Pre-Algebra Examples

$16 x^{2} - 9 y^{2} + 64 x - 54 - 161 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Move all terms not containing a variable to the right side of the equation.

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Step 1.1.1

Add

$54$ to both sides of the equation.

$16 x^{2} - 9 y^{2} + 64 x - 161 = 54$

Step 1.1.2

Add

$161$ to both sides of the equation.

$16 x^{2} - 9 y^{2} + 64 x = 54 + 161$

Step 1.1.3

Add

$54$ and

$161$ .

$16 x^{2} - 9 y^{2} + 64 x = 215$

Step 1.2

Complete the square for

$16 x^{2} + 64 x$ .

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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

$c$ .

$a = 16$

$b = 64$

$c = 0$

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

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

$d = \frac{64}{2 \cdot 16}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of

$64$ and

$2$ .

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Step 1.2.3.2.1.1

Factor

$2$ out of

$64$ .

$d = \frac{2 \cdot 32}{2 \cdot 16}$

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

$2 \cdot 16$ .

$d = \frac{2 \cdot 32}{2 (16)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 32}{2 \cdot 16}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{32}{16}$

Step 1.2.3.2.2

Cancel the common factor of

$32$ and

$16$ .

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Step 1.2.3.2.2.1

Factor

$16$ out of

$32$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor

$16$ out of

$16$ .

$d = \frac{16 \cdot 2}{16 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot 2}{16 \cdot 1}$

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

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

$e = 0 - \frac{64^{2}}{4 \cdot 16}$

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

$64$ to the power of

$2$ .

$e = 0 - \frac{4096}{4 \cdot 16}$

Step 1.2.4.2.1.2

Multiply

$4$ by

$16$ .

$e = 0 - \frac{4096}{64}$

Step 1.2.4.2.1.3

Divide

$4096$ by

$64$ .

$e = 0 - 1 \cdot 64$

Step 1.2.4.2.1.4

Multiply

$- 1$ by

$64$ .

$e = 0 - 64$

Step 1.2.4.2.2

Subtract

$64$ from

$0$ .

$e = - 64$

Step 1.2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$16 {(x + 2)}^{2} - 64$ .

$16 {(x + 2)}^{2} - 64$

Step 1.3

Substitute

$16 {(x + 2)}^{2} - 64$ for

$16 x^{2} + 64 x$ in the equation

$16 x^{2} - 9 y^{2} + 64 x = 215$ .

$16 {(x + 2)}^{2} - 64 - 9 y^{2} = 215$

Step 1.4

Move

$- 64$ to the right side of the equation by adding

$64$ to both sides.

$16 {(x + 2)}^{2} - 9 y^{2} = 215 + 64$

Step 1.5

Add

$215$ and

$64$ .

$16 {(x + 2)}^{2} - 9 y^{2} = 279$

Step 1.6

Divide each term by

$279$ to make the right side equal to one.

$\frac{16 {(x + 2)}^{2}}{279} - \frac{9 y^{2}}{279} = \frac{279}{279}$

Step 1.7

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

$1$ .

$\frac{{(x + 2)}^{2}}{\frac{279}{16}} - \frac{y^{2}}{31} = 1$

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,

$a$ .

$a = \frac{3 \sqrt{31}}{4}$

$b = \sqrt{31}$

$k = 0$

$h = - 2$

Step 4

The center of a hyperbola follows the form of

$(h, k)$ . Substitute in the values of

$h$ and

$k$ .

$(- 2, 0)$

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.

$\sqrt{{(\frac{3 \sqrt{31}}{4})}^{2} + {(\sqrt{31})}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.3.1.1

Apply the product rule to

$\frac{3 \sqrt{31}}{4}$ .

$\sqrt{\frac{{(3 \sqrt{31})}^{2}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.1.2

Apply the product rule to

$3 \sqrt{31}$ .

$\sqrt{\frac{3^{2} {\sqrt{31}}^{2}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2

Simplify the numerator.

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Step 5.3.2.1

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{9 {\sqrt{31}}^{2}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 5.3.2.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\sqrt{\frac{9 {(31^{\frac{1}{2}})}^{2}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{9 \cdot 31^{\frac{1}{2} \cdot 2}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{9 \cdot 31^{\frac{2}{2}}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2.4

Cancel the common factor of

$2$ .

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Step 5.3.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{9 \cdot 31^{\frac{2}{2}}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{9 \cdot 31^{1}}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.2.2.5

Evaluate the exponent.

$\sqrt{\frac{9 \cdot 31}{4^{2}} + {(\sqrt{31})}^{2}}$

Step 5.3.3

Simplify by cancelling exponent with radical.

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Step 5.3.3.1

Raise

$4$ to the power of

$2$ .

