Step 1.1.3

Subtract $2$ from both sides of the equation.

$x = \frac{{(y - 6)}^{2}}{12} - 2$

Step 1.1.4

Reorder terms.

$x = \frac{1}{12} \cdot {(y - 6)}^{2} - 2$

Step 1.2

Complete the square for $\frac{1}{12} \cdot {(y - 6)}^{2} - 2$ .

Tap for more steps...

Step 1.2.1

Simplify the expression.

Tap for more steps...

Step 1.2.1.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1.1

Rewrite ${(y - 6)}^{2}$ as $(y - 6) (y - 6)$ .

$\frac{1}{12} \cdot ((y - 6) (y - 6)) - 2$

Step 1.2.1.1.2

Expand $(y - 6) (y - 6)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{12} \cdot (y (y - 6) - 6 (y - 6)) - 2$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot - 6 - 6 (y - 6)) - 2$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6) - 2$

Step 1.2.1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.2.1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$\frac{1}{12} \cdot (y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6) - 2$

Step 1.2.1.1.3.1.2

Move $- 6$ to the left of $y$ .

$\frac{1}{12} \cdot (y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6) - 2$

Step 1.2.1.1.3.1.3

Multiply $- 6$ by $- 6$ .

$\frac{1}{12} \cdot (y^{2} - 6 y - 6 y + 36) - 2$

Step 1.2.1.1.3.2

Subtract $6 y$ from $- 6 y$ .

$\frac{1}{12} \cdot (y^{2} - 12 y + 36) - 2$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{12} y^{2} + \frac{1}{12} (- 12 y) + \frac{1}{12} \cdot 36 - 2$

Step 1.2.1.1.5

Simplify.

Tap for more steps...

Step 1.2.1.1.5.1

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

$\frac{y^{2}}{12} + \frac{1}{12} (- 12 y) + \frac{1}{12} \cdot 36 - 2$

Step 1.2.1.1.5.2

Cancel the common factor of $12$ .

Tap for more steps...

Step 1.2.1.1.5.2.1

Factor $12$ out of $- 12 y$ .

$\frac{y^{2}}{12} + \frac{1}{12} (12 (- y)) + \frac{1}{12} \cdot 36 - 2$

Step 1.2.1.1.5.2.2

Cancel the common factor.

$\frac{y^{2}}{12} + \frac{1}{12} (12 (- y)) + \frac{1}{12} \cdot 36 - 2$

Step 1.2.1.1.5.2.3

Rewrite the expression.

$\frac{y^{2}}{12} - y + \frac{1}{12} \cdot 36 - 2$

Step 1.2.1.1.5.3

Cancel the common factor of $12$ .

Tap for more steps...

Step 1.2.1.1.5.3.1

Factor $12$ out of $36$ .

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

Step 1.2.1.1.5.3.2

Cancel the common factor.

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

Step 1.2.1.1.5.3.3

Rewrite the expression.

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

Step 1.2.1.2

Subtract $2$ from $3$ .

$\frac{y^{2}}{12} - y + 1$

Step 1.2.2

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

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

$b = - 1$

$c = 1$

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 $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

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

$d = \frac{- 1}{\frac{2}{12}}$

Step 1.2.4.2.2

Cancel the common factor of $2$ and $12$ .

Tap for more steps...

Step 1.2.4.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.2.2.1

Factor $2$ out of $12$ .

$d = \frac{- 1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Step 1.2.4.2.2.2.2

Cancel the common factor.

$d = \frac{- 1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Step 1.2.4.2.2.2.3

Rewrite the expression.

$d = \frac{- 1}{\frac{1}{6}}$

Step 1.2.4.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 6$

Step 1.2.4.2.4

Multiply $- 1$ by $6$ .

$d = - 6$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 1.2.5.2.1.1

Raise $- 1$ to the power of $2$ .

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

Step 1.2.5.2.1.2

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

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

Step 1.2.5.2.1.3

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

Tap for more steps...

Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

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

Step 1.2.5.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1.3.2.1

Factor $4$ out of $12$ .

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

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

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

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

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

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 1 - (1 \cdot 3)$

Step 1.2.5.2.1.5

Multiply $- (1 \cdot 3)$ .

Tap for more steps...

Step 1.2.5.2.1.5.1

Multiply $3$ by $1$ .

$e = 1 - 1 \cdot 3$

Step 1.2.5.2.1.5.2

Multiply $- 1$ by $3$ .

$e = 1 - 3$

Step 1.2.5.2.2

Subtract $3$ from $1$ .

$e = - 2$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{12} {(y - 6)}^{2} - 2$ .

$\frac{1}{12} {(y - 6)}^{2} - 2$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{12} \cdot {(y - 6)}^{2} - 2$

Step 2

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

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

$h = - 2$

$k = 6$

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(- 2, 6)$

Step 5

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

Tap for more steps...

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.

Tap for more steps...

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 $12$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

$(1, 6)$

Step 7

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

$y = 6$

Step 8

Find the directrix.

Tap for more steps...

Step 8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 8.2

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

$x = - 5$

Step 9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Right

Vertex: $(- 2, 6)$

Focus: $(1, 6)$

Axis of Symmetry: $y = 6$

Directrix: $x = - 5$

Step 10