Precalculus Examples

$4 x^{2} - 9 y^{2} - 16 x - 72 y - 129 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $129$ to both sides of the equation.

$4 x^{2} - 9 y^{2} - 16 x - 72 y = 129$

Step 1.2

Complete the square for $4 x^{2} - 16 x$ .

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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 $c$ .

$a = 4$

$b = - 16$

$c = 0$

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

$d = \frac{- 16}{2 \cdot 4}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 16$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 16$ .

$d = \frac{2 \cdot - 8}{2 \cdot 4}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 8}{2 (4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 8}{2 \cdot 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 8}{4}$

Step 1.2.3.2.2

Cancel the common factor of $- 8$ and $4$ .

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Step 1.2.3.2.2.1

Factor $4$ out of $- 8$ .

$d = \frac{4 \cdot - 2}{4}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot 4}$

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

Raise $- 16$ to the power of $2$ .

$e = 0 - \frac{256}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{256}{16}$

Step 1.2.4.2.1.3

Divide $256$ by $16$ .

$e = 0 - 1 \cdot 16$

Step 1.2.4.2.1.4

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Step 1.2.4.2.2

Subtract $16$ from $0$ .

$e = - 16$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 2)}^{2} - 16$ .

$4 {(x - 2)}^{2} - 16$

Step 1.3

Substitute $4 {(x - 2)}^{2} - 16$ for $4 x^{2} - 16 x$ in the equation $4 x^{2} - 9 y^{2} - 16 x - 72 y = 129$ .

$4 {(x - 2)}^{2} - 16 - 9 y^{2} - 72 y = 129$

Step 1.4

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$4 {(x - 2)}^{2} - 9 y^{2} - 72 y = 129 + 16$

Step 1.5

Complete the square for $- 9 y^{2} - 72 y$ .

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Step 1.5.1

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

$a = - 9$

$b = - 72$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

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

Step 1.5.3

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

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Step 1.5.3.1

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

$d = \frac{- 72}{2 \cdot - 9}$

Step 1.5.3.2

Simplify the right side.

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Step 1.5.3.2.1

Cancel the common factor of $- 72$ and $2$ .

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Step 1.5.3.2.1.1

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot - 9}$

Step 1.5.3.2.1.2

Cancel the common factors.

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Step 1.5.3.2.1.2.1

Factor $2$ out of $2 \cdot - 9$ .

$d = \frac{2 \cdot - 36}{2 (- 9)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot - 9}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 36}{- 9}$

Step 1.5.3.2.2

Cancel the common factor of $- 36$ and $- 9$ .

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Step 1.5.3.2.2.1

Factor $- 9$ out of $- 36$ .

$d = \frac{- 9 (4)}{- 9}$

Step 1.5.3.2.2.2

Cancel the common factors.

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Step 1.5.3.2.2.2.1

Factor $- 9$ out of $- 9$ .

$d = \frac{- 9 \cdot 4}{- 9 \cdot 1}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 9 \cdot 4}{- 9 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $4$ by $1$ .

$d = 4$

Step 1.5.4

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

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Step 1.5.4.1

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

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot - 9}$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Simplify each term.

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Step 1.5.4.2.1.1

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot - 9}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 9$ .

$e = 0 - \frac{5184}{- 36}$

Step 1.5.4.2.1.3

Divide $5184$ by $- 36$ .

$e = 0 - - 144$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 144$ .

$e = 0 + 144$

Step 1.5.4.2.2

Add $0$ and $144$ .

$e = 144$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 9 {(y + 4)}^{2} + 144$ .

$- 9 {(y + 4)}^{2} + 144$

Step 1.6

Substitute $- 9 {(y + 4)}^{2} + 144$ for $- 9 y^{2} - 72 y$ in the equation $4 x^{2} - 9 y^{2} - 16 x - 72 y = 129$ .

$4 {(x - 2)}^{2} - 9 {(y + 4)}^{2} + 144 = 129 + 16$

Step 1.7

Move $144$ to the right side of the equation by adding $144$ to both sides.

$4 {(x - 2)}^{2} - 9 {(y + 4)}^{2} = 129 + 16 - 144$

Step 1.8

Simplify $129 + 16 - 144$ .