$\sqrt{\frac{9 \cdot 31}{16} + {(\sqrt{31})}^{2}}$

Step 5.3.3.2

Multiply

$9$ by

$31$ .

$\sqrt{\frac{279}{16} + {(\sqrt{31})}^{2}}$

Step 5.3.3.3

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 5.3.3.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\sqrt{\frac{279}{16} + {(31^{\frac{1}{2}})}^{2}}$

Step 5.3.3.3.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{279}{16} + 31^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.3.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{279}{16} + 31^{\frac{2}{2}}}$

Step 5.3.3.3.4

Cancel the common factor of

$2$ .

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Step 5.3.3.3.4.1

Cancel the common factor.

$\sqrt{\frac{279}{16} + 31^{\frac{2}{2}}}$

Step 5.3.3.3.4.2

Rewrite the expression.

$\sqrt{\frac{279}{16} + 31^{1}}$

Step 5.3.3.3.5

Evaluate the exponent.

$\sqrt{\frac{279}{16} + 31}$

Step 5.3.4

To write

$31$ as a fraction with a common denominator, multiply by

$\frac{16}{16}$ .

$\sqrt{\frac{279}{16} + 31 \cdot \frac{16}{16}}$

Step 5.3.5

Combine

$31$ and

$\frac{16}{16}$ .

$\sqrt{\frac{279}{16} + \frac{31 \cdot 16}{16}}$

Step 5.3.6

Combine the numerators over the common denominator.

$\sqrt{\frac{279 + 31 \cdot 16}{16}}$

Step 5.3.7

Simplify the numerator.

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Step 5.3.7.1

Multiply

$31$ by

$16$ .

$\sqrt{\frac{279 + 496}{16}}$

Step 5.3.7.2

Add

$279$ and

$496$ .

$\sqrt{\frac{775}{16}}$

Step 5.3.8

Rewrite

$\sqrt{\frac{775}{16}}$ as

$\frac{\sqrt{775}}{\sqrt{16}}$ .

$\frac{\sqrt{775}}{\sqrt{16}}$

Step 5.3.9

Simplify the numerator.

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Step 5.3.9.1

Rewrite

$775$ as

$5^{2} \cdot 31$ .

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Step 5.3.9.1.1

Factor

$25$ out of

$775$ .

$\frac{\sqrt{25 (31)}}{\sqrt{16}}$

Step 5.3.9.1.2

Rewrite

$25$ as

$5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 31}}{\sqrt{16}}$

Step 5.3.9.2

Pull terms out from under the radical.

$\frac{5 \sqrt{31}}{\sqrt{16}}$

Step 5.3.10

Simplify the denominator.

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Step 5.3.10.1

Rewrite

$16$ as

$4^{2}$ .

$\frac{5 \sqrt{31}}{\sqrt{4^{2}}}$

Step 5.3.10.2

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

$\frac{5 \sqrt{31}}{4}$

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.

$(- 2 + \frac{3 \sqrt{31}}{4}, 0)$

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.

$(- 2 - \frac{3 \sqrt{31}}{4}, 0)$

Step 6.5

The vertices of a hyperbola follow the form of

$(h \pm a, k)$ . Hyperbolas have two vertices.

$(- 2 + \frac{3 \sqrt{31}}{4}, 0), (- 2 - \frac{3 \sqrt{31}}{4}, 0)$

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.

$(- 2 + \frac{5 \sqrt{31}}{4}, 0)$

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.

$(- 2 - \frac{5 \sqrt{31}}{4}, 0)$

Step 7.5

The foci of a hyperbola follow the form of

$(h \pm \sqrt{a^{2} + b^{2}}, k)$ . Hyperbolas have two foci.

$(- 2 + \frac{5 \sqrt{31}}{4}, 0), (- 2 - \frac{5 \sqrt{31}}{4}, 0)$

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.

$\frac{\sqrt{{(\frac{3 \sqrt{31}}{4})}^{2} + {(\sqrt{31})}^{2}}}{\frac{3 \sqrt{31}}{4}}$

Step 8.3

Simplify.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{3 \sqrt{31}}{4})}^{2} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.2

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.3.2.1

Apply the product rule to

$\frac{3 \sqrt{31}}{4}$ .

$\sqrt{\frac{{(3 \sqrt{31})}^{2}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.2.2

Apply the product rule to

$3 \sqrt{31}$ .

$\sqrt{\frac{3^{2} {\sqrt{31}}^{2}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3

Simplify the numerator.

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Step 8.3.3.1

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{9 {\sqrt{31}}^{2}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 8.3.3.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\sqrt{\frac{9 {(31^{\frac{1}{2}})}^{2}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{9 \cdot 31^{\frac{1}{2} \cdot 2}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{9 \cdot 31^{\frac{2}{2}}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2.4

Cancel the common factor of

$2$ .

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Step 8.3.3.2.4.1

Cancel the common factor.

$\sqrt{\frac{9 \cdot 31^{\frac{2}{2}}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2.4.2

Rewrite the expression.

$\sqrt{\frac{9 \cdot 31^{1}}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.3.2.5

Evaluate the exponent.

$\sqrt{\frac{9 \cdot 31}{4^{2}} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4

Simplify by cancelling exponent with radical.

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Step 8.3.4.1

Raise

$4$ to the power of

$2$ .

$\sqrt{\frac{9 \cdot 31}{16} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.2

Multiply

$9$ by

$31$ .

$\sqrt{\frac{279}{16} + {\sqrt{31}}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 8.3.4.3.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\sqrt{\frac{279}{16} + {(31^{\frac{1}{2}})}^{2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{279}{16} + 31^{\frac{1}{2} \cdot 2}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{279}{16} + 31^{\frac{2}{2}}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3.4

Cancel the common factor of

$2$ .

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Step 8.3.4.3.4.1

Cancel the common factor.

$\sqrt{\frac{279}{16} + 31^{\frac{2}{2}}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3.4.2

Rewrite the expression.

$\sqrt{\frac{279}{16} + 31^{1}} \frac{4}{3 \sqrt{31}}$

Step 8.3.4.3.5

Evaluate the exponent.

$\sqrt{\frac{279}{16} + 31} \frac{4}{3 \sqrt{31}}$

Step 8.3.5

To write

$31$ as a fraction with a common denominator, multiply by

$\frac{16}{16}$ .

$\sqrt{\frac{279}{16} + 31 \cdot \frac{16}{16}} \frac{4}{3 \sqrt{31}}$

Step 8.3.6

Combine

$31$ and

$\frac{16}{16}$ .

$\sqrt{\frac{279}{16} + \frac{31 \cdot 16}{16}} \frac{4}{3 \sqrt{31}}$

Step 8.3.7

Combine the numerators over the common denominator.

$\sqrt{\frac{279 + 31 \cdot 16}{16}} \frac{4}{3 \sqrt{31}}$

Step 8.3.8

Simplify the numerator.

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Step 8.3.8.1

Multiply

$31$ by

$16$ .

$\sqrt{\frac{279 + 496}{16}} \frac{4}{3 \sqrt{31}}$

Step 8.3.8.2

Add

$279$ and

$496$ .

$\sqrt{\frac{775}{16}} \frac{4}{3 \sqrt{31}}$

Step 8.3.9

Rewrite

$\sqrt{\frac{775}{16}}$ as

$\frac{\sqrt{775}}{\sqrt{16}}$ .

$\frac{\sqrt{775}}{\sqrt{16}} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.10

Simplify the numerator.

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Step 8.3.10.1

Rewrite

$775$ as

$5^{2} \cdot 31$ .

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Step 8.3.10.1.1

Factor

$25$ out of

$775$ .

$\frac{\sqrt{25 (31)}}{\sqrt{16}} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.10.1.2

Rewrite

$25$ as

$5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 31}}{\sqrt{16}} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.10.2

Pull terms out from under the radical.

$\frac{5 \sqrt{31}}{\sqrt{16}} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.11

Simplify the denominator.

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Step 8.3.11.1

Rewrite

$16$ as

$4^{2}$ .

$\frac{5 \sqrt{31}}{\sqrt{4^{2}}} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.11.2

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

$\frac{5 \sqrt{31}}{4} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.12

Simplify terms.

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Step 8.3.12.1

Cancel the common factor of

$\sqrt{31}$ .

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Step 8.3.12.1.1

Factor

$\sqrt{31}$ out of

$5 \sqrt{31}$ .

$\frac{\sqrt{31} \cdot 5}{4} \cdot \frac{4}{3 \sqrt{31}}$

Step 8.3.12.1.2

Factor

$\sqrt{31}$ out of

$3 \sqrt{31}$ .

$\frac{\sqrt{31} \cdot 5}{4} \cdot \frac{4}{\sqrt{31} \cdot 3}$

Step 8.3.12.1.3

Cancel the common factor.

$\frac{\sqrt{31} \cdot 5}{4} \cdot \frac{4}{\sqrt{31} \cdot 3}$

Step 8.3.12.1.4

Rewrite the expression.

$\frac{5}{4} \cdot \frac{4}{3}$

Step 8.3.12.2

Cancel the common factor of

$4$ .

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Step 8.3.12.2.1

Cancel the common factor.

$\frac{5}{4} \cdot \frac{4}{3}$

Step 8.3.12.2.2

Rewrite the expression.