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Step 1.8.1

Add $129$ and $16$ .

$4 {(x - 2)}^{2} - 9 {(y + 4)}^{2} = 145 - 144$

Step 1.8.2

Subtract $144$ from $145$ .

$4 {(x - 2)}^{2} - 9 {(y + 4)}^{2} = 1$

Step 1.9

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(x - 2)}^{2}}{\frac{1}{4}} - \frac{{(y + 4)}^{2}}{\frac{1}{9}} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = \frac{1}{2}$

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

$k = - 4$

$h = 2$

Step 4

The center of a hyperbola follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(2, - 4)$

Step 5

Find $c$ , the distance from the center to a focus.

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Step 5.1

Find the distance from the center to a focus of the hyperbola by using the following formula.

$\sqrt{a^{2} + b^{2}}$

Step 5.2

Substitute the values of $a$ and $b$ in the formula.

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

Step 5.3

Simplify.

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Step 5.3.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.3.2

One to any power is one.

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

Step 5.3.3

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

$\sqrt{\frac{1}{4} + {(\frac{1}{3})}^{2}}$

Step 5.3.4

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{\frac{1}{4} + \frac{1^{2}}{3^{2}}}$

Step 5.3.5

One to any power is one.

$\sqrt{\frac{1}{4} + \frac{1}{3^{2}}}$

Step 5.3.6

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

$\sqrt{\frac{1}{4} + \frac{1}{9}}$

Step 5.3.7

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{1}{9}}$

Step 5.3.8

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{1}{9} \cdot \frac{4}{4}}$

Step 5.3.9

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.9.1

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{9}{4 \cdot 9} + \frac{1}{9} \cdot \frac{4}{4}}$

Step 5.3.9.2

Multiply $4$ by $9$ .

$\sqrt{\frac{9}{36} + \frac{1}{9} \cdot \frac{4}{4}}$

Step 5.3.9.3

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{9}{36} + \frac{4}{9 \cdot 4}}$

Step 5.3.9.4

Multiply $9$ by $4$ .

$\sqrt{\frac{9}{36} + \frac{4}{36}}$

Step 5.3.10

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 4}{36}}$

Step 5.3.11

Add $9$ and $4$ .

$\sqrt{\frac{13}{36}}$

Step 5.3.12

Rewrite $\sqrt{\frac{13}{36}}$ as $\frac{\sqrt{13}}{\sqrt{36}}$ .

$\frac{\sqrt{13}}{\sqrt{36}}$

Step 5.3.13

Simplify the denominator.

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Step 5.3.13.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{13}}{\sqrt{6^{2}}}$

Step 5.3.13.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{13}}{6}$

Step 6

Find the vertices.

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Step 6.1

The first vertex of a hyperbola can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

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

$(\frac{5}{2}, - 4)$

Step 6.3

The second vertex of a hyperbola can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.4

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

$(\frac{3}{2}, - 4)$

Step 6.5

The vertices of a hyperbola follow the form of $(h \pm a, k)$ . Hyperbolas have two vertices.

$(\frac{5}{2}, - 4), (\frac{3}{2}, - 4)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 7.2

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

$(2 + \frac{\sqrt{13}}{6}, - 4)$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 7.4

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

$(2 - \frac{\sqrt{13}}{6}, - 4)$

Step 7.5

The foci of a hyperbola follow the form of $(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(2 + \frac{\sqrt{13}}{6}, - 4), (2 - \frac{\sqrt{13}}{6}, - 4)$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} + b^{2}}}{a}$

Step 8.2

Substitute the values of $a$ and $b$ into the formula.

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

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{1}{2})}^{2} + {(\frac{1}{3})}^{2}} \cdot 2$

Step 8.3.2

Apply the product rule to $\frac{1}{2}$ .

$\sqrt{\frac{1^{2}}{2^{2}} + {(\frac{1}{3})}^{2}} \cdot 2$

Step 8.3.3

One to any power is one.

$\sqrt{\frac{1}{2^{2}} + {(\frac{1}{3})}^{2}} \cdot 2$

Step 8.3.4

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

$\sqrt{\frac{1}{4} + {(\frac{1}{3})}^{2}} \cdot 2$

Step 8.3.5

Apply the product rule to $\frac{1}{3}$ .

$\sqrt{\frac{1}{4} + \frac{1^{2}}{3^{2}}} \cdot 2$

Step 8.3.6

One to any power is one.

$\sqrt{\frac{1}{4} + \frac{1}{3^{2}}} \cdot 2$

Step 8.3.7

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

$\sqrt{\frac{1}{4} + \frac{1}{9}} \cdot 2$

Step 8.3.8

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{1}{9}} \cdot 2$

Step 8.3.9

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{1}{9} \cdot \frac{4}{4}} \cdot 2$

Step 8.3.10

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.10.1

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{9}{4 \cdot 9} + \frac{1}{9} \cdot \frac{4}{4}} \cdot 2$

Step 8.3.10.2

Multiply $4$ by $9$ .

$\sqrt{\frac{9}{36} + \frac{1}{9} \cdot \frac{4}{4}} \cdot 2$

Step 8.3.10.3

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{9}{36} + \frac{4}{9 \cdot 4}} \cdot 2$

Step 8.3.10.4

Multiply $9$ by $4$ .

$\sqrt{\frac{9}{36} + \frac{4}{36}} \cdot 2$

Step 8.3.11

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 4}{36}} \cdot 2$

Step 8.3.12

Add $9$ and $4$ .

$\sqrt{\frac{13}{36}} \cdot 2$

Step 8.3.13

Rewrite $\sqrt{\frac{13}{36}}$ as $\frac{\sqrt{13}}{\sqrt{36}}$ .

$\frac{\sqrt{13}}{\sqrt{36}} \cdot 2$

Step 8.3.14

Simplify the denominator.

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Step 8.3.14.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{13}}{\sqrt{6^{2}}} \cdot 2$

Step 8.3.14.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{13}}{6} \cdot 2$

Step 8.3.15

Cancel the common factor of $2$ .

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Step 8.3.15.1

Factor $2$ out of $6$ .

$\frac{\sqrt{13}}{2 (3)} \cdot 2$

Step 8.3.15.2

Cancel the common factor.

$\frac{\sqrt{13}}{2 \cdot 3} \cdot 2$

Step 8.3.15.3

Rewrite the expression.

$\frac{\sqrt{13}}{3}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

$\frac{b^{2}}{\sqrt{a^{2} + b^{2}}}$

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{\frac{\frac{1}{3^{2}}}{\sqrt{13}}}{6}$

Step 9.3

Simplify.

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Step 9.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{1}{3^{2}}}{\sqrt{13}} \cdot \frac{1}{6}$

Step 9.3.2

Combine.

$\frac{\frac{1}{3^{2}} \cdot 1}{\sqrt{13} \cdot 6}$

Step 9.3.3

Simplify the expression.

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Step 9.3.3.1

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

$\frac{\frac{1}{9} \cdot 1}{\sqrt{13} \cdot 6}$

Step 9.3.3.2

Multiply $\frac{1}{9}$ by $1$ .

$\frac{\frac{1}{9}}{\sqrt{13} \cdot 6}$

Step 9.3.3.3

Move $6$ to the left of $\sqrt{13}$ .

$\frac{\frac{1}{9}}{6 \sqrt{13}}$

Step 9.3.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{9} \cdot \frac{1}{6 \sqrt{13}}$

Step 9.3.5

Multiply $\frac{1}{6 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{1}{9} (\frac{1}{6 \sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 9.3.6

Combine and simplify the denominator.

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Step 9.3.6.1

Multiply $\frac{1}{6 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \sqrt{13} \sqrt{13}}$

Step 9.3.6.2

Move $\sqrt{13}$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 (\sqrt{13} \sqrt{13})}$

Step 9.3.6.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 ({\sqrt{13}}^{1} \sqrt{13})}$

Step 9.3.6.4

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}$

Step 9.3.6.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 {\sqrt{13}}^{1 + 1}}$

Step 9.3.6.6

Add $1$ and $1$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 {\sqrt{13}}^{2}}$

Step 9.3.6.7

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 9.3.6.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 {(13^{\frac{1}{2}})}^{2}}$

Step 9.3.6.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \cdot 13^{\frac{1}{2} \cdot 2}}$

Step 9.3.6.7.3

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

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \cdot 13^{\frac{2}{2}}}$

Step 9.3.6.7.4

Cancel the common factor of $2$ .