$5 (\frac{1}{3})$

Step 8.3.12.3

Combine

$5$ and

$\frac{1}{3}$ .

$\frac{5}{3}$

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.

$\frac{\frac{{\sqrt{31}}^{2}}{5 \sqrt{31}}}{4}$

Step 9.3

Simplify.

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Step 9.3.1

Cancel the common factor of

${\sqrt{31}}^{2}$ and

$\sqrt{31}$ .

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Step 9.3.1.1

Factor

$\sqrt{31}$ out of

${\sqrt{31}}^{2}$ .

$\frac{\frac{\sqrt{31} \sqrt{31}}{5 \sqrt{31}}}{4}$

Step 9.3.1.2

Cancel the common factors.

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Step 9.3.1.2.1

Factor

$\sqrt{31}$ out of

$5 \sqrt{31}$ .

$\frac{\frac{\sqrt{31} \sqrt{31}}{\sqrt{31} \cdot 5}}{4}$

Step 9.3.1.2.2

Cancel the common factor.

$\frac{\frac{\sqrt{31} \sqrt{31}}{\sqrt{31} \cdot 5}}{4}$

Step 9.3.1.2.3

Rewrite the expression.

$\frac{\frac{\sqrt{31}}{5}}{4}$

Step 9.3.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{31}}{5} \cdot \frac{1}{4}$

Step 9.3.3

Multiply

$\frac{\sqrt{31}}{5} \cdot \frac{1}{4}$ .

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Step 9.3.3.1

Multiply

$\frac{\sqrt{31}}{5}$ by

$\frac{1}{4}$ .

$\frac{\sqrt{31}}{5 \cdot 4}$

Step 9.3.3.2

Multiply

$5$ by

$4$ .

$\frac{\sqrt{31}}{20}$

Step 10

The asymptotes follow the form

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

$y = \pm \frac{4}{3} \cdot (x - (- 2)) + 0$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = \frac{4}{3} \cdot (x - (- 2)) + 0$

Step 11.2

Simplify

$\frac{4}{3} \cdot (x - (- 2)) + 0$ .

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

Add

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

$0$ .

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

Step 11.2.1.2

Multiply

$- 1$ by

$- 2$ .

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

Step 11.2.2

Apply the distributive property.

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

Step 11.2.3

Combine

$\frac{4}{3}$ and

$x$ .

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

Step 11.2.4

Multiply

$\frac{4}{3} \cdot 2$ .

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Step 11.2.4.1

Combine

$\frac{4}{3}$ and

$2$ .

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

Step 11.2.4.2

Multiply

$4$ by

$2$ .

$y = \frac{4 x}{3} + \frac{8}{3}$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - \frac{4}{3} \cdot (x - (- 2)) + 0$

Step 12.2

Simplify

$- \frac{4}{3} \cdot (x - (- 2)) + 0$ .

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Step 12.2.1

Simplify the expression.

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Step 12.2.1.1

Add

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

$0$ .

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

Step 12.2.1.2

Multiply

$- 1$ by

$- 2$ .

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

Step 12.2.2

Apply the distributive property.

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

Step 12.2.3

Combine

$x$ and

$\frac{4}{3}$ .

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

Step 12.2.4

Multiply

$- \frac{4}{3} \cdot 2$ .

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Step 12.2.4.1

Multiply

$2$ by

$- 1$ .

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

Step 12.2.4.2

Combine

$- 2$ and

$\frac{4}{3}$ .

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

Step 12.2.4.3

Multiply

$- 2$ by

$4$ .

$y = - \frac{x \cdot 4}{3} + \frac{- 8}{3}$

Step 12.2.5

Simplify each term.

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Step 12.2.5.1

Move

$4$ to the left of

$x$ .

$y = - \frac{4 \cdot x}{3} + \frac{- 8}{3}$

Step 12.2.5.2

Move the negative in front of the fraction.

$y = - \frac{4 x}{3} - \frac{8}{3}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{4 x}{3} + \frac{8}{3}, y = - \frac{4 x}{3} - \frac{8}{3}$

Step 14

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

Center:

$(- 2, 0)$

Vertices:

$(- 2 + \frac{3 \sqrt{31}}{4}, 0), (- 2 - \frac{3 \sqrt{31}}{4}, 0)$

Foci:

$(- 2 + \frac{5 \sqrt{31}}{4}, 0), (- 2 - \frac{5 \sqrt{31}}{4}, 0)$

Eccentricity:

$\frac{5}{3}$

Focal Parameter:

$\frac{\sqrt{31}}{20}$

Asymptotes:

$y = \frac{4 x}{3} + \frac{8}{3}$ ,

$y = - \frac{4 x}{3} - \frac{8}{3}$

Step 15