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Step 9.3.6.7.4.1

Cancel the common factor.

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \cdot 13^{\frac{2}{2}}}$

Step 9.3.6.7.4.2

Rewrite the expression.

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \cdot 13^{1}}$

Step 9.3.6.7.5

Evaluate the exponent.

$\frac{1}{9} \cdot \frac{\sqrt{13}}{6 \cdot 13}$

Step 9.3.7

Multiply $6$ by $13$ .

$\frac{1}{9} \cdot \frac{\sqrt{13}}{78}$

Step 9.3.8

Multiply $\frac{1}{9} \cdot \frac{\sqrt{13}}{78}$ .

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Step 9.3.8.1

Multiply $\frac{1}{9}$ by $\frac{\sqrt{13}}{78}$ .

$\frac{\sqrt{13}}{9 \cdot 78}$

Step 9.3.8.2

Multiply $9$ by $78$ .

$\frac{\sqrt{13}}{702}$

Step 10

The asymptotes follow the form $y = \pm \frac{b (x - h)}{a} + k$ because this hyperbola opens left and right.

$y = \pm \frac{2}{3} \cdot (x - (2)) - 4$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

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

Step 11.2

Simplify $\frac{2}{3} \cdot (x - (2)) - 4$ .

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Multiply $- 1$ by $2$ .

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

Step 11.2.1.2

Apply the distributive property.

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

Step 11.2.1.3

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

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

Step 11.2.1.4

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 11.2.1.4.1

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

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

Step 11.2.1.4.2

Multiply $2$ by $- 2$ .

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

Step 11.2.1.5

Move the negative in front of the fraction.

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

Step 11.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{2 x}{3} - \frac{4}{3} - 4 \cdot \frac{3}{3}$

Step 11.2.3

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

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

Step 11.2.4

Combine the numerators over the common denominator.

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

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Multiply $- 4$ by $3$ .

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

Step 11.2.5.2

Subtract $12$ from $- 4$ .

$y = \frac{2 x}{3} + \frac{- 16}{3}$

Step 11.2.6

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{16}{3}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

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

Step 12.2

Simplify $- \frac{2}{3} \cdot (x - (2)) - 4$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $- 1$ by $2$ .

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

Step 12.2.1.2

Apply the distributive property.

$y = - \frac{2}{3} x - \frac{2}{3} \cdot - 2 - 4$

Step 12.2.1.3

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

$y = - \frac{x \cdot 2}{3} - \frac{2}{3} \cdot - 2 - 4$

Step 12.2.1.4

Multiply $- \frac{2}{3} \cdot - 2$ .

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Step 12.2.1.4.1

Multiply $- 2$ by $- 1$ .

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

Step 12.2.1.4.2

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

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

Step 12.2.1.4.3

Multiply $2$ by $2$ .

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

Step 12.2.1.5

Move $2$ to the left of $x$ .

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

Step 12.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 12.2.3

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

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

Step 12.2.4

Combine the numerators over the common denominator.

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

Step 12.2.5

Simplify the numerator.

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Step 12.2.5.1

Multiply $- 4$ by $3$ .

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

Step 12.2.5.2

Subtract $12$ from $4$ .

$y = - \frac{2 x}{3} + \frac{- 8}{3}$

Step 12.2.6

Move the negative in front of the fraction.

$y = - \frac{2 x}{3} - \frac{8}{3}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{2 x}{3} - \frac{16}{3}, y = - \frac{2 x}{3} - \frac{8}{3}$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(2, - 4)$

Vertices: $(\frac{5}{2}, - 4), (\frac{3}{2}, - 4)$

Foci: $(2 + \frac{\sqrt{13}}{6}, - 4), (2 - \frac{\sqrt{13}}{6}, - 4)$

Eccentricity: $\frac{\sqrt{13}}{3}$

Focal Parameter: $\frac{\sqrt{13}}{702}$

Asymptotes: $y = \frac{2 x}{3} - \frac{16}{3}$ , $y = - \frac{2 x}{3} - \frac{8}{3}$

Step 15